Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 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 |
| prime radical | C2×C22⋊D193 | C22 |
| hypercenter | #126 | D193 |
| hypocenter | C193 | #563 |
| 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. #20259899 | C2 | semidirect |
| 2. #20259898 | C2 | semidirect |
| 3. #20259894 | C2 | semidirect |
| 4. #20259893 | C2 | semidirect |
| 5. #20259892 | C2 | semidirect |
| 6. #20259890 | C2 | semidirect |
| 7. #20259889 | C2 | semidirect |
| 11. #20259879 | C2 | semidirect |
| 12. #20259877 | C2 | semidirect |
| 13. #20259876 | C2 | semidirect |
| 14. #20259876 | C2 | semidirect |
| 15. #126×C193 | C2 | semidirect |
| 97. C193 | #563 | semidirect |
| 104. #126 | D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259899 | 2 |
| #20259898 | 2 |
| #20259894 | 2 |
| #20259893 | 2 |
| #20259892 | 2 |
| #20259890 | 2 |
| #20259889 | 2 |
| #20259885 | 2 |
| #20259883 | 2 |
| #20259881 | 2 |
| #20259879 | 2 |
| #20259877 | 2 |
| #20259876 | 2 |
| #126×C193 | 2 |
| #563 | 193 |