Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
460 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 205 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 | C22×C251 | C23×C4 |
| Fitting | #376×C251 | C2 |
| socle, prime radical, cosocle | C23×C251 | C24 |
| hypercenter | #376 | D251 |
| hypocenter | C251 | #2331 |
| LCS | G ▷ C22×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #376 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #22548715 | C4 | semidirect |
| 40. C42×C251 | Q8 | semidirect |
| 79. C251 | #2331 | semidirect |
| 120. C42 | Dic502 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548715 | 2 |
| #30631591 | 2 |
| #30631585 | 2 |
| #30631570 | 2 |
| #30631553 | 2 |
| C4×C4⋊Dic251 | 2 |
| #376×C251 | 2 |
| #2331 | 251 |