Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
500 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 204 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 | C23×#6157 |
| commutator | C2×C53 | C23×C8 |
| Fitting | C2×C22⋊C4×C53 | C4 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C2×C22⋊C4 | #6157 |
| hypocenter | C53 | #2972 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811650 | C2 | semidirect |
| 2. #16811640 | C2 | semidirect |
| 3. D53⋊(C22⋊C8) | C2 | semidirect |
| 4. C2×C22⋊#88324 | C2 | semidirect |
| 6. C2×D53⋊C8 | C22 | semidirect |
| 7. C22×#88324 | C22 | semidirect |
| 8. #13133284 | C22 | semidirect |
| 9. #13133274 | C22 | semidirect |
| 11. C22⋊#88324 | C22 | semidirect |
| 12. C4×#88324 | C22 | semidirect |
| 26. C2×C22⋊D53 | C8 | semidirect |
| 47. C23×C53 | C2×C8 | semidirect |
| 52. C22×D53 | C2×C8 | semidirect |
| 58. C22⋊D53 | C2×C8 | semidirect |
| 63. C2×Dic53 | C2×C8 | semidirect |
| 101. C53 | #2972 | semidirect |
| 147. C23 | D53⋊C8 | semidirect |