Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
208 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 208 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, socle | C22×C251 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C251 |
| Frattini, cosocle | C22×C4 | C23×C251 |
| prime radical | C23×C251 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C2×C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C4⋊Q8×C251 | C4 | semidirect |
| 6. C4⋊Q8×C251 | C4 | semidirect |
| 11. C4×Q8×C251 | C4 | semidirect |
| 74. C251 | #1515 | direct (N × H) |
| 75. #1515 | C251 | direct (N × H) |
| 80. C4⋊Q8 | C4×C251 | semidirect |
| 81. C4⋊Q8 | C4×C251 | semidirect |
| 86. C4×Q8 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533×C251 | 2 |
| #432×C251 | 2 |
| #383×C251 | 2 |
| #328×C251 | 2 |
| #1515 | 251 |