Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
310 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 109 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 | #532 |
| 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. #20259898 | C2 | semidirect |
| 3. #20259895 | C2 | semidirect |
| 4. #20259885 | C2 | semidirect |
| 5. #20259873 | C2 | semidirect |
| 6. #20259871 | C2 | semidirect |
| 7. D4×Dic193 | C2 | semidirect |
| 8. #20259861 | C2 | semidirect |
| 9. #20259859 | C2 | semidirect |
| 10. C2×C4×Dic193 | C2 | semidirect |
| 11. #121×C193 | C2 | semidirect |
| 44. C2×Dic193 | D4 | semidirect |
| 71. C4×C193 | #41 | semidirect |
| 76. Dic193 | #41 | semidirect |
| 83. C193 | #532 | semidirect |
| 91. #121 | D193 | semidirect |
| 160. C4 | #15690554 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690554 | 2 |
| #20259898 | 2 |
| #20259895 | 2 |
| #20259885 | 2 |
| #20259873 | 2 |
| #20259871 | 2 |
| D4×Dic193 | 2 |
| #20259861 | 2 |
| #20259859 | 2 |
| C2×C4×Dic193 | 2 |
| #121×C193 | 2 |
| #532 | 193 |