Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C2×C4×Q8×C53 | C2 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | #2632 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810558 | C2 | semidirect |
| 3. #16810543 | C2 | semidirect |
| 5. #16810534 | C2 | semidirect |
| 6. #16810514 | C2 | semidirect |
| 7. #16810491 | C2 | semidirect |
| 8. C2×C4×Q8×C53 | C2 | semidirect |
| 96. C2×C4×Q8 | D53 | semidirect |
| 103. C53 | #2632 | semidirect |