Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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 | #563 |
| 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. #21439958 | C2 | semidirect |
| 2. #21439955 | C2 | semidirect |
| 3. #21439954 | C2 | semidirect |
| 4. #21439950 | C2 | semidirect |
| 7. #21439938 | C2 | semidirect |
| 8. #21439936 | C2 | semidirect |
| 9. C4⋊Q8×C239 | C2 | semidirect |
| 59. C239 | #563 | semidirect |
| 62. C4⋊Q8 | D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439958 | 2 |
| #21439955 | 2 |
| #21439954 | 2 |
| #21439950 | 2 |
| #21439949 | 2 |
| #21439945 | 2 |
| #21439938 | 2 |
| #21439936 | 2 |
| C4⋊Q8×C239 | 2 |
| #563 | 239 |