Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #120×C193 | C2 |
| hypercenter | #120 | D193 |
| hypocenter | C193 | #538 |
| LCS | G ▷ C22×C193 ▷ C193 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259892 | C2 | semidirect |
| 2. #20259892 | C2 | semidirect |
| 3. #20259886 | C2 | semidirect |
| 4. #20259885 | C2 | semidirect |
| 5. #20259881 | C2 | semidirect |
| 6. #20259880 | C2 | semidirect |
| 7. #20259876 | C2 | semidirect |
| 8. #20259871 | C2 | semidirect |
| 9. #20259871 | C2 | semidirect |
| 10. #20259870 | C2 | semidirect |
| 11. #20259868 | C2 | semidirect |
| 12. D4×Dic193 | C2 | semidirect |
| 13. #20259859 | C2 | semidirect |
| 14. C2×C22⋊Dic193 | C2 | semidirect |
| 15. #120×C193 | C2 | semidirect |
| 16. C2×Dic386 | C22 | semidirect |
| 20. C2×C22⋊D193 | C22 | semidirect |
| 23. C22×Dic193 | C22 | semidirect |
| 24. C2×C4×D193 | C22 | semidirect |
| 25. C4⋊Dic193 | C22 | semidirect |
| 26. C4⋊Dic193 | C22 | semidirect |
| 27. #15690534 | C22 | semidirect |
| 28. #15690534 | C22 | semidirect |
| 29. #15690534 | C22 | semidirect |
| 30. #15690534 | C22 | semidirect |
| 38. C22⋊Dic193 | C22 | semidirect |
| 39. C22⋊Dic193 | C22 | semidirect |
| 40. #15690532 | C22 | semidirect |
| 41. #15690532 | C22 | semidirect |
| 42. C4×Dic193 | C22 | semidirect |
| 43. C4×Dic193 | C22 | semidirect |
| 116. C193 | #538 | semidirect |
| 129. #120 | D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259892 | 2 |
| #20259886 | 2 |
| #20259885 | 2 |
| #20259881 | 2 |
| #20259880 | 2 |
| #20259876 | 2 |
| #20259871 | 2 |
| #20259870 | 2 |
| #20259868 | 2 |
| D4×Dic193 | 2 |
| #20259859 | 2 |
| C2×C22⋊Dic193 | 2 |
| #120×C193 | 2 |
| #538 | 193 |