Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
350 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C251 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C251 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C251 ◁ #41×C251 ◁ #131×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #578×C251 | C2 | semidirect |
| 2. #510×C251 | C2 | semidirect |
| 3. #508×C251 | C2 | semidirect |
| 4. C2×Q32×C251 | C2 | semidirect |
| 5. #504×C251 | C2 | semidirect |
| 49. C251 | #3417 | direct (N × H) |
| 50. #3417 | C251 | direct (N × H) |
| 51. #578 | C2×C251 | semidirect |
| 52. #510 | C2×C251 | semidirect |
| 53. #508 | C2×C251 | semidirect |
| 54. C2×Q32 | C2×C251 | semidirect |
| 55. #504 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #578×C251 | 2 |
| #510×C251 | 2 |
| #508×C251 | 2 |
| C2×Q32×C251 | 2 |
| #504×C251 | 2 |
| #3417 | 251 |