Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D251 |
| commutator, socle, cosocle | C23×C251 | C24 |
| Fitting | #391×C251 | C2 |
| prime radical | C22×D1004 | C22 |
| hypercenter | #391 | D251 |
| hypocenter | C251 | #2808 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30631614 | C2 | semidirect |
| 3. #30631598 | C2 | semidirect |
| 5. #30631585 | C2 | semidirect |
| 6. #30631579 | C2 | semidirect |
| 7. #30631577 | C2 | semidirect |
| 8. #30631574 | C2 | semidirect |
| 9. #391×C251 | C2 | semidirect |
| 123. C251 | #2808 | semidirect |
| 128. #391 | D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30631614 | 2 |
| #30631605 | 2 |
| #30631598 | 2 |
| #30631589 | 2 |
| #30631585 | 2 |
| #30631579 | 2 |
| #30631577 | 2 |
| #30631574 | 2 |
| #391×C251 | 2 |
| #2808 | 251 |