Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D509 |
| commutator, socle, cosocle | C22×C509 | C24 |
| Fitting | #117×C509 | C2 |
| prime radical | C23×C509 | C23 |
| hypercenter | #117 | D509 |
| hypocenter | C509 | #544 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654215 | C2 | semidirect |
| 2. #30654214 | C2 | semidirect |
| 3. #30654207 | C2 | semidirect |
| 4. #30654206 | C2 | semidirect |
| 31. C2×C4×C509 | Q8 | semidirect |
| 51. C509 | #544 | semidirect |
| 83. C2×C4 | Dic1018 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654215 | 2 |
| #30654214 | 2 |
| #30654207 | 2 |
| #30654206 | 2 |
| #30654192 | 2 |
| #30654174 | 2 |
| C2×C4⋊Dic509 | 2 |
| #117×C509 | 2 |
| #544 | 509 |