Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #555 |
| 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. #20259898 | C2 | semidirect |
| 2. #20259887 | C2 | semidirect |
| 3. #20259885 | C2 | semidirect |
| 4. #20259874 | C2 | semidirect |
| 5. #20259873 | C2 | semidirect |
| 6. #20259872 | C2 | semidirect |
| 7. #20259861 | C2 | semidirect |
| 8. C4×C22⋊D193 | C2 | semidirect |
| 9. #20259857 | C2 | semidirect |
| 10. #20259856 | C2 | semidirect |
| 11. #117×C193 | C2 | semidirect |
| 78. C193 | #555 | semidirect |
| 90. #117 | D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259898 | 2 |
| #20259887 | 2 |
| #20259885 | 2 |
| #20259874 | 2 |
| #20259873 | 2 |
| #20259872 | 2 |
| #20259861 | 2 |
| C4×C22⋊D193 | 2 |
| #20259857 | 2 |
| #20259856 | 2 |
| #117×C193 | 2 |
| #555 | 193 |