Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 334 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×D193 |
| commutator, socle, cosocle | C22×C193 | C24 |
| Fitting | #117×C193 | C2 |
| hypercenter | #117 | D193 |
| hypocenter | C193 | #535 |
| LCS | G ▷ C22×C193 ▷ C193 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690555 | C2 | semidirect |
| 2. C22×D772 | C2 | semidirect |
| 3. #20259896 | C2 | semidirect |
| 4. #20259887 | C2 | semidirect |
| 5. #20259874 | C2 | semidirect |
| 6. #20259869 | C2 | semidirect |
| 7. #117×C193 | C2 | semidirect |
| 10. C2×Dic386 | C22 | semidirect |
| 14. C2×D772 | C22 | semidirect |
| 15. C2×D772 | C22 | semidirect |
| 17. C2×C4×D193 | C22 | semidirect |
| 18. #15690532 | C22 | semidirect |
| 20. C4⋊C4×C193 | C22 | semidirect |
| 22. C42×C193 | C22 | semidirect |
| 36. C2×C4×C193 | D4 | semidirect |
| 57. C193 | #535 | semidirect |
| 65. #117 | D193 | semidirect |
| 77. C4⋊C4 | C2×D193 | semidirect |
| 80. C42 | C2×D193 | semidirect |
| 94. C2×C4 | D772 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690555 | 2 |
| C22×D772 | 2 |
| #20259896 | 2 |
| #20259887 | 2 |
| #20259874 | 2 |
| #20259869 | 2 |
| #117×C193 | 2 |
| #535 | 193 |