Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
144 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 144 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 | C2×C53 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| prime radical | C53×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ #131×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SD32×C53 | C2 | semidirect |
| 2. C53×#504 | C2 | semidirect |
| 3. C53×#473 | C2 | semidirect |
| 5. C53×#368 | C2 | semidirect |
| 6. C53×#362 | C2 | semidirect |
| 7. C53×#361 | C2 | semidirect |
| 67. #2223 | C53 | direct (N × H) |
| 72. C2×SD32 | C2×C53 | semidirect |
| 73. #504 | C2×C53 | semidirect |
| 74. #473 | C2×C53 | semidirect |
| 76. #368 | C2×C53 | semidirect |
| 77. #362 | C2×C53 | semidirect |
| 78. #361 | C2×C53 | semidirect |
| 79. C53 | #2223 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD32×C53 | 2 |
| C53×#504 | 2 |
| C53×#473 | 2 |
| C53×#472 | 2 |
| C53×#368 | 2 |
| C53×#362 | 2 |
| C53×#361 | 2 |
| #2223 | 53 |