Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C251 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C251 |
| prime radical | #41×C251 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C251 ◁ C2×C4×C251 ◁ #131×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q32×C251 | C2 | semidirect |
| 4. #368×C251 | C2 | semidirect |
| 5. #362×C251 | C2 | semidirect |
| 52. C251 | #2222 | direct (N × H) |
| 53. #2222 | C251 | direct (N × H) |
| 54. C2×Q32 | C2×C251 | semidirect |
| 57. #368 | C2×C251 | semidirect |
| 58. #362 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q32×C251 | 2 |
| #504×C251 | 2 |
| #473×C251 | 2 |
| #368×C251 | 2 |
| #362×C251 | 2 |
| #2222 | 251 |