Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 144 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 | C4×C53 | #106 |
| commutator | C8 | C22×C4×C53 |
| Frattini, cosocle | C2×C8 | C23×C53 |
| socle | C2×C53 | C2×#106 |
| prime radical | C23×C53 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#365 | C2 | semidirect |
| 8. M32×C53 | C4 | semidirect |
| 10. C2×C16×C53 | C4 | semidirect |
| 51. #2155 | C53 | direct (N × H) |
| 59. #365 | C2×C53 | semidirect |
| 60. C53 | #2155 | direct (N × H) |
| 64. M32 | C4×C53 | semidirect |
| 66. C2×C16 | C4×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 |
| C2×#108×C53 | 2 |
| C53×#427 | 2 |
| C53×#365 | 2 |
| #2155 | 53 |