Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 630 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 | C2×#126×C251 | C2 |
| hypercenter | C2×#126 | D251 |
| hypocenter | C251 | #2818 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22548712 | C2 | semidirect |
| 2. #30631604 | C2 | semidirect |
| 3. #30631596 | C2 | semidirect |
| 4. #30631555 | C2 | semidirect |
| 5. C2×#126×C251 | C2 | semidirect |
| 26. C42×C251 | D4 | semidirect |
| 54. C251 | #2818 | semidirect |
| 57. C2×#126 | D251 | semidirect |
| 81. C42 | C22⋊D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548712 | 2 |
| #30631604 | 2 |
| #30631596 | 2 |
| #30631555 | 2 |
| C2×#126×C251 | 2 |
| #2818 | 251 |