Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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 | C24×D271 |
| commutator, socle, cosocle | C22×C271 | C25 |
| Fitting | #543×C271 | C2 |
| prime radical | C24×C271 | C23 |
| hypercenter | #543 | D271 |
| hypocenter | C271 | #3458 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #31774526 | C2 | semidirect |
| 3. #31774449 | C2 | semidirect |
| 7. #22576159 | C22 | semidirect |
| 8. #22576158 | C22 | semidirect |
| 9. #22576150 | C22 | semidirect |
| 12. Q8×Dic271 | C22 | semidirect |
| 13. #22576125 | C22 | semidirect |
| 22. C22×Dic271 | Q8 | semidirect |
| 61. C271 | #3458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31774526 | 2 |
| #31774523 | 2 |
| #31774449 | 2 |
| C2×#22576116 | 2 |
| #543×C271 | 2 |
| #3458 | 271 |