Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
172 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C251 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C251 |
| prime radical | C2×D4×C251 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #494×C251 | C2 | semidirect |
| 4. #335×C251 | C2 | semidirect |
| 5. #332×C251 | C2 | semidirect |
| 36. D4×C251 | Q16 | semidirect |
| 63. C251 | #1669 | direct (N × H) |
| 64. #1669 | C251 | direct (N × H) |
| 65. #494 | C2×C251 | semidirect |
| 68. #335 | C2×C251 | semidirect |
| 69. #332 | C2×C251 | semidirect |
| 100. D4 | Q16×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #494×C251 | 2 |
| #474×C251 | 2 |
| C8×D4×C251 | 2 |
| #335×C251 | 2 |
| #332×C251 | 2 |
| #1669 | 251 |