Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
394 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 248 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 | C24 |
| commutator | C22 | C23×C4×C53 |
| Frattini, cosocle | C2×C4 | C24×C53 |
| socle | C22×C53 | C23×C4 |
| prime radical | C53×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#435 | C2 | semidirect |
| 3. C53×#432 | C2 | semidirect |
| 4. C53×#408 | C2 | semidirect |
| 61. #2980 | C53 | direct (N × H) |
| 67. #435 | C2×C53 | semidirect |
| 68. #432 | C2×C53 | semidirect |
| 69. #408 | C2×C53 | semidirect |
| 70. C53 | #2980 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#532 | 2 |
| C53×#435 | 2 |
| C53×#432 | 2 |
| C53×#408 | 2 |
| #2980 | 53 |