Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
142 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| prime radical | C2×D8×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#500 | C2 | semidirect |
| 3. C53×#367 | C2 | semidirect |
| 4. C53×C4⋊C16 | C2 | semidirect |
| 19. D8×C53 | Q8 | semidirect |
| 51. #2226 | C53 | direct (N × H) |
| 56. #500 | C2×C53 | semidirect |
| 58. #367 | C2×C53 | semidirect |
| 59. C4⋊C16 | C2×C53 | semidirect |
| 61. C53 | #2226 | direct (N × H) |
| 75. D8 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#500 | 2 |
| C4×D8×C53 | 2 |
| C53×#367 | 2 |
| C53×C4⋊C16 | 2 |
| C53×#357 | 2 |
| #2226 | 53 |