Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
392 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | Q82×C53 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | Q82 | D53 |
| hypocenter | C53 | #3362 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810971 | C2 | semidirect |
| 3. #16810910 | C2 | semidirect |
| 4. #16810823 | C2 | semidirect |
| 5. Q8×#88322 | C2 | semidirect |
| 6. C4×#11719924 | C2 | semidirect |
| 7. Q82×C53 | C2 | semidirect |
| 46. Q8×C53 | SD16 | semidirect |
| 47. Q8×C53 | #41 | semidirect |
| 77. Q82 | D53 | semidirect |
| 84. C53 | #3362 | semidirect |
| 123. Q8 | #11719924 | semidirect |
| 124. Q8 | #11719949 | semidirect |