Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 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 | #537 |
| 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. C2×#15690554 | C2 | semidirect |
| 2. #20259877 | C2 | semidirect |
| 3. #20259876 | C2 | semidirect |
| 4. #20259873 | C2 | semidirect |
| 5. #20259872 | C2 | semidirect |
| 6. #20259871 | C2 | semidirect |
| 7. #20259871 | C2 | semidirect |
| 8. C2×#15690534 | C2 | semidirect |
| 9. #121×C193 | C2 | semidirect |
| 40. C2×Dic193 | D4 | semidirect |
| 75. C193 | #537 | semidirect |
| 84. #121 | D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690554 | 2 |
| #20259877 | 2 |
| #20259876 | 2 |
| #20259873 | 2 |
| #20259872 | 2 |
| #20259871 | 2 |
| C2×#15690534 | 2 |
| #121×C193 | 2 |
| #537 | 193 |