Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
360 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #120 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C22×D4×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#419 | C2 | semidirect |
| 5. C53×#339 | C2 | semidirect |
| 10. #124×C53 | C4 | semidirect |
| 22. C2×Q8×C53 | C2×C4 | semidirect |
| 40. C42×C53 | C2×C4 | semidirect |
| 75. #1885 | C53 | direct (N × H) |
| 83. #419 | C2×C53 | semidirect |
| 85. #339 | C2×C53 | semidirect |
| 86. C53 | #1885 | direct (N × H) |
| 91. #124 | C4×C53 | semidirect |
| 103. C2×Q8 | C2×C4×C53 | semidirect |
| 121. C42 | C2×C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#124×C53 | 2 |
| C53×#518 | 2 |
| C53×#419 | 2 |
| C2×#100×C53 | 2 |
| C53×#339 | 2 |
| #1885 | 53 |