Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
456 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 225 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D53 |
| commutator | C2×C53 | C23×C8 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C8×Q8×C53 | C2 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C8×Q8 | D53 |
| hypocenter | C53 | C8×#41 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C8×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×#88322 | C2 | semidirect |
| 3. C8×D212 | C2 | semidirect |
| 4. #16810688 | C2 | semidirect |
| 5. #16810675 | C2 | semidirect |
| 6. C4×C8×D53 | C2 | semidirect |
| 7. C8×Q8×C53 | C2 | semidirect |
| 25. #11719949 | C8 | direct (N × H) |
| 44. Q8×C53 | C2×C8 | semidirect |
| 51. D212 | C2×C8 | semidirect |
| 55. C4×D53 | C2×C8 | semidirect |
| 57. C8×C53 | #41 | semidirect |
| 59. #88322 | #41 | semidirect |
| 78. C8×Q8 | D53 | semidirect |
| 83. C53 | C8×#41 | semidirect |
| 120. Q8 | C8×D53 | semidirect |
| 135. C8 | #11719949 | direct (N × H) |