Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
360 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 156 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 | C22⋊C4 |
| commutator | C22 | C2×C42×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C22×C53 | C2×C22⋊C4 |
| prime radical | C22×D4×C53 | C22 |
| LCS | G ▷ C22 ▷ 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 |
|---|---|---|
| 3. C4×M16×C53 | C2 | semidirect |
| 4. C53×#343 | C2 | semidirect |
| 13. #100×C53 | C4 | direct (N × H) |
| 28. M16×C53 | C2×C4 | semidirect |
| 67. C4×C53 | #100 | direct (N × H) |
| 69. C4×#100 | C53 | direct (N × H) |
| 78. C4×M16 | C2×C53 | semidirect |
| 79. #343 | C2×C53 | semidirect |
| 80. C53 | C4×#100 | direct (N × H) |
| 89. #100 | C4×C53 | direct (N × H) |
| 104. M16 | C2×C4×C53 | semidirect |
| 143. C4 | #100×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D4×C53 | 2 |
| C2×#100×C53 | 2 |
| C4×M16×C53 | 2 |
| C53×#343 | 2 |
| C4×#100 | 53 |