Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
616 conjugacy classes of subgroups; only the 253 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 253 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator | C2×C53 | C23×C8 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C2×C4⋊C8×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C4⋊C8 | D53 |
| hypocenter | C53 | C2×C8×D4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4⋊C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C8×D53 | C2 | semidirect |
| 3. #16810688 | C2 | direct (N × H) |
| 4. C2×#13133129 | C2 | semidirect |
| 5. C2×C4×#88322 | C2 | semidirect |
| 6. C2×C4⋊C8×C53 | C2 | semidirect |
| 12. C2×C8×D53 | C22 | semidirect |
| 17. #13133129 | C22 | semidirect |
| 18. C4×#88322 | C22 | semidirect |
| 21. C4⋊C8×C53 | C22 | semidirect |
| 27. C2×D212 | C8 | semidirect |
| 40. C2×#88322 | D4 | semidirect |
| 47. C22×D53 | C2×C8 | semidirect |
| 50. C2×C4×C53 | C2×C8 | semidirect |
| 59. D212 | C2×C8 | semidirect |
| 65. #88322 | C2×D4 | semidirect |
| 73. C2×D53 | C22×C8 | semidirect |
| 76. C4×C53 | C22×C8 | semidirect |
| 81. C2×C53 | C8×D4 | semidirect |
| 89. C2×C4⋊C8 | D53 | semidirect |
| 92. C53 | C2×C8×D4 | semidirect |
| 105. C4⋊C8 | C2×D53 | semidirect |
| 142. C2×C4 | C8×D53 | semidirect |
| 163. C4 | C2×C8×D53 | semidirect |
| 167. C2 | #16810688 | direct (N × H) |