Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
230 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 96 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 | C2×D4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C22×C53 | C4×D4 |
| prime radical | C2×#41×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#424 | C2 | semidirect |
| 5. C2×C4×C8×C53 | C2 | semidirect |
| 6. C53×#341 | C2 | semidirect |
| 7. C53×#339 | C2 | semidirect |
| 17. #103×C53 | C4 | semidirect |
| 18. #102×C53 | C4 | semidirect |
| 85. #1937 | C53 | direct (N × H) |
| 91. #424 | C2×C53 | semidirect |
| 95. C2×C4×C8 | C2×C53 | semidirect |
| 96. #341 | C2×C53 | semidirect |
| 97. #339 | C2×C53 | semidirect |
| 98. C53 | #1937 | direct (N × H) |
| 108. #103 | C4×C53 | semidirect |
| 109. #102 | C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#424 | 2 |
| C2×#104×C53 | 2 |
| C53×#416 | 2 |
| C53×#408 | 2 |
| C2×C4×C8×C53 | 2 |
| C53×#341 | 2 |
| C53×#339 | 2 |
| #1937 | 53 |