Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
432 conjugacy classes of subgroups; only the 297 normal subgroups are shown.
This group has 432 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 | C4×#41×C53 | C2 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C4×#41 | D53 |
| hypocenter | C53 | #2988 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×#88322 | C2 | semidirect |
| 2. D4×#88322 | C2 | semidirect |
| 3. #16810676 | C2 | semidirect |
| 4. C2×C4⋊#88322 | C2 | semidirect |
| 19. #41×C53 | C8 | semidirect |
| 36. Q8×C53 | C2×C8 | semidirect |
| 37. D4×C53 | C2×C8 | semidirect |
| 41. C2×C4×C53 | C2×C8 | semidirect |
| 71. C53 | #2988 | semidirect |
| 85. #41 | #88322 | semidirect |
| 102. Q8 | C2×#88322 | semidirect |
| 103. D4 | C2×#88322 | semidirect |
| 107. C2×C4 | C2×#88322 | semidirect |