Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
500 conjugacy classes of subgroups; only the 297 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 297 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator | C2×C53 | C23×C8 |
| Fitting | C2×C4×D4×C53 | C2 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C2×C4×D4 | D53 |
| hypocenter | C53 | #2972 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×#88322 | C2 | semidirect |
| 2. #16810676 | C2 | semidirect |
| 3. C2×C22⋊#88322 | C2 | semidirect |
| 5. C22×#88322 | C22 | semidirect |
| 6. C4⋊#88322 | C22 | semidirect |
| 8. C22⋊#88322 | C22 | semidirect |
| 9. C4×#88322 | C22 | semidirect |
| 18. C2×D4×C53 | C8 | semidirect |
| 35. C23×C53 | C2×C8 | semidirect |
| 38. D4×C53 | C2×C8 | semidirect |
| 44. C2×C4×C53 | C2×C8 | semidirect |
| 74. C53 | #2972 | semidirect |
| 88. C2×D4 | #88322 | semidirect |
| 105. C23 | C2×#88322 | semidirect |
| 108. D4 | C2×#88322 | semidirect |
| 114. C2×C4 | C2×#88322 | semidirect |