Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
456 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 185 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D212 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#405 | C2 |
| socle | C2×C53 | C2×C4×D4 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #2955 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ #405 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#13133223 | C2 | semidirect |
| 3. #16810746 | C2 | semidirect |
| 4. #16810741 | C2 | semidirect |
| 5. #16810672 | C2 | semidirect |
| 6. #16810641 | C2 | semidirect |
| 7. C53×#405 | C2 | semidirect |
| 9. #13133250 | C4 | semidirect |
| 10. #13133245 | C4 | semidirect |
| 14. #13133223 | C22 | semidirect |
| 15. #13133223 | C22 | semidirect |
| 19. #13133153 | C22 | semidirect |
| 20. #13133145 | C22 | semidirect |
| 24. #97×C53 | C22 | semidirect |
| 25. C4×C8×C53 | C22 | semidirect |
| 96. #405 | D53 | semidirect |
| 97. C53 | #2955 | semidirect |
| 114. #97 | C2×D53 | semidirect |
| 115. C4×C8 | C2×D53 | semidirect |