Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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 | C22⋊Q8×C239 | C2 |
| hypercenter | C22⋊Q8 | D239 |
| hypocenter | C239 | #538 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21439955 | C2 | semidirect |
| 2. #21439949 | C2 | semidirect |
| 3. #21439945 | C2 | semidirect |
| 4. #21439935 | C2 | semidirect |
| 5. #21439934 | C2 | semidirect |
| 6. #21439930 | C2 | semidirect |
| 7. (C22⋊D239)⋊C4 | C2 | semidirect |
| 8. C2×(C2×D239)⋊C4 | C2 | semidirect |
| 9. C22⋊Q8×C239 | C2 | semidirect |
| 13. C2×D956 | C22 | semidirect |
| 16. C2×C4×D239 | C22 | semidirect |
| 17. C4⋊Dic239 | C22 | semidirect |
| 18. Dic239⋊C4 | C22 | semidirect |
| 23. (C2×D239)⋊C4 | C22 | semidirect |
| 24. (C2×D239)⋊C4 | C22 | semidirect |
| 25. C4×Dic239 | C22 | semidirect |
| 26. C2×Q8×C239 | C22 | semidirect |
| 28. C4⋊C4×C239 | C22 | semidirect |
| 29. C4⋊C4×C239 | C22 | semidirect |
| 74. C239 | #538 | semidirect |
| 79. C22⋊Q8 | D239 | semidirect |
| 86. C2×Q8 | C2×D239 | semidirect |
| 96. C4⋊C4 | C2×D239 | semidirect |
| 97. C4⋊C4 | C2×D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439955 | 2 |
| #21439949 | 2 |
| #21439945 | 2 |
| #21439935 | 2 |
| #21439934 | 2 |
| #21439930 | 2 |
| (C22⋊D239)⋊C4 | 2 |
| C2×(C2×D239)⋊C4 | 2 |
| C22⋊Q8×C239 | 2 |
| #538 | 239 |