Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
202 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 92 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 | C42 | C23×C53 |
| socle | C22×C53 | C2×C22⋊C4 |
| prime radical | C23×C53 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×Q8⋊C8 | C2 | semidirect |
| 67. #1497 | C53 | direct (N × H) |
| 75. Q8⋊C8 | C2×C53 | semidirect |
| 76. C53 | #1497 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#533 | 2 |
| C53×#432 | 2 |
| C53×#431 | 2 |
| C53×Q8⋊C8 | 2 |
| #1497 | 53 |