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The Hoek, Mann and Jahns system is a system of taxonomy of algae. It was first published in Algae: An Introduction to Phycology by Cambridge University Press in 1995.

Division Cyanophyta (= Cyanobacteria)

Main article: Cyanophyta

Class Cyanophyceae

Order Chroococcales (e.g., Cyanothece, Aphanothece, Merismopedia, Chroococcus, Gloeocapsa, Microcystis, Chamaesiphon)
Order Pleurocapsales (e.g., Cyanocystis, Pleurocapsa)
Order Oscillatoriales (e.g., Oscillatoria, Lyngbya, Microcoleus)
Order Nostocales (e.g., Nostoc, Anabaena, Aphanizomenon, Scytonema, Rivularia)
Order Stigonematales (e.g., Stigonema)

Division Prochlorophyta

Main article: Prochlorophyta

Class Prochlorophyceae (e.g., Prochloron, Prochlorococcus, Prochlorothrix)

Division Glaucophyta

Main article: Glaucophyta

Class Glaucophyceae (e.g., Cyanophora, Glaucocystis)

Division Rhodophyta

Main article: Rhodophyta

Class Bangiophyceae

Order Porphyridiales (e.g., Porphyridium, Chroodactylon)
Order Rhodochaetales (e.g., Rhodochaete)
Order Erythropeltidales (e.g., Erythrotrichia)
Order Compsogonales
Order Bangiales (e.g., Porphyra, Bangia)

Class Florideophyceae

Order Acrochaetiales (e.g., Audouinella)
Order Palmariales (e.g., Palmaria, Rhodophysema)
Order Nemaliales (e.g., Nemalion, Galaxaura)
Order Batrachospermales (e.g., Lemanea, Batrachospermum)
Order Corallinales (e.g., Lithophyllum, Jania)
Order Hildebrandiales
Order Bonnemaisoniales
Order Gelidiales (e.g., Gelidium)
Order Gigartinales (e.g., Acrosymphyton, Dumontia, Chondrus, Mastocarpus)
Order Gracilariales
Order Ahnfeltiales
Order Rhodymeniales (e.g., Chylocladia)
Order Ceramiales (e.g., Polysiphonia, Callithamnion, Ceramium, Hypoglossum)

Division Heterokontophyta

Main article: Heterokontophyta

Class Chrysophyceae

Order Ochromonadales (e.g., Ochromonas, Pseudokephyrion, Dinobryon)
Order Mallomonadales (= Class Synurophyceae, e.g., Mallomonas, Synura)
Order Pedinellales (= Class Pedinellophyceae, e.g., Pedinella)
Order Chrysamoebidales (e.g., Rhizochrysis, Chrysarachnion)
Order Chrysocapsales (e.g., Chrysocapsa, Hydrurus)
Order Chrysosphaerales (e.g., Chrysosphaera)
Order Phaeothamniales (e.g., Phaeothamnion, Thallochrysis)

Class Parmophyceae

Order Parmales (e.g., Pentalamina)

Class Sarcinochrysidophyceae

Order Sarcinochrysidales (e.g., Ankylochrysis, Sarcinochrysis, Nematochrysis)

Genus Pelagococcus (relative of the Chrysophyceae)
Class Xanthophyceae

Order Chloramoebales (e.g., Chloromeson)
Order Rhizochloridales (e.g., Rhizochloris, Myxochloris)
Order Heterogloeales (e.g., Gloeochloris)
Order Mischococcales (e.g., Chloridella, Botrydiopsis, Characiopsis, Ophiocytium)
Order Tribonematales (e.g., Tribonema, Heterococcus, Heterodendron)
Order Botrydiales (e.g., Botrydium)
Order Vaucheriales (e.g., Vaucheria)

Class Eustigmatophyceae (e.g., Ellipsoidion, Eustigmatos [= Pleurochloris], Polyedriella, Vischeria, Chlorobotrys, Nannochloropsis)
Class Bacillariophyceae (= Diatomophyceae)

Order Pennales (e.g., Navicula, Gyrosigma, Achnanthes, Tabellaria, Nitzchia, Surirella)
Order Centrales (e.g., Coscinodiscus, Thalassiosira, Stephanodiscus, Cerataulus, Triceratium, Odontella, Rhizosolenia)

Class Raphidophyceae (e.g., Chatonella, Gonyostomum, Merotrichia, Vacuolaria, Fibrocapsa)
Class Dictyochophyceae (e.g., Dictyocha)
Class Phaeophyceae

Order Ectocarpales (e.g., Phaeostroma, Spongonema, Ectocarpus)
Order Sphacelariales (e.g., Sphacelaria)
Order Syringodermatales
Order Dictyotales (e.g., Dictyota)
Order Scytosiphonales (e.g., Scytosiphon)
Order Cutleriales (e.g., Cutleria)
Order Dictyosiphonales (e.g., Dictyosiphon)
Order Chordariales (e.g., Sphaerotrichia, Liebmania, Elachista)
Order Sporochnales
Order Desmarestiales (e.g., Desmarestia)
Order Laminariales (e.g., Laminaria, Chorda)
Order Fucales (e.g., Fucus, Hormosira)
Order Durvillaeales (e.g., Durvillaea)
Order Ascoseirales

Class Bicocoecida (e.g., Pseudobodo, Cafeteria)
Class Oomycetes (e.g., Phytophthora)
Class Hyphochytridiomycetes
Class Labyrinthulomycetes

Division Haptophyta (= Prymnesiophyta)

Main article: Haptophyta

Class Haptophyceae

Order Prymnesiales (e.g., Chrysochromulina, Prymnesium, Corymbellus, Phaeocystis)
Order Isochrysidales (e.g., Chrysotila, Pleurochrysis [= Hymenomonas])
Order Coccolithophorales (e.g., Emiliania, Syracosphaera, Discosphaera)
Order Pavlovales (e.g., Pavlova)

Division Cryptophyta

Main article: Cryptophyta

Class Cryptophyceae (e.g., Cryptomonas, Chilomonas, Bjornbergiella)

Division Dinophyta

Main article: Dinophyta

Class Dinophyceae

Order Gymnodiniales (e.g., Gymnodinium, Symbiodinium, Polykrikos)
Order Gloeodiniales (e.g., Gloeodinium)
Order Thoracosphaerales (e.g., Thoracosphaera)
Order Phytodiniales (= Dinococcales, e.g., Phytodinium)
Order Dinotrichales (e.g., Dinoclonium)
Order Dinamoebidales (e.g., Stylodinium, Dinamoebidium)
Order Noctilucales (e.g., Noctiluca)
Order Blastodiniales (e.g., Blastodinium)
Order Syndiniales
Order Peridiniales (e.g., Peridinium, Protoperidinium, Ceratium, Gonyaulax)
Order Dynophysiales (e.g., Dinophysis, Triposolenia, Amphisolenia, Ornithocercus, Histioneis)
Order Prorocentrales (e.g., Prorocentrum)

Division Euglenophyta

Main article: Euglenophyta

Class Euglenophyceae

Order Euglenales (e.g., Euglena, Astasia)
Order Eutreptiales
Order Euglenamorphales
Order Rhabdomonadales
Order Sphenomonadales
Order Heteronematales

Division Chlorarachniophyta

Main article: Chlorarachniophyta

Class Chlorarachniophyceae (e.g., Chlorarachnion)

Division Chlorophyta

Main article: Chlorophyta

Class Prasinophyceae

Order Mamiellales (e.g., Mamiella, Bathycoccus)
Order Pseudoscourfieldiales (e.g., Nephroselmis)
Order Pyramimonadales (e.g., Pyramimonas, Halosphaera, Pachysphaera [?], Pterosperma [?], Mesostigma [?])
Genus Pedinomonas (sometimes classified in the class Pedinophyceae, or Loxophyceae)
Order Chlorodendrales (e.g., Tetraselmis)

Class Chlorophyceae

Order Volvocales [including the Tetrasporales] (e.g., Chlamydomonas, Chlorogonium, Hyalogonium, Gloeomonas, Diplostauron, Polytoma, Carteria, Brachiomonas, Lobomonas, Sphaerellopsis, Haematococcus, Phacotus, Pteromonas, Stephanosphaera, Gonium, Pandorina, Eudorina, Volvox, Pseudosphaerocystis)
Order Chlorococcales (e.g., Chlorococcum, Golenkinia, Chlorella, Chodatella, Oocystis, Kirchneriella, Nephrocytium, Actinastrum, Crucigenia, Sphaerocystis, Coccomyxa, Scenedesmus, Hydrodictyon, Pediastrum, Chlorosarcinopsis, Cylindrocapsa, Geminella, Binuclearia, Radiofilum, Sphaeroplea, Characiosiphon, Atractomorpha)
Order Chaetophorales (e.g., Uronema, Stigeoclonium, Draparnaldia, Schizomeris)
Order Oedogoniales (e.g., Oedogonium)

Class Ulvophyceae

Order Codiolales (e.g., Chlorocystis, Ulothrix, Spongomorpha, Urospora, Acrosiphonia, Monostroma)
Order Ulvales (e.g., Ulva, Enteromorpha, Ulvaria, Acrochaete)

Class Cladophorophyceae

Order Cladophorales (e.g., Cladophora, Chaetomorpha, Rhizoclonium, Struvea, Valonia, Siphonocladus, Dictyosphaeria)

Class Bryopsidophyceae

Order Bryopsidales (e.g., Bryopsis, Pseudobryopsis, Derbesia, Codium)
Order Halimedales (e.g., Udotea, Penicillus, Halimeda, Caulerpa)

Class Dasycladophyceae

Order Dasycladales (e.g., Acetabularia, Batophora, Dasycladus, Neomeris, Cymopolia)

Class Trentepohliophyceae

Order Trentepohliales (e.g., Trentepohlia, Cephaleuros)

Class Pleurastrophyceae

Order Pleurastrales (e.g., Trebouxia, Myrmecia, Friedmannia, Pleurastrosarcina, Pleurastrum, Microthamnion)

Position uncertain:

Order Prasiolales (e.g., Prasiola)

Class Klebsormidiophyceae

Order Klebsormidiales (e.g., Chlorokybus, Raphidonema, Klebsormidium)
Order Coleochaetales (e.g., Chaetosphaeridium, Coleochaete)

Class Zygnematophyceae

Order Zygnematales (e.g., Spirogyra, Mougeotia, Zygnema, Spirotaenia, Cylindrocystis, Netrium)
Order Desmidiales (e.g., Cosmarium, Gonatozygon, Closterium, Pleurotaenium, Tetmemorus, Euastrum, Micrasterias, Cosmocladium, Xanthidium, Staurastrum, Desmidium, Hyalotheca, Spondylosium, Sphaerozosma, Onychonema)

Class Charophyceae

Order Charales (e.g., Chara, Lamprothamnium, Nitellopsis, Nitella, Tolypella)

Note added in proof

In a note added in proof, an alternative classification is presented for the class Chlorophyceae:

Class Chlamydophyceae

Order Volvocales
Order Chaetophorales
Order Chlorococcales (e.g., Chlorococcum, Chlorosarcinopsis, Cylindrocapsa [?])

Class Chlorophyceae

Order Chlorellales (e.g., Chlorella, Kirchneriella, Scenedesmus, Hydrodictyon, Pediastrum, Sphaeroplea, Atractomorpha)

Class Oedogoniophyceae

Order Oedogoniales (e.g., Oedogonium)

Many examples of genus given in the book could not be assigned to one of these classes, because the ultrastructural characteristics needed for their classification have not yet been studied.