Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
186 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C53 |
| prime radical | #127×C53 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#500 | C2 | semidirect |
| 3. C53×#487 | C2 | semidirect |
| 4. C53×#487 | C2 | semidirect |
| 7. C53×#322 | C2 | semidirect |
| 64. #1719 | C53 | direct (N × H) |
| 69. #500 | C2×C53 | semidirect |
| 71. #487 | C2×C53 | semidirect |
| 72. #487 | C2×C53 | semidirect |
| 75. #322 | C2×C53 | semidirect |
| 76. C53 | #1719 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#500 | 2 |
| C53×#492 | 2 |
| C53×#487 | 2 |
| C53×#486 | 2 |
| C53×#322 | 2 |
| #1719 | 53 |