Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
576 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 250 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 | C2×C42×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C42 | #6157 |
| hypocenter | C53 | C2×#423 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811627 | C2 | direct (N × H) |
| 3. C2×C4⋊#88324 | C2 | semidirect |
| 8. C4⋊#88324 | C22 | semidirect |
| 39. C2×C4×C53 | M16 | semidirect |
| 60. C4×C53 | C2×M16 | semidirect |
| 66. C2×C53 | #423 | semidirect |
| 74. C53 | C2×#423 | semidirect |
| 118. C2×C4 | #11719936 | semidirect |
| 134. C4 | C2×#11719936 | semidirect |
| 136. C2 | #16811627 | direct (N × H) |