Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
576 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C23×C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C23×C4 | #6157 |
| hypocenter | C53 | C2×#423 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811626 | C2 | direct (N × H) |
| 3. C2×#13133284 | C2 | semidirect |
| 4. C2×C4⋊#88324 | C2 | semidirect |
| 9. #13133284 | C22 | semidirect |
| 10. C4⋊#88324 | C22 | semidirect |
| 51. C2×C4×C53 | M16 | semidirect |
| 58. C2×Dic53 | M16 | semidirect |
| 78. C4×C53 | C2×M16 | semidirect |
| 82. Dic53 | C2×M16 | semidirect |
| 87. C2×C53 | #423 | semidirect |
| 96. C53 | C2×#423 | semidirect |
| 156. C2×C4 | #11719940 | semidirect |
| 175. C4 | C2×#11719940 | semidirect |
| 177. C2 | #16811626 | direct (N × H) |