Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1144 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C23×C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C23×C4 | #6157 |
| hypocenter | C53 | C2×#381 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811566 | C2 | direct (N × H) |
| 3. C2×(C2×#6157)⋊C4 | C2 | semidirect |
| 8. (C2×#6157)⋊C4 | C22 | semidirect |
| 10. C23×Dic53 | C4 | semidirect |
| 12. C22×C4×D53 | C4 | semidirect |
| 13. C22×C4×D53 | C4 | semidirect |
| 16. C23×C4×C53 | C4 | semidirect |
| 29. C22×C4×C53 | C2×C4 | semidirect |
| 32. C22×Dic53 | C2×C4 | semidirect |
| 38. C2×C4×D53 | C2×C4 | semidirect |
| 90. C2×C53 | #381 | semidirect |
| 94. D53 | #381 | semidirect |
| 100. C53 | C2×#381 | semidirect |
| 109. C23×C4 | #6157 | semidirect |
| 135. C22×C4 | C2×#6157 | semidirect |
| 188. C2 | #16811566 | direct (N × H) |