Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#500 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #500 | D53 |
| hypocenter | C53 | #3334 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #500 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811276 | C2 | semidirect |
| 2. #16810938 | C2 | semidirect |
| 3. #16810900 | C2 | semidirect |
| 4. #16810860 | C2 | semidirect |
| 5. C4×D424 | C2 | semidirect |
| 6. C8×D212 | C2 | semidirect |
| 7. #107×D53 | C2 | semidirect |
| 8. C53×#500 | C2 | semidirect |
| 62. D212 | D8 | semidirect |
| 68. C8×C53 | #41 | semidirect |
| 91. #500 | D53 | semidirect |
| 98. C53 | #3334 | semidirect |
| 159. C8 | #11719949 | semidirect |