Finite Group DatabasebetaFinite Group DatabasebetaOrder 15424.
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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×D241 |
| commutator, socle, cosocle | C22×C241 | C24 |
| Fitting | #123×C241 | C2 |
| hypercenter | #123 | D241 |
| hypocenter | C241 | #547 |
| LCS | G ▷ C22×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16838770 | C2 | semidirect |
| 2. C2×#16838769 | C2 | semidirect |
| 3. #21440641 | C2 | semidirect |
| 4. #21440627 | C2 | semidirect |
| 5. #21440626 | C2 | semidirect |
| 6. #21440624 | C2 | semidirect |
| 7. #21440620 | C2 | semidirect |
| 8. #21440619 | C2 | semidirect |
| 9. #21440615 | C2 | semidirect |
| 10. #21440614 | C2 | semidirect |
| 11. #21440602 | C2 | semidirect |
| 12. #21440602 | C2 | semidirect |
| 13. C2×#16838749 | C2 | semidirect |
| 14. C4⋊C4×D241 | C2 | semidirect |
| 15. #123×C241 | C2 | semidirect |
| 61. C22⋊D241 | D4 | semidirect |
| 107. Dic241 | #41 | semidirect |
| 115. C241 | #547 | semidirect |
| 123. #123 | D241 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16838770 | 2 |
| C2×#16838769 | 2 |
| #21440641 | 2 |
| #21440627 | 2 |
| #21440626 | 2 |
| #21440624 | 2 |
| #21440620 | 2 |
| #21440619 | 2 |
| #21440615 | 2 |
| #21440614 | 2 |
| #21440602 | 2 |
| C2×#16838749 | 2 |
| C4⋊C4×D241 | 2 |
| #123×C241 | 2 |
| #547 | 241 |