Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
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×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | #123×C191 | C2 |
| hypercenter | #123 | D191 |
| hypocenter | C191 | #547 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690287 | C2 | semidirect |
| 2. C2×#15690286 | C2 | semidirect |
| 3. #20259321 | C2 | semidirect |
| 4. #20259307 | C2 | semidirect |
| 5. #20259306 | C2 | semidirect |
| 6. #20259304 | C2 | semidirect |
| 7. #20259300 | C2 | semidirect |
| 8. #20259299 | C2 | semidirect |
| 9. #20259295 | C2 | semidirect |
| 10. #20259294 | C2 | semidirect |
| 11. (C22⋊D191)⋊C4 | C2 | semidirect |
| 12. (C22⋊D191)⋊C4 | C2 | semidirect |
| 13. C2×Dic191⋊C4 | C2 | semidirect |
| 14. C4⋊C4×D191 | C2 | semidirect |
| 15. #123×C191 | C2 | semidirect |
| 61. C22⋊D191 | D4 | semidirect |
| 107. Dic191 | #41 | semidirect |
| 115. C191 | #547 | semidirect |
| 123. #123 | D191 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690287 | 2 |
| C2×#15690286 | 2 |
| #20259321 | 2 |
| #20259307 | 2 |
| #20259306 | 2 |
| #20259304 | 2 |
| #20259300 | 2 |
| #20259299 | 2 |
| #20259295 | 2 |
| #20259294 | 2 |
| (C22⋊D191)⋊C4 | 2 |
| C2×Dic191⋊C4 | 2 |
| C4⋊C4×D191 | 2 |
| #123×C191 | 2 |
| #547 | 191 |