Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 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 | #121×C193 | C2 |
| hypercenter | #121 | D193 |
| hypocenter | C193 | #562 |
| LCS | G ▷ C22×C193 ▷ C193 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259889 | C2 | semidirect |
| 2. #20259886 | C2 | semidirect |
| 3. #20259885 | C2 | semidirect |
| 4. #20259882 | C2 | semidirect |
| 5. #20259881 | C2 | semidirect |
| 6. #20259880 | C2 | semidirect |
| 7. #20259880 | C2 | semidirect |
| 8. #20259878 | C2 | semidirect |
| 9. #20259877 | C2 | semidirect |
| 10. #20259876 | C2 | semidirect |
| 11. #20259876 | C2 | semidirect |
| 12. #20259871 | C2 | semidirect |
| 13. #20259871 | C2 | semidirect |
| 14. #20259870 | C2 | semidirect |
| 15. #121×C193 | C2 | semidirect |
| 109. C193 | #562 | semidirect |
| 125. #121 | D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259889 | 2 |
| #20259886 | 2 |
| #20259885 | 2 |
| #20259882 | 2 |
| #20259881 | 2 |
| #20259880 | 2 |
| #20259878 | 2 |
| #20259877 | 2 |
| #20259876 | 2 |
| #20259871 | 2 |
| #20259870 | 2 |
| #121×C193 | 2 |
| #562 | 193 |