Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 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×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting, prime radical | #120×C191 | C2 |
| hypercenter | #120 | D191 |
| hypocenter | C191 | #551 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259315 | C2 | semidirect |
| 2. #20259309 | C2 | semidirect |
| 3. #20259303 | C2 | semidirect |
| 4. #20259303 | C2 | semidirect |
| 5. #20259299 | C2 | semidirect |
| 6. D4×Dic191 | C2 | semidirect |
| 7. #20259278 | C2 | semidirect |
| 8. C2×C22⋊Dic191 | C2 | semidirect |
| 10. C2×Dic382 | C22 | semidirect |
| 13. C22×Dic191 | C22 | semidirect |
| 14. C4⋊Dic191 | C22 | semidirect |
| 15. Dic191⋊C4 | C22 | semidirect |
| 16. Dic191⋊C4 | C22 | semidirect |
| 17. Dic191⋊C4 | C22 | semidirect |
| 22. C22⋊Dic191 | C22 | semidirect |
| 23. C22⋊Dic191 | C22 | semidirect |
| 24. C4×Dic191 | C22 | semidirect |
| 25. C4×Dic191 | C22 | semidirect |
| 68. C191 | #551 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259315 | 2 |
| #20259309 | 2 |
| #20259303 | 2 |
| #20259299 | 2 |
| D4×Dic191 | 2 |
| #20259278 | 2 |
| C2×C22⋊Dic191 | 2 |
| #120×C191 | 2 |
| #551 | 191 |