Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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 | #125×C509 | C2 |
| prime radical | #30654190 | C2 |
| hypercenter | #125 | D509 |
| hypocenter | C509 | #529 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654210 | C2 | semidirect |
| 2. #30654203 | C2 | semidirect |
| 3. #30654200 | C2 | semidirect |
| 4. #30654195 | C2 | semidirect |
| 6. #30654180 | C2 | semidirect |
| 7. #30654177 | C2 | semidirect |
| 8. #30654177 | C2 | semidirect |
| 9. #30654173 | C2 | semidirect |
| 10. C42×D509 | C2 | semidirect |
| 11. #125×C509 | C2 | semidirect |
| 74. C509 | #529 | semidirect |
| 77. #125 | D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654210 | 2 |
| #30654203 | 2 |
| #30654200 | 2 |
| #30654195 | 2 |
| #30654190 | 2 |
| #30654180 | 2 |
| #30654177 | 2 |
| #30654173 | 2 |
| C42×D509 | 2 |
| #125×C509 | 2 |
| #529 | 509 |