Finite Group DatabasebetaFinite Group DatabasebetaOrder 23226.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #26042479 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×D553 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C72×C79 | C2×C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | #15693307 | — | 7 | 1 | no | maximal |
| 5 | #15693299 | — | 7 | 1 | no | maximal |
| 6 | #13110306 | — | 7 | 1 | no | — |
| 7 | #13110306 | — | 7 | 1 | no | — |
| 8 | #13110306 | — | 14 | 3 | no | — |
| 9 | D553 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 10 | C7×D79 | — | 21 | 1 | no | — |
| 11 | C7×C79 | — | 2 | 3 | no | — |
| 12 | C7×C79 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 13 | C7×C79 | C2×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 14 | D79⋊C3 | — | 49 | 1 | no | Hall |
| 15 | #64423 | — | 79 | 1 | no | maximal, Hall |
| 16 | C79⋊C3 | — | 49 | 1 | no | Hall |
| 17 | D79 | — | 7 | 1 | no | Hall |
| 18 | #3861 | — | 79 | 1 | no | Hall |
| 19 | C7×D7 | — | 237 | 1 | no | Hall |
| 20 | C79 | #64423 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C72 | D79⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 22 | C2×ΓL(1,𝔽8) | — | 553 | 1 | no | — |
| 23 | AGL(1,𝔽7) | — | 553 | 1 | no | — |
| 24 | ΓL(1,𝔽8) | — | 553 | 1 | no | — |
| 25 | ΓL(1,𝔽8) | — | 553 | 1 | no | — |
| 26 | ΓL(1,𝔽8) | — | 1106 | 3 | no | — |
| 27 | C2×C7 | — | 1659 | 1 | no | — |
| 28 | D7 | — | 237 | 1 | no | — |
| 29 | C7 | — | 2 | 3 | no | — |
| 30 | C7 | #15693299 | 1 | 1 | yes | characteristic, complemented |
| 31 | C7 | #15693307 | 1 | 1 | yes | characteristic, complemented |
| 32 | C2×C3 | — | 3871 | 1 | no | Hall |
| 33 | C3 | — | 3871 | 1 | no | Sylow |
| 34 | C2 | — | 1659 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C79 | C2×C3 |
| LCS | G ▷ C72×C79 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26042479 | C2 | semidirect |
| 2. C7×D553 | C3 | semidirect |
| 3. C72×C79 | C2×C3 | semidirect |
| 9. D553 | ΓL(1,𝔽8) | semidirect |
| 12. C7×C79 | AGL(1,𝔽7) | semidirect |
| 13. C7×C79 | C2×ΓL(1,𝔽8) | semidirect |
| 20. C79 | #64423 | semidirect |
| 21. C72 | D79⋊C3 | semidirect |
| 30. C7 | #15693299 | semidirect |
| 31. C7 | #15693307 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26042479 | 2 |
| C7×D553 | 3 |
| #15693307 | 7 |
| #15693299 | 7 |
| #64423 | 79 |