Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
820 conjugacy classes of subgroups; only the 253 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 253 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 | C22×C53 | C23×C4 |
| Fitting | C53×#380 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #380 | D53 |
| hypocenter | C53 | Q8×C22⋊C4 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#13133182 | C2 | semidirect |
| 3. #16810528 | C2 | semidirect |
| 4. #16810527 | C2 | semidirect |
| 10. C22×Dic106 | C4 | semidirect |
| 50. C22⋊Dic53 | Q8 | semidirect |
| 78. C2×Dic53 | C2×Q8 | semidirect |
| 81. Dic106 | C22⋊C4 | semidirect |
| 86. C22×C53 | C4×Q8 | semidirect |
| 114. C53 | Q8×C22⋊C4 | semidirect |
| 192. C22 | #13133182 | semidirect |