Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1320 conjugacy classes of subgroups; only the 565 normal subgroups are shown.
This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 565 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×#126 |
| 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. C2×#13133212 | C2 | direct (N × H) |
| 2. C22×C4⋊Dic53 | C2 | semidirect |
| 3. C22×#11719919 | C2 | semidirect |
| 7. C22×C4×Dic53 | C2 | semidirect |
| 8. C22×C4⋊C4×C53 | C2 | semidirect |
| 13. #13133212 | C22 | direct (N × H) |
| 14. C2×C4⋊Dic53 | C22 | semidirect |
| 15. C2×#11719919 | C22 | semidirect |
| 19. C2×C4×Dic53 | C22 | semidirect |
| 23. C2×C4⋊C4×C53 | C22 | semidirect |
| 41. C4⋊C4×C53 | C23 | semidirect |
| 45. C4⋊Dic53 | C23 | semidirect |
| 46. #11719919 | C23 | semidirect |
| 47. C4×Dic53 | C23 | semidirect |
| 69. C22×C53 | #126 | semidirect |
| 85. C2×C53 | C2×#126 | semidirect |
| 93. C22×C4⋊C4 | D53 | semidirect |
| 98. C53 | C22×#126 | semidirect |
| 109. C2×C4⋊C4 | C2×D53 | semidirect |
| 133. C4⋊C4 | C22×D53 | semidirect |
| 163. C22 | #13133212 | direct (N × H) |
| 175. C2 | C2×#13133212 | direct (N × H) |