Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
232 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 152 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 | D4 |
| commutator | C4 | C2×C42×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C22×C53 | C4×D4 |
| prime radical | C23×C53 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C22×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. #108×C53 | C4 | semidirect |
| 12. #107×C53 | C4 | semidirect |
| 13. #106×C53 | C4 | semidirect |
| 16. C4×C8×C53 | C4 | semidirect |
| 60. #1772 | C53 | direct (N × H) |
| 70. C53 | #1772 | direct (N × H) |
| 76. #108 | C4×C53 | semidirect |
| 77. #107 | C4×C53 | semidirect |
| 78. #106 | C4×C53 | semidirect |
| 81. C4×C8 | C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#108×C53 | 2 |
| C53×#427 | 2 |
| C53×#405 | 2 |
| C2×C4×C8×C53 | 2 |
| C53×#339 | 2 |
| #1772 | 53 |