Finite Group DatabasebetaFinite Group DatabasebetaOrder 22336.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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×D349 |
| commutator, socle, cosocle | C22×C349 | C24 |
| Fitting | #125×C349 | C2 |
| prime radical | #26013614 | C2 |
| hypercenter | #125 | D349 |
| hypocenter | C349 | #529 |
| LCS | G ▷ C22×C349 ▷ C349 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26013634 | C2 | semidirect |
| 2. #26013627 | C2 | semidirect |
| 3. #26013624 | C2 | semidirect |
| 4. #26013619 | C2 | semidirect |
| 6. #26013604 | C2 | semidirect |
| 7. #26013601 | C2 | semidirect |
| 8. #26013601 | C2 | semidirect |
| 9. #26013597 | C2 | semidirect |
| 10. C42×D349 | C2 | semidirect |
| 11. #125×C349 | C2 | semidirect |
| 74. C349 | #529 | semidirect |
| 77. #125 | D349 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26013634 | 2 |
| #26013627 | 2 |
| #26013624 | 2 |
| #26013619 | 2 |
| #26013614 | 2 |
| #26013604 | 2 |
| #26013601 | 2 |
| #26013597 | 2 |
| C42×D349 | 2 |
| #125×C349 | 2 |
| #529 | 349 |