Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
262 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #120 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C2×C53 | #385 |
| prime radical | C2×#41×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#430 | C2 | semidirect |
| 3. C53×#423 | C2 | semidirect |
| 6. C53×#344 | C2 | semidirect |
| 7. C53×#339 | C2 | semidirect |
| 21. #104×C53 | C4 | semidirect |
| 22. #104×C53 | C4 | semidirect |
| 92. #1861 | C53 | direct (N × H) |
| 99. #430 | C2×C53 | semidirect |
| 100. #423 | C2×C53 | semidirect |
| 103. #344 | C2×C53 | semidirect |
| 104. #339 | C2×C53 | semidirect |
| 105. C53 | #1861 | direct (N × H) |
| 119. #104 | C4×C53 | semidirect |
| 120. #104 | C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#568 | 2 |
| C53×#430 | 2 |
| C53×#423 | 2 |
| C53×#421 | 2 |
| C2×#104×C53 | 2 |
| C53×#344 | 2 |
| C53×#339 | 2 |
| #1861 | 53 |