Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C2×C4⋊C4×C191 | C2 |
| hypercenter | C2×C4⋊C4 | D191 |
| hypocenter | C191 | #547 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690287 | C2 | semidirect |
| 2. #20259321 | C2 | semidirect |
| 3. #20259307 | C2 | semidirect |
| 4. #20259305 | C2 | semidirect |
| 5. #20259300 | C2 | semidirect |
| 6. #20259297 | C2 | semidirect |
| 7. #20259296 | C2 | semidirect |
| 8. #20259284 | C2 | semidirect |
| 9. C4×C22⋊D191 | C2 | semidirect |
| 10. C4⋊C4×D191 | C2 | semidirect |
| 11. C2×C4⋊C4×C191 | C2 | semidirect |
| 42. C22⋊D191 | D4 | semidirect |
| 70. C4×C191 | #41 | semidirect |
| 81. C191 | #547 | semidirect |
| 93. C2×C4⋊C4 | D191 | semidirect |
| 156. C4 | #15690287 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690287 | 2 |
| #20259321 | 2 |
| #20259307 | 2 |
| #20259305 | 2 |
| #20259300 | 2 |
| #20259297 | 2 |
| #20259296 | 2 |
| #20259284 | 2 |
| C4×C22⋊D191 | 2 |
| C4⋊C4×D191 | 2 |
| C2×C4⋊C4×C191 | 2 |
| #547 | 191 |