Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
38 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic2768 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #26011210 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D32×C173 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Dic1384 | — | 2 | 1 | no | — |
| 5 | D16×C173 | — | 2 | 1 | no | — |
| 6 | C32×C173 | C22 | 1 | 1 | yes | characteristic |
| 7 | Dic692 | — | 4 | 1 | no | — |
| 8 | D8×C173 | — | 4 | 1 | no | — |
| 9 | C16×C173 | D4 | 1 | 1 | yes | characteristic |
| 10 | Dic346 | — | 8 | 1 | no | — |
| 11 | D4×C173 | — | 8 | 1 | no | — |
| 12 | C8×C173 | D8 | 1 | 1 | yes | characteristic |
| 13 | C22×C173 | — | 16 | 1 | no | — |
| 14 | C4×C173 | D16 | 1 | 1 | yes | characteristic |
| 15 | Dic173 | — | 16 | 1 | no | — |
| 16 | C2×C173 | — | 32 | 1 | no | — |
| 17 | C2×C173 | D32 | 1 | 1 | yes | characteristic |
| 18 | C173 | SD128 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | SD128 | — | 173 | 1 | no | maximal, Sylow |
| 20 | Q64 | — | 173 | 1 | no | — |
| 21 | D32 | D173 | 1 | 1 | yes | characteristic |
| 22 | C64 | — | 173 | 1 | no | — |
| 23 | Q32 | — | 346 | 1 | no | — |
| 24 | D16 | — | 2 | 1 | no | — |
| 25 | C32 | C2×D173 | 1 | 1 | yes | characteristic |
| 26 | Q16 | — | 692 | 1 | no | — |
| 27 | D8 | — | 4 | 1 | no | — |
| 28 | C16 | C22⋊D173 | 1 | 1 | yes | characteristic |
| 29 | Q8 | — | 1384 | 1 | no | — |
| 30 | D4 | — | 8 | 1 | no | — |
| 31 | C8 | #14583811 | 1 | 1 | yes | characteristic |
| 32 | C22 | — | 16 | 1 | no | — |
| 33 | C4 | — | 2768 | 1 | no | — |
| 34 | C4 | #19149620 | 1 | 1 | yes | characteristic |
| 35 | C2 | — | 32 | 1 | no | — |
| 36 | C2 | #26011326 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #26011326 |
| commutator, cosocle | C32×C173 | C22 |
| Frattini | C32 | C2×D173 |
| Fitting, prime radical | D32×C173 | C2 |
| socle | C2×C173 | D32 |
| hypercenter | D32 | D173 |
| hypocenter | C173 | SD128 |
| LCS | G ▷ C32×C173 ▷ C16×C173 ▷ C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic2768 | C2 | semidirect |
| 2. #26011210 | C2 | semidirect |
| 18. C173 | SD128 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Dic2768 | 2 |
| #26011210 | 2 |
| D32×C173 | 2 |
| SD128 | 173 |