Finite Group DatabasebetaFinite Group DatabasebetaOrder 10112.
38 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic1264 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D2528 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C64×C79 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | Dic632 | — | 2 | 1 | no | — |
| 5 | D1264 | — | 2 | 1 | no | — |
| 6 | C32×C79 | C22 | 1 | 1 | yes | characteristic |
| 7 | Dic316 | — | 4 | 1 | no | — |
| 8 | D632 | — | 4 | 1 | no | — |
| 9 | C16×C79 | D4 | 1 | 1 | yes | characteristic |
| 10 | D316 | — | 8 | 1 | no | — |
| 11 | Dic158 | — | 8 | 1 | no | — |
| 12 | C8×C79 | D8 | 1 | 1 | yes | characteristic |
| 13 | C2×D79 | — | 16 | 1 | no | — |
| 14 | C4×C79 | D16 | 1 | 1 | yes | characteristic |
| 15 | Dic79 | — | 16 | 1 | no | — |
| 16 | C2×C79 | D32 | 1 | 1 | yes | characteristic |
| 17 | D79 | — | 32 | 1 | no | — |
| 18 | SD128 | — | 79 | 1 | no | maximal, Sylow |
| 19 | C79 | SD128 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 20 | Q64 | — | 79 | 1 | no | — |
| 21 | D32 | — | 79 | 1 | no | — |
| 22 | C64 | D79 | 1 | 1 | yes | characteristic, complemented |
| 23 | Q32 | — | 158 | 1 | no | — |
| 24 | D16 | — | 158 | 1 | no | — |
| 25 | C32 | C2×D79 | 1 | 1 | yes | characteristic |
| 26 | Q16 | — | 316 | 1 | no | — |
| 27 | D8 | — | 316 | 1 | no | — |
| 28 | C16 | D316 | 1 | 1 | yes | characteristic |
| 29 | Q8 | — | 632 | 1 | no | — |
| 30 | D4 | — | 632 | 1 | no | — |
| 31 | C8 | D632 | 1 | 1 | yes | characteristic |
| 32 | C22 | — | 1264 | 1 | no | — |
| 33 | C4 | — | 1264 | 1 | no | — |
| 34 | C4 | D1264 | 1 | 1 | yes | characteristic |
| 35 | C2 | — | 2528 | 1 | no | — |
| 36 | C2 | D2528 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D2528 |
| commutator, cosocle | C32×C79 | C22 |
| Frattini | C32 | C2×D79 |
| Fitting | C64×C79 | C2 |
| socle | C2×C79 | D32 |
| prime radical | D2528 | C2 |
| hypercenter | C64 | D79 |
| hypocenter | C79 | SD128 |
| LCS | G ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic1264 | C2 | semidirect |
| 3. C64×C79 | C2 | semidirect |
| 19. C79 | SD128 | semidirect |
| 22. C64 | D79 | semidirect |