Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
322 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D193 |
| commutator, socle, cosocle | C23×C193 | C23 |
| Fitting | C2×C22⋊C4×C193 | C2 |
| hypercenter | C2×C22⋊C4 | D193 |
| hypocenter | C193 | #392 |
| LCS | G ▷ C23×C193 ▷ C193 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D772 | C2 | semidirect |
| 2. C22×C22⋊D193 | C2 | semidirect |
| 3. C2×#15690532 | C2 | semidirect |
| 4. #20259806 | C2 | semidirect |
| 5. C2×C22⋊C4×C193 | C2 | semidirect |
| 24. C22×D193 | D4 | semidirect |
| 29. C2×Dic193 | D4 | semidirect |
| 34. C23×C193 | D4 | semidirect |
| 39. C2×C4×C193 | D4 | semidirect |
| 67. C193 | #392 | semidirect |
| 72. C2×C22⋊C4 | D193 | semidirect |
| 97. C23 | D772 | semidirect |
| 107. C2×C4 | C22⋊D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D772 | 2 |
| C22×C22⋊D193 | 2 |
| C2×#15690532 | 2 |
| #20259806 | 2 |
| C2×C22⋊C4×C193 | 2 |
| #392 | 193 |