Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
644 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #381×C251 | C2 |
| hypercenter | #381 | D251 |
| hypocenter | C251 | #2815 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #381 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22548689 | C2 | semidirect |
| 2. #30631614 | C2 | semidirect |
| 3. #30631597 | C2 | semidirect |
| 4. #30631596 | C2 | semidirect |
| 5. #381×C251 | C2 | semidirect |
| 30. C22×C4×C251 | D4 | semidirect |
| 72. C251 | #2815 | semidirect |
| 78. #381 | D251 | semidirect |
| 100. C22×C4 | D1004 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548689 | 2 |
| #30631614 | 2 |
| #30631597 | 2 |
| #30631596 | 2 |
| #381×C251 | 2 |
| #2815 | 251 |