Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
162 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C23 |
| commutator | C22 | C22×C8×C53 |
| Frattini, cosocle | C2×C8 | C23×C53 |
| socle | C22×C53 | C22×C8 |
| prime radical | C2×D4×C53 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×M32×C53 | C2 | semidirect |
| 3. C53×C4⋊C16 | C2 | semidirect |
| 4. C53×C22⋊C16 | C2 | semidirect |
| 5. C4×C16×C53 | C2 | semidirect |
| 27. C16×C53 | D4 | semidirect |
| 46. C4×C53 | M32 | semidirect |
| 49. #2169 | C53 | direct (N × H) |
| 54. C2×M32 | C2×C53 | semidirect |
| 56. C4⋊C16 | C2×C53 | semidirect |
| 57. C22⋊C16 | C2×C53 | semidirect |
| 58. C4×C16 | C2×C53 | semidirect |
| 59. C53 | #2169 | direct (N × H) |
| 81. C16 | D4×C53 | semidirect |
| 100. C4 | M32×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M32×C53 | 2 |
| C8×D4×C53 | 2 |
| C53×C4⋊C16 | 2 |
| C53×C22⋊C16 | 2 |
| C4×C16×C53 | 2 |
| #2169 | 53 |