Finite Group DatabasebetaFinite Group DatabasebetaOrder 7712.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic964 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #16838745 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D8×C241 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Dic482 | — | 2 | 1 | no | — |
| 5 | D4×C241 | — | 2 | 1 | no | — |
| 6 | C8×C241 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×C241 | — | 4 | 1 | no | — |
| 8 | C4×C241 | D4 | 1 | 1 | yes | characteristic |
| 9 | Dic241 | — | 4 | 1 | no | — |
| 10 | C2×C241 | — | 8 | 1 | no | — |
| 11 | C2×C241 | D8 | 1 | 1 | yes | characteristic |
| 12 | C241 | SD32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | SD32 | — | 241 | 1 | no | maximal, Sylow |
| 14 | Q16 | — | 241 | 1 | no | — |
| 15 | D8 | D241 | 1 | 1 | yes | characteristic |
| 16 | C16 | — | 241 | 1 | no | — |
| 17 | Q8 | — | 482 | 1 | no | — |
| 18 | D4 | — | 2 | 1 | no | — |
| 19 | C8 | C2×D241 | 1 | 1 | yes | characteristic |
| 20 | C22 | — | 4 | 1 | no | — |
| 21 | C4 | — | 964 | 1 | no | — |
| 22 | C4 | C22⋊D241 | 1 | 1 | yes | characteristic |
| 23 | C2 | — | 8 | 1 | no | — |
| 24 | C2 | #16838755 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16838755 |
| commutator | C8×C241 | C22 |
| Frattini | C8 | C2×D241 |
| Fitting | D8×C241 | C2 |
| socle | C2×C241 | D8 |
| hypercenter | D8 | D241 |
| hypocenter | C241 | SD32 |
| LCS | G ▷ C8×C241 ▷ C4×C241 ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic964 | C2 | semidirect |
| 2. #16838745 | C2 | semidirect |
| 12. C241 | SD32 | semidirect |