Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
374 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 374 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×#498 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #498 | D53 |
| hypocenter | C53 | #3384 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #498 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811334 | C2 | semidirect |
| 2. #16811288 | C2 | semidirect |
| 3. #16811030 | C2 | semidirect |
| 4. #16810922 | C2 | semidirect |
| 5. #16810912 | C2 | semidirect |
| 6. #16810910 | C2 | semidirect |
| 7. #16810900 | C2 | semidirect |
| 8. #16810753 | C2 | semidirect |
| 11. C53×#498 | C2 | semidirect |
| 72. D424 | Q8 | semidirect |
| 89. D212 | C2×Q8 | semidirect |
| 97. C8×C53 | C2×Q8 | semidirect |
| 125. #498 | D53 | semidirect |
| 133. C53 | #3384 | semidirect |
| 220. C8 | Q8×D53 | semidirect |