Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
520 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 176 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 | #392×C251 | C2 |
| 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 |
| 2. #393×C251 | C2 | semidirect |
| 4. #386×C251 | C2 | semidirect |
| 5. #381×C251 | C2 | semidirect |
| 62. C251 | #2760 | direct (N × H) |
| 63. #2760 | C251 | direct (N × H) |
| 64. #401 | C2×C251 | semidirect |
| 65. #393 | C2×C251 | semidirect |
| 67. #386 | C2×C251 | semidirect |
| 68. #381 | 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 |
| #393×C251 | 2 |
| #392×C251 | 2 |
| #386×C251 | 2 |
| #381×C251 | 2 |
| #2760 | 251 |