A system of taxonomy of the cryptogams, the Smith system was published in:

Smith, G.M. (1938). Cryptogamic Botany, vol. 1. Algae and fungi. McGraw-Hill, New York.

Smith, G.M. (1955). Cryptogamic Botany, vol. 2. Bryophytes and pteridophytes. 2nd ed. McGraw-Hill, New York.

Division Chlorophyta

Further information: Chlorophyta

Class 1. Chlorophyceae

Order 1. Volvocales

Family 1. Chlamydomonadaceae
Family 2. Volvocaceae

Order 2. Tetrasporales
Order 3. Ulotrichales

Family 1. Ulotrichaceae
Family 2. Microsporaceae
Family 3. Cylindrocapsaceae
Family 4. Chaetophoraceae
Family 5. Protococcaceae
Family 6. Coleochaetaceae
Family 7. Trentepohliaceae

Order 4. Ulvales

Family 1. Ulvaceae
Family 2 Schizomeridaceae

Order 5. Schizogoniales

Family Schizogoniaceae

Order 6. Cladophorales

Family 1. Cladophoraceae
Family 2. Sphaeropleaceae

Order 7. Oedogoniales

Family Oedogoniaceae

Order 8. Zygnematales

Family 1. Zygnemataceae
Family 2. Mesotaeniaceae
Family 3. Desmidiaceae

Order 9. Chlorococcales

Family 1. Chlorococcaceae
Family 2. Endosphaeraceae
Family 3. Characiaceae
Family 4. Protosiphonaceae
Family 5. Hydrodictyaceae
Family 6. Oöcystaceae
Family 7. Scenedesmaceae

Order 10. Siphonales

Family 1. Bryopsidaceae
Family 2. Caulerpaceae
Family 3. Halicystaceae
Family 4. Codiaceae
Family 5. Derbesiaceae
Family 6. Vaucheriaceae
Family 7. Phyllosiphonaceae

Order 11. Siphonocladiales

Family 1. Valoniaceae
Family 2. Dasycladaceae

Class 2. Charophyceae

Order Charales

Family Characeae

Division Euglenophyta

Further information: Euglenophyta

Order 1. Euglenales
Order 2. Colaciales (Euglenocapsales)

Division Pyrrophyta

Further information: Pyrrophyta

Class 1. Cryptophyceae
Class 2. Desmokontae
Class 3. Dinophyceae

Order 1. Gymnodiniales
Order 2. Peridiniales
Order 3. Dinophysidales
Order 4. Rhyzodiniales
Order 5. Dinocapsales
Order 6. Dinotrichales
Order 7. Dinococcales

Division Chrysophyta

Further information: Chrysophyta

Class 1. Xanthophyceae (Heterokontae)

Order 1. Heterochloridales
Order 2. Rhizochloridales
Order 3. Heterocapsales
Order 4. Heterotrichales
Order 5. Heterococcales
Order 6. Heterosiphonales

Class 2. Chrysophyceae

Order 1. Chrysomonadales

Suborder 1. Cromulinae
Suborder 2. Isochrysidineae
Suborder 3. Ochromonadineae

Order 2. Rhizochrysidales
Order 3. Chrysocapsales
Order 4. Chrysotrichales
Order 5. Chrysosphaerales

Class 3. Bacillariophyceae

Order 1. Centrales
Order 2. Pennales

Division Phaeophyta

Further information: Phaeophyta

Class 1. Isogeneratae

Order 1. Ectocarpales
Order 2. Sphacelariales
Order 3. Tilopteridales
Order 4. Cutleriales
Order 5. Dictyotales

Class 2. Heterogeneratae

Subclass 1. Haplostichineae

Order 1. Chordariales
Order 2. Sporochnales
Order 3. Desmarestiales

Subclass 2. Polystichineae

Order 1. Punctariales
Order 2. Dictyosiphonales
Order 3. Laminariales

Class 3. Cyclosporeae

Order Fucales

Division Cyanophyta

Further information: Cyanophyta

Class Myxophyceae (Cyanophyceae)

Order 1. Chroococcales
Order 2. Chamaesiphonales
Order 3. Hormogonales

Division Rhodophyta

Further information: Rhodophyta

Class Rhodophyceae

Subclass 1. Bangioideae

Order Bangiales

Subclass 2. Florideae

Order 1. Nemalionales
Order 2. Gelidiales
Order 3. Cryptonemiales
Order 4. Gigartinales
Order 5. Rhodymeniales
Order 6. Ceramiales

Division Myxothallophyta

Class 1. Myxomycetae

Subclass 1. Endosporeae
Subclass 2. Exosporeae

Class 2. Phytomyxinae
Class 3. Acrasieae

Division Eumycetae

Class 1. Phycomycetae

Order 1. Chytridiales

Family 1. Rhizidiaceae
Family 2. Olpidiaceae
Family 3. Synchytriaceae
Family 4. Cladochytriaceae
Family 5. Woroninaceae

Order 2. Blastocladiales
Order 3. Monoblepharidales
Order 4. Ancylistales
Order 5. Saprolegniales

Family 1. Saprolegniaceae
Family 2. Leptomitaceae
Family 3. Pythiaceae

Order 6. Peronosporales
Order 7. Mucorales
Order 8. Entomophthorales

Class 2. Ascomycetae

Subclass 1. Protoascomycetae
Subclass 2. Euascomycetae

Order 1. Aspergillales
Order 2. Erysiphales
Order 3. Hysteriales
Order 4. Phacidiales
Order 5. Pezizales
Order 6. Tuberales
Order 7. Helvellales
Order 8. Exoascales
Order 9. Hypocreales
Order 10. Sphaeriales
Order 11. Dothidiales
Order 12. Laboulbeniales

Class 3. Basidiomycetae

Subclass 1. Eubasidii

Order 1. Agaricales (Hymenomyceatae)
Order 2. Lycoperdales (Gasteromycetae)
Order 3. Dacryomycetales
Order 4. Tremellales
Order 5. Auriculariales

Subclass 2. Hemibasidii

Order 1. Urediniales
Order 2. Ustilaginales

Family 1. Ustilaginaceae
Family 2. Tilletiaceae

Class 4. Fungi imperfecti
Lichens

Subclass 1. Ascholichenes
Subclass 2. Basidiolichenes

Division Bryophyta

Further information: Bryophyte

Class 1. Hepaticae (liverworts)

Order 1. Sphaerocarpales

Family 1. Sphaerocarpaceae
Family 2. Riellaceae

Order 2. Marchantiales

Family 1. Ricciaceae
Family 2. Corsiniaceae
Family 3. Targioniaceae
Family 4. Monocleaceae
Family 5. Marchantiaceae

Order 3. Jungermanniales

Suborder 1. Metzgerineae
Suborder 2. Jungermannineae

Order 4. Calobryales

Class 2. Anthocerotae (hornworts)

Order 1. Anthocerotales

Class 3. Musci (mosses)

Subclass 1. Sphagnobrya
Subclass 2. Andreaeobrya
Subclass 3. Eubrya

Division Psilophyta

Further information: Psilophyta

Class 1. Psilophytinae (psilophytes)

Order 1. Psilophytales

Family 1. Rhyniaceae
Family 2. Psilophytaceae
Family 3. Pseudosporochnaceae
Family 4. Zosterophyllaceae
Family 5. Asteroxylaceae

Order 2. Psilotales

Division Lepidophyta

Class 1. Lycopodinae (lycopods)

Order 1. Lycopodiales

Family 1. Protolepidodendraceae
Family 2. Lycopodiaceae

Order 2. Selaginellales

Family 1. Selaginellaceae
Family 2. Miadesmiaceae

Order 3. Lepidodendrales

Family 1. Lepidodendraceae
Family 2. Lepidocarpaceae
Family 3. Bothrodendraceae
Family 4. Sigillariaceae

Order 4. Isoetales

Family 1. Pleuromeiaceae
Family 2. Isoetaceae

Division Calamophyta

Further information: Calamophyta

Class 1. Equisetinae (horsetails)

Order 1. Hyeniales
Order 2. Sphenophyllales
Order 3. Equisetales

Family 1. Calamitaceae
Family 2. Equisetaceae

Division Pterophyta

Further information: Pterophyta

Class 1. Filicinae (ferns)

Subclass 1. Primofilices (or Inversicatenales, Coenopterideae, Palaeopteridales)

Order 1. Protopteridales

Family 1. Protopteridaceae
Family 2. Cladoxylaceae

Order 2. Coenopteridales

Family 1. Zygopteridaceae
Family 2. Botryopteridaceae
Family 3. Anachoropteridaceae

Order 3. Archaeopteridales

Subclass 2. Eusporangitae

Order 1. Ophioglossales
Order 2. Marattiales

Subclass 3. Leptosporangiatae

Order 1. Filicales

Family 1. Osmundaceae
Family 2. Schizaeaceae
Family 3. Gleicheniaceae
Family 4. Matoniaceae
Family 5. Dipteridaceae
Family 6. Hymenophyllaceae
Family 7. Cyatheaceae
Family 8. Dicksoniaceae
Family 9. Polypodiaceae
Family 10. Parkeriaceae

Order 2. Marsileales
Order 3. Salviniales

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