Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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×D193 |
| commutator, socle, cosocle | C22×C193 | C24 |
| Fitting | #126×C193 | C2 |
| hypercenter | #126 | D193 |
| hypocenter | C193 | #560 |
| LCS | G ▷ C22×C193 ▷ C193 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259896 | C2 | semidirect |
| 2. #20259893 | C2 | semidirect |
| 3. #20259884 | C2 | semidirect |
| 4. #20259874 | C2 | semidirect |
| 5. #20259872 | C2 | semidirect |
| 6. #20259869 | C2 | semidirect |
| 7. #126×C193 | C2 | semidirect |
| 11. C2×D772 | C22 | semidirect |
| 13. C2×C4×D193 | C22 | semidirect |
| 14. #15690534 | C22 | semidirect |
| 17. #15690532 | C22 | semidirect |
| 18. C4⋊C4×C193 | C22 | semidirect |
| 20. C42×C193 | C22 | semidirect |
| 50. C193 | #560 | semidirect |
| 54. #126 | D193 | semidirect |
| 66. C4⋊C4 | C2×D193 | semidirect |
| 71. C42 | C2×D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259896 | 2 |
| #20259893 | 2 |
| #20259884 | 2 |
| #20259874 | 2 |
| #20259872 | 2 |
| #20259869 | 2 |
| #126×C193 | 2 |
| #560 | 193 |