Finite Group DatabasebetaFinite Group DatabasebetaOrder 12642.
34 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#11744749 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×D301 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C72×C43 | C2×C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | #14268588 | — | 7 | 1 | no | maximal |
| 5 | (C43⋊C3)⋊D7 | — | 7 | 1 | no | maximal |
| 6 | #11744749 | D7 | 1 | 1 | yes | characteristic, complemented |
| 7 | C7×C43⋊C3 | — | 7 | 1 | no | — |
| 8 | D301 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 9 | C7×D43 | — | 7 | 1 | no | — |
| 10 | C7×C43 | — | 6 | 1 | no | — |
| 11 | C7×C43 | C2×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 12 | C7×C43 | C3×D7 | 1 | 1 | yes | characteristic, complemented |
| 13 | D7×ΓL(1,𝔽8) | — | 43 | 1 | no | maximal, Hall |
| 14 | D43⋊C3 | — | 49 | 1 | no | Hall |
| 15 | C7×ΓL(1,𝔽8) | — | 43 | 1 | no | Hall |
| 16 | C43⋊C3 | — | 7 | 1 | no | Hall |
| 17 | C7×D7 | — | 43 | 1 | no | Hall |
| 18 | D43 | — | 7 | 1 | no | Hall |
| 19 | C72 | D43⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 20 | C43 | D7×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C3×D7 | — | 301 | 1 | no | — |
| 22 | C2×ΓL(1,𝔽8) | — | 301 | 1 | no | — |
| 23 | C3×C7 | — | 301 | 1 | no | — |
| 24 | ΓL(1,𝔽8) | — | 43 | 1 | no | — |
| 25 | C2×C7 | — | 301 | 1 | no | — |
| 26 | D7 | — | 43 | 1 | no | — |
| 27 | C7 | — | 6 | 1 | no | — |
| 28 | C7 | (C43⋊C3)⋊D7 | 1 | 1 | yes | characteristic, complemented |
| 29 | C7 | #14268588 | 1 | 1 | yes | characteristic, complemented |
| 30 | C2×C3 | — | 2107 | 1 | no | Hall |
| 31 | C3 | — | 301 | 1 | no | Sylow |
| 32 | C2 | — | 301 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C43 | C2×C3 |
| LCS | G ▷ C72×C43 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#11744749 | C2 | semidirect |
| 2. C7×D301 | C3 | semidirect |
| 3. C72×C43 | C2×C3 | semidirect |
| 6. #11744749 | D7 | semidirect |
| 8. D301 | ΓL(1,𝔽8) | semidirect |
| 11. C7×C43 | C2×ΓL(1,𝔽8) | semidirect |
| 12. C7×C43 | C3×D7 | semidirect |
| 19. C72 | D43⋊C3 | semidirect |
| 20. C43 | D7×ΓL(1,𝔽8) | semidirect |
| 28. C7 | (C43⋊C3)⋊D7 | semidirect |
| 29. C7 | #14268588 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#11744749 | 2 |
| C7×D301 | 3 |
| #14268588 | 7 |
| (C43⋊C3)⋊D7 | 7 |
| D7×ΓL(1,𝔽8) | 43 |