Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
232 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C251 | D4 |
| commutator | C4 | C2×C42×C251 |
| Frattini, cosocle | C22×C4 | C23×C251 |
| socle | C22×C251 | C4×D4 |
| prime radical | C23×C251 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C251 ◁ C22×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. #108×C251 | C4 | semidirect |
| 12. #107×C251 | C4 | semidirect |
| 13. #106×C251 | C4 | semidirect |
| 16. C4×C8×C251 | C4 | semidirect |
| 64. C251 | #1772 | direct (N × H) |
| 65. #1772 | C251 | direct (N × H) |
| 76. #108 | C4×C251 | semidirect |
| 77. #107 | C4×C251 | semidirect |
| 78. #106 | C4×C251 | semidirect |
| 81. C4×C8 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#108×C251 | 2 |
| #427×C251 | 2 |
| #405×C251 | 2 |
| C2×C4×C8×C251 | 2 |
| #339×C251 | 2 |
| #1772 | 251 |