Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
372 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C251 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C251 |
| prime radical | C24×C251 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #401×C251 | C2 | semidirect |
| 3. #398×C251 | C2 | semidirect |
| 54. C251 | #2834 | direct (N × H) |
| 55. #2834 | C251 | direct (N × H) |
| 56. #401 | C2×C251 | semidirect |
| 58. #398 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #401×C251 | 2 |
| #399×C251 | 2 |
| #398×C251 | 2 |
| #397×C251 | 2 |
| #385×C251 | 2 |
| #2834 | 251 |