Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #124×C509 | C2 |
| prime radical | #30654186 | C2 |
| hypercenter | #124 | D509 |
| hypocenter | C509 | #542 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654215 | C2 | semidirect |
| 2. #30654208 | C2 | semidirect |
| 3. #30654201 | C2 | semidirect |
| 4. #30654199 | C2 | semidirect |
| 5. #30654192 | C2 | semidirect |
| 6. #30654191 | C2 | semidirect |
| 8. Q8×Dic509 | C2 | semidirect |
| 9. D4×Dic509 | C2 | semidirect |
| 10. #30654172 | C2 | semidirect |
| 11. #124×C509 | C2 | semidirect |
| 73. C509 | #542 | semidirect |
| 78. #124 | D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654215 | 2 |
| #30654208 | 2 |
| #30654201 | 2 |
| #30654199 | 2 |
| #30654192 | 2 |
| #30654191 | 2 |
| #30654186 | 2 |
| Q8×Dic509 | 2 |
| D4×Dic509 | 2 |
| #30654172 | 2 |
| #124×C509 | 2 |
| #542 | 509 |