Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
272 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 92 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 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C53×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#432 | C2 | semidirect |
| 3. C53×#376 | C2 | semidirect |
| 4. C53×#327 | C2 | semidirect |
| 6. C4⋊Q8×C53 | C4 | semidirect |
| 7. #127×C53 | C4 | semidirect |
| 12. C4×D4×C53 | C4 | semidirect |
| 78. #1505 | C53 | direct (N × H) |
| 85. #432 | C2×C53 | semidirect |
| 86. #376 | C2×C53 | semidirect |
| 87. #327 | C2×C53 | semidirect |
| 88. C53 | #1505 | direct (N × H) |
| 90. C4⋊Q8 | C4×C53 | semidirect |
| 91. #127 | C4×C53 | semidirect |
| 96. C4×D4 | C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#532 | 2 |
| C53×#432 | 2 |
| C53×#376 | 2 |
| C53×#327 | 2 |
| #1505 | 53 |