Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
110 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C251 | D8 |
| commutator | C8 | C2×C8×C251 |
| Frattini, cosocle | C4×C8 | C22×C251 |
| socle | C22×C251 | #102 |
| prime radical | C2×D8×C251 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C251 ◁ C42×C251 ◁ C4×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C16×C251 | C2 | semidirect |
| 3. #335×C251 | C2 | semidirect |
| 14. D8×C251 | C8 | semidirect |
| 49. C251 | #1331 | direct (N × H) |
| 50. #1331 | C251 | direct (N × H) |
| 52. C4×C16 | C2×C251 | semidirect |
| 53. #335 | C2×C251 | semidirect |
| 64. D8 | C8×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D8×C251 | 2 |
| C4×C16×C251 | 2 |
| #335×C251 | 2 |
| #1331 | 251 |