Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #16810458 |
| commutator, cosocle | C32×C53 | C23 |
| Frattini | C32 | C22×D53 |
| Fitting | C2×C53×D32 | C2 |
| socle | C22×C53 | D32 |
| hypercenter | C2×D32 | D53 |
| hypocenter | C53 | C2×D64 |
| LCS | G ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×D32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D1696 | C2 | semidirect |
| 2. #20272927 | C2 | direct (N × H) |
| 3. C2×#16810342 | C2 | semidirect |
| 4. C2×C53×D32 | C2 | semidirect |
| 7. D1696 | C22 | semidirect |
| 8. #16810342 | C22 | semidirect |
| 11. C53×D32 | C22 | semidirect |
| 49. C2×D32 | D53 | semidirect |
| 53. C2×C53 | D64 | semidirect |
| 61. D32 | C2×D53 | semidirect |
| 64. C53 | C2×D64 | semidirect |
| 101. C2 | #20272927 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D1696 | 2 |
| #20272927 | 2 |
| C2×#16810342 | 2 |
| C2×C53×D32 | 2 |
| C2×D64 | 53 |