Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
836 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 836 conjugacy classes of subgroups — too many to draw. Only its 173 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 | C4×C22⋊C4×C53 | C2 |
| hypercenter | C4×C22⋊C4 | D53 |
| hypocenter | C53 | #2432 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#13133217 | C2 | semidirect |
| 3. C2×#13133205 | C2 | semidirect |
| 4. C2×#13133196 | C2 | semidirect |
| 5. #16810539 | C2 | semidirect |
| 6. #16810529 | C2 | semidirect |
| 7. #16810525 | C2 | semidirect |
| 8. #16810497 | C2 | semidirect |
| 9. C4×C22⋊C4×C53 | C2 | semidirect |
| 93. C2×C4×C53 | #41 | semidirect |
| 145. C4×C22⋊C4 | D53 | semidirect |
| 146. C53 | #2432 | semidirect |
| 241. C2×C4 | #11719945 | semidirect |