Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
168 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C42 |
| commutator | C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C8 | C23×C53 |
| socle | C22×C53 | #97 |
| prime radical | C23×C53 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#347 | C2 | direct (N × H) |
| 7. M32×C53 | C4 | semidirect |
| 9. C2×C16×C53 | C4 | semidirect |
| 23. C16×C53 | C2×C4 | semidirect |
| 45. C2×#347 | C53 | direct (N × H) |
| 48. C2×C53 | #347 | direct (N × H) |
| 52. #347 | C2×C53 | direct (N × H) |
| 53. C53 | C2×#347 | direct (N × H) |
| 57. M32 | C4×C53 | semidirect |
| 59. C2×C16 | C4×C53 | semidirect |
| 73. C16 | C2×C4×C53 | semidirect |
| 97. C2 | C53×#347 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M32×C53 | 2 |
| C2×#97×C53 | 2 |
| C53×#347 | 2 |
| C2×#347 | 53 |