Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
840 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 205 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C23×Q8×C53 | C2 |
| hypercenter | C23×Q8 | D53 |
| hypocenter | C53 | #2409 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133193 | C2 | semidirect |
| 3. #16810542 | C2 | semidirect |
| 4. #16810534 | C2 | semidirect |
| 5. #16810530 | C2 | semidirect |
| 7. #16810496 | C2 | semidirect |
| 8. #16810483 | C2 | semidirect |
| 9. C23×Q8×C53 | C2 | semidirect |
| 38. C2×Q8×C53 | D4 | semidirect |
| 48. C2×C22⋊D53 | Q8 | semidirect |
| 73. C22×D53 | C2×Q8 | semidirect |
| 95. C2×Dic53 | C2×Q8 | semidirect |
| 101. C22×C53 | C22⋊Q8 | semidirect |
| 127. C23×Q8 | D53 | semidirect |
| 136. C53 | #2409 | semidirect |
| 165. C2×Q8 | C22⋊D53 | semidirect |
| 233. C22 | #13133193 | semidirect |