Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
516 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 165 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 | C43×C251 | C2 |
| prime radical | C22×D1004 | C22 |
| hypercenter | C43 | D251 |
| hypocenter | C251 | #2860 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22548709 | C2 | semidirect |
| 3. #30631585 | C2 | semidirect |
| 4. #30631578 | C2 | semidirect |
| 5. C43×C251 | C2 | semidirect |
| 28. C42×C251 | D4 | semidirect |
| 52. C4×C251 | #126 | semidirect |
| 58. C251 | #2860 | semidirect |
| 64. C43 | D251 | semidirect |
| 87. C42 | D1004 | semidirect |
| 112. C4 | #22548709 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548709 | 2 |
| #30631589 | 2 |
| #30631585 | 2 |
| #30631578 | 2 |
| C43×C251 | 2 |
| #2860 | 251 |