Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 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 | #125×C191 | C2 |
| prime radical | C22×D191 | C23 |
| hypercenter | #125 | D191 |
| hypocenter | C191 | #541 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259321 | C2 | semidirect |
| 6. #20259313 | C2 | semidirect |
| 7. #20259312 | C2 | semidirect |
| 8. #20259312 | C2 | semidirect |
| 9. #20259312 | C2 | semidirect |
| 12. C4×Dic382 | C2 | semidirect |
| 13. #20259287 | C2 | semidirect |
| 15. #125×C191 | C2 | semidirect |
| 91. C191 | #541 | semidirect |
| 102. #125 | D191 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259321 | 2 |
| #20259317 | 2 |
| #20259316 | 2 |
| #20259313 | 2 |
| #20259312 | 2 |
| #20259300 | 2 |
| C4×Dic382 | 2 |
| #20259287 | 2 |
| #20259280 | 2 |
| #125×C191 | 2 |
| #541 | 191 |