Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
234 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #126×C191 | C2 |
| prime radical | #20259295 | C2 |
| hypercenter | #126 | D191 |
| hypocenter | C191 | #529 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259316 | C2 | semidirect |
| 2. #20259312 | C2 | semidirect |
| 3. #20259308 | C2 | semidirect |
| 4. #20259307 | C2 | semidirect |
| 5. #20259304 | C2 | semidirect |
| 6. #20259300 | C2 | semidirect |
| 8. #20259287 | C2 | semidirect |
| 9. #20259284 | C2 | semidirect |
| 10. (C22⋊D191)⋊C4 | C2 | semidirect |
| 11. (C22⋊D191)⋊C4 | C2 | semidirect |
| 12. #20259280 | C2 | semidirect |
| 13. #20259278 | C2 | semidirect |
| 14. C42×D191 | C2 | semidirect |
| 15. #126×C191 | C2 | semidirect |
| 103. C191 | #529 | semidirect |
| 106. #126 | D191 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259316 | 2 |
| #20259312 | 2 |
| #20259308 | 2 |
| #20259307 | 2 |
| #20259304 | 2 |
| #20259300 | 2 |
| #20259295 | 2 |
| #20259287 | 2 |
| #20259284 | 2 |
| (C22⋊D191)⋊C4 | 2 |
| #20259280 | 2 |
| #20259278 | 2 |
| C42×D191 | 2 |
| #126×C191 | 2 |
| #529 | 191 |