Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
432 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 157 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, prime radical, cosocle | C23×C251 | C24 |
| Fitting | #391×C251 | C2 |
| hypercenter | #391 | D251 |
| hypocenter | C251 | #2811 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 25. C2×Dic502 | Q8 | semidirect |
| 34. C2×Q8×C251 | Q8 | semidirect |
| 78. C251 | #2811 | semidirect |
| 107. C2×Q8 | Dic502 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30631608 | 2 |
| #30631607 | 2 |
| #30631594 | 2 |
| #30631593 | 2 |
| #30631591 | 2 |
| #391×C251 | 2 |
| #2811 | 251 |