Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
244 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 112 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 | C2×D4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C22×C53 | C2×C22⋊C4 |
| prime radical | C23×C53 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C4⋊Q8×C53 | C4 | semidirect |
| 7. #125×C53 | C4 | semidirect |
| 30. C42×C53 | Q8 | semidirect |
| 54. #1995 | C53 | direct (N × H) |
| 62. C53 | #1995 | direct (N × H) |
| 64. C4⋊Q8 | C4×C53 | semidirect |
| 65. #125 | C4×C53 | semidirect |
| 88. C42 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#533 | 2 |
| C53×#423 | 2 |
| C53×#376 | 2 |
| C53×#339 | 2 |
| #1995 | 53 |