Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
238 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 99 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 | C4×D4×C191 | C2 |
| hypercenter | C4×D4 | D191 |
| hypocenter | C191 | #553 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259317 | C2 | semidirect |
| 2. #20259314 | C2 | semidirect |
| 3. #20259304 | C2 | semidirect |
| 4. #20259296 | C2 | semidirect |
| 5. #20259296 | C2 | semidirect |
| 6. #20259294 | C2 | semidirect |
| 7. C4×D764 | C2 | semidirect |
| 8. (C22⋊D191)⋊C4 | C2 | semidirect |
| 9. #20259281 | C2 | semidirect |
| 10. C2×C4⋊Dic191 | C2 | semidirect |
| 11. C4×D4×C191 | C2 | semidirect |
| 78. C191 | #553 | semidirect |
| 88. C4×D4 | D191 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259317 | 2 |
| #20259314 | 2 |
| #20259304 | 2 |
| #20259296 | 2 |
| #20259294 | 2 |
| C4×D764 | 2 |
| (C22⋊D191)⋊C4 | 2 |
| #20259281 | 2 |
| C2×C4⋊Dic191 | 2 |
| C4×D4×C191 | 2 |
| #553 | 191 |