Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1840 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1840 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C22×C4⋊C4×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C22×C4⋊C4 | D53 |
| hypocenter | C53 | C22×C22⋊Q8 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic106 | C2 | semidirect |
| 3. C2×#13133195 | C2 | direct (N × H) |
| 4. C22×#11719919 | C2 | semidirect |
| 5. C22×#11719919 | C2 | semidirect |
| 8. C22×C4⋊C4×C53 | C2 | semidirect |
| 11. C22×Dic106 | C22 | semidirect |
| 17. #13133195 | C22 | direct (N × H) |
| 18. C2×#11719919 | C22 | semidirect |
| 19. C2×#11719919 | C22 | semidirect |
| 25. C2×C4⋊C4×C53 | C22 | semidirect |
| 29. C23×D53 | Q8 | semidirect |
| 46. C2×Dic106 | C23 | semidirect |
| 47. C4⋊C4×C53 | C23 | semidirect |
| 50. #11719919 | C23 | semidirect |
| 51. #11719919 | C23 | semidirect |
| 59. C22×D53 | C2×Q8 | semidirect |
| 75. C22×C53 | C22⋊Q8 | semidirect |
| 83. C2×D53 | C22×Q8 | semidirect |
| 91. C2×C53 | C2×C22⋊Q8 | semidirect |
| 101. C22×C4⋊C4 | D53 | semidirect |
| 104. C53 | C22×C22⋊Q8 | semidirect |
| 119. C2×C4⋊C4 | C2×D53 | semidirect |
| 145. C4⋊C4 | C22×D53 | semidirect |
| 175. C22 | #13133195 | direct (N × H) |
| 187. C2 | C2×#13133195 | direct (N × H) |