Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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 | #2434 |
| 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. C2×#13133215 | C2 | semidirect |
| 4. #16810543 | C2 | semidirect |
| 5. #16810530 | C2 | semidirect |
| 6. #16810529 | C2 | semidirect |
| 9. C2×C4⋊Q8×C53 | C2 | semidirect |
| 64. #11719921 | Q8 | semidirect |
| 88. C2×C4×C53 | C2×Q8 | semidirect |
| 99. C2×Dic53 | C2×Q8 | semidirect |
| 113. C2×D53 | C4⋊Q8 | semidirect |
| 131. C2×C4⋊Q8 | D53 | semidirect |
| 138. C53 | #2434 | semidirect |
| 230. C2×C4 | Q8×D53 | semidirect |