Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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 | #123×C509 | C2 |
| prime radical | C2×#22550860 | C2 |
| hypercenter | #123 | D509 |
| hypocenter | C509 | #536 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Dic1018 | C2 | semidirect |
| 3. #30654214 | C2 | semidirect |
| 4. #30654201 | C2 | semidirect |
| 5. #30654199 | C2 | semidirect |
| 6. #30654198 | C2 | semidirect |
| 7. #30654196 | C2 | semidirect |
| 8. #30654193 | C2 | semidirect |
| 9. #30654192 | C2 | semidirect |
| 10. #30654171 | C2 | semidirect |
| 11. #123×C509 | C2 | semidirect |
| 79. C509 | #536 | semidirect |
| 88. #123 | D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22550860 | 2 |
| C22×Dic1018 | 2 |
| #30654214 | 2 |
| #30654201 | 2 |
| #30654199 | 2 |
| #30654198 | 2 |
| #30654196 | 2 |
| #30654193 | 2 |
| #30654192 | 2 |
| #30654171 | 2 |
| #123×C509 | 2 |
| #536 | 509 |