Finite Group DatabasebetaFinite Group DatabasebetaOrder 5696.
30 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic712 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #15667918 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | Q32×C89 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Dic356 | — | 2 | 1 | no | — |
| 5 | Q16×C89 | — | 2 | 1 | no | — |
| 6 | C16×C89 | C22 | 1 | 1 | yes | characteristic |
| 7 | Q8×C89 | — | 4 | 1 | no | — |
| 8 | Dic178 | — | 4 | 1 | no | — |
| 9 | C8×C89 | D4 | 1 | 1 | yes | characteristic |
| 10 | C4×C89 | — | 8 | 1 | no | — |
| 11 | C4×C89 | D8 | 1 | 1 | yes | characteristic |
| 12 | Dic89 | — | 8 | 1 | no | — |
| 13 | C2×C89 | D16 | 1 | 1 | yes | characteristic |
| 14 | C89 | Q64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | Q64 | — | 89 | 1 | no | maximal, Sylow |
| 16 | Q32 | — | 89 | 1 | no | — |
| 17 | Q32 | D89 | 1 | 1 | yes | characteristic |
| 18 | C32 | — | 89 | 1 | no | — |
| 19 | Q16 | — | 2 | 1 | no | — |
| 20 | Q16 | — | 178 | 1 | no | — |
| 21 | C16 | C2×D89 | 1 | 1 | yes | characteristic |
| 22 | Q8 | — | 4 | 1 | no | — |
| 23 | Q8 | — | 356 | 1 | no | — |
| 24 | C8 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 25 | C4 | — | 8 | 1 | no | — |
| 26 | C4 | — | 712 | 1 | no | — |
| 27 | C4 | #13083559 | 1 | 1 | yes | characteristic |
| 28 | C2 | #15667953 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #15667953 |
| commutator | C16×C89 | C22 |
| Frattini | C16 | C2×D89 |
| Fitting | Q32×C89 | C2 |
| socle | C2×C89 | D16 |
| hypercenter | Q32 | D89 |
| hypocenter | C89 | Q64 |
| LCS | G ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. C89 | Q64 | semidirect |