Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
150 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 150 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, prime radical | C22×C251 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C251 |
| 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 |
|---|---|---|
| 56. C8×C251 | Q16 | semidirect |
| 70. C251 | #1672 | direct (N × H) |
| 71. #1672 | C251 | direct (N × H) |
| 127. C8 | Q16×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #498×C251 | 2 |
| #477×C251 | 2 |
| #462×C251 | 2 |
| C8×Q8×C251 | 2 |
| #334×C251 | 2 |
| #333×C251 | 2 |
| #332×C251 | 2 |
| #1672 | 251 |