Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D239 |
| commutator, socle, cosocle | C22×C239 | C24 |
| Fitting | C4×Q8×C239 | C2 |
| prime radical | C2×D956 | C22 |
| hypercenter | C4×Q8 | D239 |
| hypocenter | C239 | #554 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21439960 | C2 | semidirect |
| 2. #21439952 | C2 | semidirect |
| 3. #21439938 | C2 | semidirect |
| 4. #21439936 | C2 | semidirect |
| 7. C4⋊C4×D239 | C2 | semidirect |
| 8. C4×Q8×C239 | C2 | semidirect |
| 27. D956 | Q8 | semidirect |
| 53. C239 | #554 | semidirect |
| 59. C4×Q8 | D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439960 | 2 |
| #21439952 | 2 |
| #21439938 | 2 |
| #21439936 | 2 |
| C4×D956 | 2 |
| C4⋊C4×D239 | 2 |
| C4×Q8×C239 | 2 |
| #554 | 239 |