Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1088 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 181 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×C22⋊Q8×C53 | C2 |
| hypercenter | C2×C22⋊Q8 | D53 |
| hypocenter | C53 | #2435 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#13133206 | C2 | semidirect |
| 3. #16810540 | C2 | semidirect |
| 4. #16810524 | C2 | semidirect |
| 5. #16810478 | C2 | semidirect |
| 6. C4×C22⋊Dic53 | C2 | semidirect |
| 7. C2×C22⋊Q8×C53 | C2 | semidirect |
| 15. C2×#11719921 | C22 | semidirect |
| 16. C2×#11719921 | C22 | semidirect |
| 17. C2×C4×Dic53 | C22 | semidirect |
| 18. #13133124 | C22 | semidirect |
| 19. C22×Q8×C53 | C22 | semidirect |
| 22. C2×C4⋊C4×C53 | C22 | semidirect |
| 23. C2×C4⋊C4×C53 | C22 | semidirect |
| 85. C22×C53 | #124 | semidirect |
| 110. C2×C22⋊Q8 | D53 | semidirect |
| 115. C53 | #2435 | semidirect |
| 117. C22×Q8 | C2×D53 | semidirect |
| 125. C2×C4⋊C4 | C2×D53 | semidirect |
| 126. C2×C4⋊C4 | C2×D53 | semidirect |
| 199. C22 | #13133206 | semidirect |