Finite Group DatabasebetaFinite Group DatabasebetaOrder 4352.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic544 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14526603 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C17×D64 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Dic272 | — | 2 | 1 | no | — |
| 5 | C17×D32 | — | 2 | 1 | no | — |
| 6 | C17×C64 | C22 | 1 | 1 | yes | characteristic |
| 7 | C17×D16 | — | 4 | 1 | no | — |
| 8 | Dic136 | — | 4 | 1 | no | — |
| 9 | C17×C32 | D4 | 1 | 1 | yes | characteristic |
| 10 | D8×C17 | — | 8 | 1 | no | — |
| 11 | Dic68 | — | 8 | 1 | no | — |
| 12 | C16×C17 | D8 | 1 | 1 | yes | characteristic |
| 13 | SD256 | — | 17 | 1 | no | maximal, Sylow |
| 14 | D4×C17 | — | 16 | 1 | no | — |
| 15 | Dic34 | — | 16 | 1 | no | — |
| 16 | C8×C17 | D16 | 1 | 1 | yes | characteristic |
| 17 | Q128 | — | 17 | 1 | no | — |
| 18 | D64 | D17 | 1 | 1 | yes | characteristic |
| 19 | C128 | — | 17 | 1 | no | — |
| 20 | C22×C17 | — | 32 | 1 | no | — |
| 21 | C4×C17 | D32 | 1 | 1 | yes | characteristic |
| 22 | Dic17 | — | 32 | 1 | no | — |
| 23 | Q64 | — | 34 | 1 | no | — |
| 24 | D32 | — | 2 | 1 | no | — |
| 25 | C64 | C2×D17 | 1 | 1 | yes | characteristic |
| 26 | C2×C17 | — | 64 | 1 | no | — |
| 27 | C2×C17 | D64 | 1 | 1 | yes | characteristic |
| 28 | Q32 | — | 68 | 1 | no | — |
| 29 | D16 | — | 4 | 1 | no | — |
| 30 | C32 | C22⋊D17 | 1 | 1 | yes | characteristic |
| 31 | C17 | SD256 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 32 | Q16 | — | 136 | 1 | no | — |
| 33 | D8 | — | 8 | 1 | no | — |
| 34 | C16 | #63228 | 1 | 1 | yes | characteristic |
| 35 | Q8 | — | 272 | 1 | no | — |
| 36 | D4 | — | 16 | 1 | no | — |
| 37 | C8 | #10587497 | 1 | 1 | yes | characteristic |
| 38 | C22 | — | 32 | 1 | no | — |
| 39 | C4 | — | 544 | 1 | no | — |
| 40 | C4 | #11762644 | 1 | 1 | yes | characteristic |
| 41 | C2 | — | 64 | 1 | no | — |
| 42 | C2 | #14526996 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #14526996 |
| commutator, cosocle | C17×C64 | C22 |
| Frattini | C64 | C2×D17 |
| Fitting, prime radical | C17×D64 | C2 |
| socle | C2×C17 | D64 |
| hypercenter | D64 | D17 |
| hypocenter | C17 | SD256 |
| LCS | G ▷ C17×C64 ▷ C17×C32 ▷ C16×C17 ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ D64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic544 | C2 | semidirect |
| 2. #14526603 | C2 | semidirect |
| 31. C17 | SD256 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Dic544 | 2 |
| #14526603 | 2 |
| C17×D64 | 2 |
| SD256 | 17 |