Finite Group DatabasebetaFinite Group DatabasebetaOrder 7712.
28 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16838784 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 2 | C2×#14515358 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14515358 | C4 | 1 | 1 | yes | characteristic |
| 4 | #14515358 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×#11757210 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×D241 | C8 | 1 | 1 | yes | characteristic |
| 7 | #11757210 | C8 | 1 | 1 | yes | characteristic |
| 8 | #11757210 | C2×C4 | 1 | 1 | yes | characteristic |
| 9 | C2×C241 | C16 | 1 | 1 | yes | characteristic, complemented |
| 10 | D241 | C16 | 1 | 1 | yes | characteristic, complemented |
| 11 | D241 | C2×C8 | 1 | 1 | yes | characteristic |
| 12 | C241 | C2×C16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C2×C16 | — | 241 | 1 | no | maximal, Sylow |
| 14 | C2×C8 | — | 241 | 1 | no | — |
| 15 | C16 | — | 241 | 2 | no | — |
| 16 | C2×C4 | — | 241 | 1 | no | — |
| 17 | C8 | — | 241 | 1 | no | — |
| 18 | C8 | — | 241 | 1 | no | — |
| 19 | C22 | — | 241 | 1 | no | — |
| 20 | C4 | — | 241 | 1 | no | — |
| 21 | C4 | — | 241 | 1 | no | — |
| 22 | C2 | — | 241 | 1 | no | — |
| 23 | C2 | — | 241 | 1 | no | — |
| 24 | C2 | #16838784 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #16838784 |
| commutator, hypocenter | C241 | C2×C16 |
| Frattini | C1 | G |
| Fitting, socle | C2×C241 | C16 |
| LCS | G ▷ C241 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16838784 | C2 | direct (N × H) |
| 9. C2×C241 | C16 | semidirect |
| 10. D241 | C16 | semidirect |
| 12. C241 | C2×C16 | semidirect |
| 24. C2 | #16838784 | direct (N × H) |