Finite Group DatabasebetaFinite Group DatabasebetaOrder 5536.
60 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #14583805 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #14583803 | C2 | 1 | 3 | yes | maximal |
| 3 | C42×C173 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×D173 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×Dic173 | C22 | 1 | 3 | yes | — |
| 6 | C2×C4×C173 | C22 | 1 | 3 | yes | — |
| 7 | C22×C173 | C23 | 1 | 1 | yes | characteristic |
| 8 | C2×D173 | — | 2 | 3 | no | — |
| 9 | C4×C173 | — | 2 | 3 | no | — |
| 10 | Dic173 | — | 2 | 3 | no | — |
| 11 | C2×C173 | #41 | 1 | 3 | yes | — |
| 12 | D173 | — | 4 | 1 | no | — |
| 13 | C173 | #126 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | #126 | — | 173 | 1 | no | maximal, Sylow |
| 15 | C4⋊C4 | — | 173 | 3 | no | — |
| 16 | C22⋊C4 | — | 173 | 3 | no | — |
| 17 | C42 | D173 | 1 | 1 | yes | characteristic, complemented |
| 18 | C23 | — | 173 | 1 | no | — |
| 19 | C2×C4 | — | 173 | 3 | no | — |
| 20 | C2×C4 | C2×D173 | 1 | 3 | yes | — |
| 21 | C22 | — | 346 | 3 | no | — |
| 22 | C22 | C22×D173 | 1 | 1 | yes | characteristic |
| 23 | C4 | — | 2 | 3 | no | — |
| 24 | C4 | — | 346 | 3 | no | — |
| 25 | C2 | — | 692 | 1 | no | — |
| 26 | C2 | #14583826 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D173 |
| commutator, socle | C22×C173 | C23 |
| Fitting | C42×C173 | C2 |
| hypercenter | C42 | D173 |
| hypocenter | C173 | #126 |
| LCS | G ▷ C22×C173 ▷ C173 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14583805 | C2 | semidirect |
| 3. C42×C173 | C2 | semidirect |
| 13. C173 | #126 | semidirect |
| 17. C42 | D173 | semidirect |