Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
326 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 84 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, Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C53×#535 | C2 |
| LCS | G ▷ C22×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 |
|---|---|---|
| 2. C53×#428 | C2 | semidirect |
| 3. C2×#104×C53 | C2 | semidirect |
| 4. C53×#417 | C2 | semidirect |
| 5. C2×#100×C53 | C2 | semidirect |
| 6. C53×#390 | C2 | semidirect |
| 7. C53×#339 | C2 | semidirect |
| 50. M16×C53 | D4 | semidirect |
| 123. #2037 | C53 | direct (N × H) |
| 132. #428 | C2×C53 | semidirect |
| 133. C2×#104 | C2×C53 | semidirect |
| 134. #417 | C2×C53 | semidirect |
| 135. C2×#100 | C2×C53 | semidirect |
| 136. #390 | C2×C53 | semidirect |
| 137. #339 | C2×C53 | semidirect |
| 138. C53 | #2037 | direct (N × H) |
| 181. M16 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#535 | 2 |
| C53×#428 | 2 |
| C2×#104×C53 | 2 |
| C53×#417 | 2 |
| C2×#100×C53 | 2 |
| C53×#390 | 2 |
| C53×#339 | 2 |
| #2037 | 53 |