Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| prime radical | #135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#494 | C2 | semidirect |
| 4. C53×#364 | C2 | semidirect |
| 5. C53×#362 | C2 | semidirect |
| 55. #2214 | C53 | direct (N × H) |
| 60. #494 | C2×C53 | semidirect |
| 62. #364 | C2×C53 | semidirect |
| 63. #362 | C2×C53 | semidirect |
| 64. C53 | #2214 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#510 | 2 |
| C53×#494 | 2 |
| C53×#443 | 2 |
| C53×#364 | 2 |
| C53×#362 | 2 |
| #2214 | 53 |