Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
964 conjugacy classes of subgroups.
The subgroup lattice has 142 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C53 |
| prime radical | C23×C53 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q82×C53 | C2 | semidirect |
| 2. C53×#557 | C2 | semidirect |
| 3. C53×#557 | C2 | semidirect |
| 25. C2×Q8×C53 | Q8 | semidirect |
| 66. #3502 | C53 | direct (N × H) |
| 71. Q82 | C2×C53 | semidirect |
| 72. #557 | C2×C53 | semidirect |
| 73. #557 | C2×C53 | semidirect |
| 78. C53 | #3502 | direct (N × H) |
| 96. C2×Q8 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q82×C53 | 2 |
| C53×#557 | 2 |
| C53×#544 | 2 |
| C53×#533 | 2 |
| C53×#519 | 2 |
| C2×C4×Q8×C53 | 2 |
| #3502 | 53 |