Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
136 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 88 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 | C4×C8×C251 |
| Frattini, cosocle | C22×C8 | C22×C251 |
| socle | C22×C251 | C22⋊C8 |
| prime radical | C23×C251 | M16 |
| 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 |
|---|---|---|
| 8. M32×C251 | C4 | semidirect |
| 12. C2×C16×C251 | C4 | semidirect |
| 56. C251 | #1379 | direct (N × H) |
| 57. #1379 | C251 | direct (N × H) |
| 65. M32 | C4×C251 | semidirect |
| 69. C2×C16 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M32×C251 | 2 |
| C22×C16×C251 | 2 |
| #405×C251 | 2 |
| #1379 | 251 |