Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C53×#543 | C2 |
| hypercenter | #543 | D53 |
| hypocenter | C53 | #3459 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811222 | C2 | semidirect |
| 2. #16811213 | C2 | semidirect |
| 3. #16811185 | C2 | semidirect |
| 4. #16811159 | C2 | semidirect |
| 5. #16811136 | C2 | semidirect |
| 6. C2×#13133177 | C2 | semidirect |
| 7. C53×#543 | C2 | semidirect |
| 8. C2×#11719949 | C22 | semidirect |
| 11. C22×C4×D53 | C22 | semidirect |
| 12. #13133212 | C22 | semidirect |
| 13. #13133208 | C22 | semidirect |
| 14. #13133206 | C22 | semidirect |
| 15. #13133202 | C22 | semidirect |
| 16. #13133201 | C22 | semidirect |
| 17. #13133193 | C22 | semidirect |
| 18. #13133192 | C22 | semidirect |
| 21. #13133187 | C22 | semidirect |
| 22. Q8×Dic53 | C22 | semidirect |
| 23. #13133179 | C22 | semidirect |
| 24. #13133177 | C22 | semidirect |
| 25. #13133177 | C22 | semidirect |
| 26. #13133175 | C22 | semidirect |
| 27. C2×#11719919 | C22 | semidirect |
| 28. C2×#11719921 | C22 | semidirect |
| 29. C2×C4×Dic53 | C22 | semidirect |
| 30. C22⋊Q8×C53 | C22 | semidirect |
| 31. C22⋊Q8×C53 | C22 | semidirect |
| 64. C23×C53 | #41 | semidirect |
| 107. #543 | D53 | semidirect |
| 114. C53 | #3459 | semidirect |
| 125. C22⋊Q8 | C2×D53 | semidirect |
| 126. C22⋊Q8 | C2×D53 | semidirect |
| 175. C23 | #11719949 | semidirect |