Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
134 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D53 |
| commutator, cosocle | C32×C53 | C23 |
| Frattini | C32 | C22×D53 |
| Fitting | C53×SD128 | C2 |
| socle | C2×C53 | C2×D32 |
| hypercenter | SD128 | D53 |
| hypocenter | C53 | #13742 |
| LCS | G ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ SD128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20277025 | C2 | semidirect |
| 2. D32×D53 | C2 | semidirect |
| 3. #20272929 | C2 | semidirect |
| 4. D3392 | C2 | semidirect |
| 5. #20272927 | C2 | semidirect |
| 6. #20272925 | C2 | semidirect |
| 7. C53×SD128 | C2 | semidirect |
| 12. D1696 | C22 | semidirect |
| 15. #16810342 | C22 | semidirect |
| 16. C53×Q64 | C22 | semidirect |
| 18. C53×C64 | C22 | semidirect |
| 64. SD128 | D53 | semidirect |
| 75. Q64 | C2×D53 | semidirect |
| 83. C64 | C2×D53 | semidirect |
| 84. C53 | #13742 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20277025 | 2 |
| D32×D53 | 2 |
| #20272929 | 2 |
| D3392 | 2 |
| #20272927 | 2 |
| #20272925 | 2 |
| C53×SD128 | 2 |
| #13742 | 53 |