Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1348 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 213 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 | #2433 |
| 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 |
|---|---|---|
| 1. C23×D212 | C2 | semidirect |
| 2. C2×#13133214 | C2 | semidirect |
| 3. C2×#13133191 | C2 | semidirect |
| 4. C2×#13133187 | C2 | semidirect |
| 5. #16810533 | C2 | semidirect |
| 6. #16810525 | C2 | semidirect |
| 7. C4×C22⋊C4×C53 | C2 | semidirect |
| 13. C22×D212 | C22 | semidirect |
| 14. C22×D212 | C22 | semidirect |
| 19. C2×C4⋊Dic53 | C22 | semidirect |
| 20. C2×#11719921 | C22 | semidirect |
| 21. C2×#11719921 | C22 | semidirect |
| 24. C2×C42×C53 | C22 | semidirect |
| 25. #95×C53 | C22 | semidirect |
| 34. C22×C4×C53 | D4 | semidirect |
| 51. C2×D212 | D4 | semidirect |
| 52. C22⋊C4×C53 | D4 | semidirect |
| 82. C2×C4×C53 | C2×D4 | semidirect |
| 93. C22×C53 | #127 | semidirect |
| 110. C4×C53 | #120 | semidirect |
| 126. C4×C22⋊C4 | D53 | semidirect |
| 127. C53 | #2433 | semidirect |
| 144. C2×C42 | C2×D53 | semidirect |
| 145. #95 | C2×D53 | semidirect |
| 168. C22×C4 | D212 | semidirect |
| 175. C22⋊C4 | D212 | semidirect |
| 208. C2×C4 | C2×D212 | semidirect |
| 223. C22 | #13133214 | semidirect |
| 231. C4 | #13133187 | semidirect |