Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
134 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#16810458 |
| commutator, cosocle | C32×C53 | C23 |
| Frattini | C32 | C22×D53 |
| Fitting | C53×#2262 | C2 |
| socle | C2×C53 | C2×D32 |
| hypercenter | #2262 | D53 |
| hypocenter | C53 | #13742 |
| LCS | G ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ #2262 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D1696 | C2 | semidirect |
| 2. #20272929 | C2 | semidirect |
| 3. #20272927 | C2 | semidirect |
| 4. #20272926 | C2 | semidirect |
| 5. C53×#2262 | C2 | semidirect |
| 9. #16810342 | C22 | semidirect |
| 11. C53×Q64 | C22 | semidirect |
| 13. C53×D32 | C22 | semidirect |
| 53. #2262 | D53 | semidirect |
| 64. Q64 | C2×D53 | semidirect |
| 68. D32 | C2×D53 | semidirect |
| 71. C53 | #13742 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D1696 | 2 |
| #20272929 | 2 |
| #20272927 | 2 |
| #20272926 | 2 |
| C53×#2262 | 2 |
| #13742 | 53 |