Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1068 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 529 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | Q82×C53 | C2 |
| hypercenter | Q82 | D53 |
| hypocenter | C53 | Q8×#41 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811352 | C2 | semidirect |
| 2. #16811334 | C2 | semidirect |
| 3. Q8×D212 | C2 | semidirect |
| 4. #16811178 | C2 | semidirect |
| 6. C4×Q8×D53 | C2 | semidirect |
| 7. Q82×C53 | C2 | semidirect |
| 30. #11719949 | Q8 | direct (N × H) |
| 50. Q8×C53 | C2×Q8 | semidirect |
| 51. Q8×C53 | #41 | semidirect |
| 58. D212 | C2×Q8 | semidirect |
| 61. C4×D53 | C2×Q8 | semidirect |
| 78. Q82 | D53 | semidirect |
| 85. C53 | Q8×#41 | semidirect |
| 129. Q8 | Q8×D53 | semidirect |
| 130. Q8 | #11719949 | direct (N × H) |