Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D2656 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C83⋊C64 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D32×C83 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D1328 | — | 2 | 1 | no | — |
| 5 | D16×C83 | — | 2 | 1 | no | — |
| 6 | C32×C83 | C22 | 1 | 1 | yes | characteristic |
| 7 | D664 | — | 4 | 1 | no | — |
| 8 | D8×C83 | — | 4 | 1 | no | — |
| 9 | C16×C83 | D4 | 1 | 1 | yes | characteristic |
| 10 | D4×C83 | — | 8 | 1 | no | — |
| 11 | D332 | — | 8 | 1 | no | — |
| 12 | C8×C83 | D8 | 1 | 1 | yes | characteristic |
| 13 | C22×C83 | — | 16 | 1 | no | — |
| 14 | C2×D83 | — | 16 | 1 | no | — |
| 15 | C4×C83 | D16 | 1 | 1 | yes | characteristic |
| 16 | C2×C83 | — | 32 | 1 | no | — |
| 17 | C2×C83 | D32 | 1 | 1 | yes | characteristic |
| 18 | D83 | — | 32 | 1 | no | — |
| 19 | D64 | — | 83 | 1 | no | maximal, Sylow |
| 20 | C83 | D64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | D32 | — | 83 | 1 | no | — |
| 22 | D32 | D83 | 1 | 1 | yes | characteristic, complemented |
| 23 | C64 | — | 83 | 1 | no | — |
| 24 | D16 | — | 2 | 1 | no | — |
| 25 | D16 | — | 166 | 1 | no | — |
| 26 | C32 | C2×D83 | 1 | 1 | yes | characteristic |
| 27 | D8 | — | 4 | 1 | no | — |
| 28 | D8 | — | 332 | 1 | no | — |
| 29 | C16 | C22⋊D83 | 1 | 1 | yes | characteristic |
| 30 | D4 | — | 8 | 1 | no | — |
| 31 | D4 | — | 664 | 1 | no | — |
| 32 | C8 | #13050575 | 1 | 1 | yes | characteristic |
| 33 | C22 | — | 16 | 1 | no | — |
| 34 | C22 | — | 1328 | 1 | no | — |
| 35 | C4 | #14581733 | 1 | 1 | yes | characteristic |
| 36 | C2 | — | 32 | 1 | no | — |
| 37 | C2 | — | 2656 | 1 | no | — |
| 38 | C2 | #19127536 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #19127536 |
| commutator, cosocle | C32×C83 | C22 |
| Frattini | C32 | C2×D83 |
| Fitting | D32×C83 | C2 |
| socle | C2×C83 | D32 |
| hypercenter | D32 | D83 |
| hypocenter | C83 | D64 |
| LCS | G ▷ C32×C83 ▷ C16×C83 ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D2656 | C2 | semidirect |
| 2. C83⋊C64 | C2 | semidirect |
| 3. D32×C83 | C2 | semidirect |
| 20. C83 | D64 | semidirect |
| 22. D32 | D83 | semidirect |