Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 334 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 | #127×C509 | C2 |
| hypercenter | #127 | D509 |
| hypocenter | C509 | #535 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22550860 | C2 | semidirect |
| 2. C2×D4×D509 | C2 | semidirect |
| 3. #30654211 | C2 | semidirect |
| 4. #30654203 | C2 | semidirect |
| 5. #30654202 | C2 | semidirect |
| 6. #30654187 | C2 | semidirect |
| 7. #30654183 | C2 | semidirect |
| 8. #30654172 | C2 | semidirect |
| 9. #127×C509 | C2 | semidirect |
| 12. C2×Dic1018 | C22 | semidirect |
| 16. C2×C22⋊D509 | C22 | semidirect |
| 18. C22×Dic509 | C22 | semidirect |
| 20. #22550840 | C22 | semidirect |
| 21. C22⋊Dic509 | C22 | semidirect |
| 23. #22550838 | C22 | semidirect |
| 24. C4×Dic509 | C22 | semidirect |
| 26. C2×D4×C509 | C22 | semidirect |
| 27. C2×D4×C509 | C22 | semidirect |
| 28. C42×C509 | C22 | semidirect |
| 42. C4×D509 | D4 | semidirect |
| 71. C509 | #535 | semidirect |
| 76. #127 | D509 | semidirect |
| 91. C2×D4 | C2×D509 | semidirect |
| 92. C2×D4 | C2×D509 | semidirect |
| 100. C42 | C2×D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22550860 | 2 |
| C2×D4×D509 | 2 |
| #30654211 | 2 |
| #30654203 | 2 |
| #30654202 | 2 |
| #30654187 | 2 |
| #30654183 | 2 |
| #30654172 | 2 |
| #127×C509 | 2 |
| #535 | 509 |