Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
132 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 60 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×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| prime radical | C23×C53 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#367 | C2 | semidirect |
| 4. C53×#366 | C2 | semidirect |
| 5. C53×#358 | C2 | semidirect |
| 6. C53×#358 | C2 | semidirect |
| 61. #2237 | C53 | direct (N × H) |
| 68. #367 | C2×C53 | semidirect |
| 69. #366 | C2×C53 | semidirect |
| 70. #358 | C2×C53 | semidirect |
| 71. #358 | C2×C53 | semidirect |
| 73. C53 | #2237 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#467 | 2 |
| C53×#427 | 2 |
| C53×#367 | 2 |
| C53×#366 | 2 |
| C53×#358 | 2 |
| C53×C22⋊C16 | 2 |
| #2237 | 53 |