Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
330 conjugacy classes of subgroups; only the 256 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 256 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C24 |
| commutator | C22 | C23×C4×C53 |
| Frattini, cosocle | C2×C4 | C24×C53 |
| socle | C22×C53 | C23×C4 |
| prime radical | C23×C53 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#446 | C2 | semidirect |
| 3. C8×Q8×C53 | C2 | semidirect |
| 51. #2986 | C53 | direct (N × H) |
| 56. #446 | C2×C53 | semidirect |
| 57. C8×Q8 | C2×C53 | semidirect |
| 60. C53 | #2986 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#533 | 2 |
| C53×#446 | 2 |
| C8×Q8×C53 | 2 |
| C53×#433 | 2 |
| C53×#431 | 2 |
| #2986 | 53 |