Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C251 | D4 |
| commutator | C4 | C2×C42×C251 |
| Frattini, cosocle | C22×C4 | C23×C251 |
| socle | C23×C251 | C2×D4 |
| prime radical | C24×C251 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C251 ◁ C23×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #340×C251 | C2 | semidirect |
| 6. C22×C8×C251 | C4 | semidirect |
| 9. #117×C251 | C4 | semidirect |
| 59. C251 | #1735 | direct (N × H) |
| 60. #1735 | C251 | direct (N × H) |
| 63. #340 | C2×C251 | semidirect |
| 66. C22×C8 | C4×C251 | semidirect |
| 69. #117 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C8×C251 | 2 |
| C2×#117×C251 | 2 |
| #340×C251 | 2 |
| #1735 | 251 |