Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
680 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22⋊C4 |
| commutator | C22 | C2×C42×C53 |
| Frattini, cosocle | C24 | C23×C53 |
| prime radical | C25×C53 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C25×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#342 | C2 | direct (N × H) |
| 6. C23×C4×C53 | C4 | semidirect |
| 11. C2×C22⋊C4×C53 | C4 | semidirect |
| 12. C2×C22⋊C4×C53 | C4 | semidirect |
| 30. C22×C4×C53 | C2×C4 | semidirect |
| 35. C22⋊C4×C53 | C2×C4 | semidirect |
| 93. C2×#342 | C53 | direct (N × H) |
| 100. C2×C53 | #342 | direct (N × H) |
| 105. #342 | C2×C53 | direct (N × H) |
| 106. C53 | C2×#342 | direct (N × H) |
| 109. C23×C4 | C4×C53 | semidirect |
| 114. C2×C22⋊C4 | C4×C53 | semidirect |
| 115. C2×C22⋊C4 | C4×C53 | semidirect |
| 133. C22×C4 | C2×C4×C53 | semidirect |
| 138. C22⋊C4 | C2×C4×C53 | semidirect |
| 202. C2 | C53×#342 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊C4×C53 | 2 |
| C53×#342 | 2 |
| C2×#342 | 53 |