Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
128 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C251 | #99 |
| commutator | C23 | C2×C8×C251 |
| Frattini, cosocle | C2×M16 | C22×C251 |
| socle | C2×C251 | #323 |
| prime radical | C24×C251 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C251 ◁ C2×C4×C251 ◁ C2×M16×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #349×C251 | C2 | semidirect |
| 51. C251 | #1320 | direct (N × H) |
| 52. #1320 | C251 | direct (N × H) |
| 54. #349 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #407×C251 | 2 |
| #349×C251 | 2 |
| #1320 | 251 |