Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 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×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D53 |
| hypocenter | C53 | #3549 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811365 | C2 | semidirect |
| 2. #16811335 | C2 | semidirect |
| 3. #16811255 | C2 | semidirect |
| 4. #16811191 | C2 | semidirect |
| 5. #16811188 | C2 | semidirect |
| 6. #16811172 | C2 | semidirect |
| 7. C53×H(3,𝔽4) | C2 | semidirect |
| 8. C2×#11719949 | C22 | semidirect |
| 9. C2×#11719947 | C22 | semidirect |
| 10. C22×C22⋊D53 | C22 | semidirect |
| 11. C2×D4×D53 | C22 | semidirect |
| 12. #13133215 | C22 | semidirect |
| 13. #13133214 | C22 | semidirect |
| 15. #13133213 | C22 | semidirect |
| 16. #13133206 | C22 | semidirect |
| 17. #13133205 | C22 | semidirect |
| 18. #13133203 | C22 | semidirect |
| 19. #13133190 | C22 | semidirect |
| 20. #13133190 | C22 | semidirect |
| 21. #13133189 | C22 | semidirect |
| 22. #13133187 | C22 | semidirect |
| 23. #13133187 | C22 | semidirect |
| 24. #13133186 | C22 | semidirect |
| 25. #13133185 | C22 | semidirect |
| 26. #13133177 | C22 | semidirect |
| 27. #13133174 | C22 | semidirect |
| 28. #13133173 | C22 | semidirect |
| 29. C42×D53 | C22 | semidirect |
| 30. #124×C53 | C22 | semidirect |
| 31. #124×C53 | C22 | semidirect |
| 32. #120×C53 | C22 | semidirect |
| 103. H(3,𝔽4) | D53 | semidirect |
| 109. C53 | #3549 | semidirect |
| 121. #124 | C2×D53 | semidirect |
| 122. #124 | C2×D53 | semidirect |
| 130. #120 | C2×D53 | semidirect |