Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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 | C4×D4×C193 | C2 |
| hypercenter | C4×D4 | D193 |
| hypocenter | C193 | #555 |
| LCS | G ▷ C22×C193 ▷ C193 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259899 | C2 | semidirect |
| 2. #20259885 | C2 | semidirect |
| 3. #20259873 | C2 | semidirect |
| 4. #20259870 | C2 | semidirect |
| 5. C4×D772 | C2 | semidirect |
| 6. #20259858 | C2 | semidirect |
| 7. #20259857 | C2 | semidirect |
| 8. C4×D4×C193 | C2 | semidirect |
| 57. C193 | #555 | semidirect |
| 64. C4×D4 | D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259899 | 2 |
| #20259885 | 2 |
| #20259873 | 2 |
| #20259870 | 2 |
| C4×D772 | 2 |
| #20259858 | 2 |
| #20259857 | 2 |
| C4×D4×C193 | 2 |
| #555 | 193 |