Finite Group DatabasebetaFinite Group DatabasebetaOrder 32896.
432 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 139 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×D257 |
| commutator, socle, cosocle | C23×C257 | C24 |
| Fitting | #396×C257 | C2 |
| prime radical | C24×C257 | C23 |
| hypercenter | #396 | D257 |
| hypocenter | C257 | #2401 |
| LCS | G ▷ C23×C257 ▷ C257 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C257⋊#401 | C2 | semidirect |
| 2. #30656676 | C2 | semidirect |
| 4. C4⋊C4×Dic257 | C2 | semidirect |
| 6. #30656582 | C2 | semidirect |
| 70. C257 | #2401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C257⋊#401 | 2 |
| #30656676 | 2 |
| #30656662 | 2 |
| C4⋊C4×Dic257 | 2 |
| C22⋊C4×Dic257 | 2 |
| #30656582 | 2 |
| #396×C257 | 2 |
| #2401 | 257 |