Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
310 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 111 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 | #127×C191 | C2 |
| hypercenter | #127 | D191 |
| hypocenter | C191 | #532 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690286 | C2 | semidirect |
| 2. #20259322 | C2 | semidirect |
| 3. #20259309 | C2 | semidirect |
| 4. #20259293 | C2 | semidirect |
| 5. D4×Dic191 | C2 | semidirect |
| 6. C42×D191 | C2 | semidirect |
| 7. #127×C191 | C2 | semidirect |
| 29. C4×D191 | D4 | semidirect |
| 41. C4×C191 | #41 | semidirect |
| 49. C191 | #532 | semidirect |
| 53. #127 | D191 | semidirect |
| 94. C4 | #15690286 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690286 | 2 |
| #20259322 | 2 |
| #20259309 | 2 |
| #20259293 | 2 |
| D4×Dic191 | 2 |
| C42×D191 | 2 |
| #127×C191 | 2 |
| #532 | 191 |