Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
752 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 181 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 | C2×#124×C53 | C2 |
| hypercenter | C2×#124 | D53 |
| hypocenter | C53 | #2858 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133215 | C2 | semidirect |
| 3. C2×#13133206 | C2 | semidirect |
| 4. C2×#13133205 | C2 | semidirect |
| 5. C2×#13133204 | C2 | semidirect |
| 6. #16810523 | C2 | semidirect |
| 7. C42×Dic53 | C2 | semidirect |
| 8. C2×#124×C53 | C2 | semidirect |
| 52. C42×C53 | D4 | semidirect |
| 59. C4×Dic53 | D4 | semidirect |
| 106. Dic53 | #124 | semidirect |
| 119. C2×#124 | D53 | semidirect |
| 122. C53 | #2858 | semidirect |
| 173. C42 | C22⋊D53 | semidirect |