Finite Group DatabasebetaFinite Group DatabasebetaOrder 17536.
840 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D137 |
| commutator, socle, cosocle | C23×C137 | C24 |
| Fitting | #380×C137 | C2 |
| prime radical | C22×C22⋊D137 | C22 |
| hypercenter | #380 | D137 |
| hypocenter | C137 | #2409 |
| LCS | G ▷ C23×C137 ▷ C137 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic274 | C2 | semidirect |
| 2. C2×#17984068 | C2 | semidirect |
| 4. #22577324 | C2 | semidirect |
| 5. #22577316 | C2 | semidirect |
| 6. #22577308 | C2 | semidirect |
| 8. #22577278 | C2 | semidirect |
| 9. #380×C137 | C2 | semidirect |
| 37. C2×Dic274 | D4 | semidirect |
| 40. C2×C22⋊D137 | Q8 | semidirect |
| 67. C22×D137 | C2×Q8 | semidirect |
| 81. C2×Dic137 | C2×Q8 | semidirect |
| 101. C22×C137 | C22⋊Q8 | semidirect |
| 126. C137 | #2409 | semidirect |
| 136. #380 | D137 | semidirect |
| 233. C22 | #17984068 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic274 | 2 |
| C2×#17984068 | 2 |
| C2×C4×C22⋊D137 | 2 |
| #22577324 | 2 |
| #22577316 | 2 |
| #22577308 | 2 |
| #22577280 | 2 |
| #22577278 | 2 |
| #380×C137 | 2 |
| #2409 | 137 |