Finite Group DatabasebetaFinite Group DatabasebetaOrder 8768.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #17984178 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#14555513 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14555513 | C4 | 1 | 1 | yes | characteristic |
| 4 | #14555513 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×#11764040 | C4 | 1 | 1 | yes | characteristic |
| 6 | #11764040 | C8 | 1 | 1 | yes | characteristic |
| 7 | #11764040 | C2×C4 | 1 | 1 | yes | characteristic |
| 8 | C2×C4×C137 | C8 | 1 | 1 | yes | characteristic |
| 9 | C22×C137 | C16 | 1 | 1 | yes | characteristic |
| 10 | C4×C137 | C16 | 1 | 1 | yes | characteristic |
| 11 | C4×C137 | C2×C8 | 1 | 1 | yes | characteristic |
| 12 | C2×C137 | — | 2 | 1 | no | — |
| 13 | C2×C137 | C2×C16 | 1 | 1 | yes | characteristic |
| 14 | C137 | M64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | M64 | — | 137 | 1 | no | maximal, Sylow |
| 16 | C2×C16 | — | 137 | 1 | no | — |
| 17 | C32 | — | 137 | 2 | no | — |
| 18 | C2×C8 | — | 137 | 1 | no | — |
| 19 | C16 | — | 137 | 1 | no | — |
| 20 | C16 | — | 137 | 1 | no | — |
| 21 | C2×C4 | #11764049 | 1 | 1 | yes | characteristic |
| 22 | C8 | — | 137 | 1 | no | — |
| 23 | C8 | — | 137 | 1 | no | — |
| 24 | C22 | #14555522 | 1 | 1 | yes | characteristic |
| 25 | C4 | #14555522 | 1 | 1 | yes | characteristic |
| 26 | C4 | C2×#11764049 | 1 | 1 | yes | characteristic |
| 27 | C2 | — | 2 | 1 | no | — |
| 28 | C2 | C2×#14555522 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#11764049 |
| commutator, socle | C2×C137 | C2×C16 |
| Fitting | C2×C4×C137 | C8 |
| hypercenter | C2×C4 | #11764049 |
| hypocenter | C137 | M64 |
| LCS | G ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #17984178 | C2 | semidirect |
| 14. C137 | M64 | semidirect |