Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
420 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 119 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 | #401×C251 | C2 |
| prime radical | C22×D1004 | C22 |
| hypercenter | #401 | D251 |
| hypocenter | C251 | #2801 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #401 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30631624 | C2 | semidirect |
| 2. #30631611 | C2 | semidirect |
| 4. #30631584 | C2 | semidirect |
| 5. #30631579 | C2 | semidirect |
| 6. #30631576 | C2 | semidirect |
| 7. #401×C251 | C2 | semidirect |
| 77. C251 | #2801 | semidirect |
| 79. #401 | D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30631624 | 2 |
| #30631611 | 2 |
| #30631605 | 2 |
| #30631584 | 2 |
| #30631579 | 2 |
| #30631576 | 2 |
| #401×C251 | 2 |
| #2801 | 251 |