Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
392 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C24 |
| commutator | C22 | C23×C4×C53 |
| Frattini, cosocle | C2×C4 | C24×C53 |
| socle | C22×C53 | C23×C4 |
| prime radical | C24×C53 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#433 | C2 | semidirect |
| 3. C53×#423 | C2 | semidirect |
| 5. C4×M16×C53 | C2 | semidirect |
| 14. C4⋊C8×C53 | C22 | semidirect |
| 16. #97×C53 | C22 | semidirect |
| 17. C4×C8×C53 | C22 | semidirect |
| 61. #2921 | C53 | direct (N × H) |
| 67. #433 | C2×C53 | semidirect |
| 68. #423 | C2×C53 | semidirect |
| 70. C4×M16 | C2×C53 | semidirect |
| 71. C53 | #2921 | direct (N × H) |
| 80. C4⋊C8 | C22×C53 | semidirect |
| 82. #97 | C22×C53 | semidirect |
| 83. C4×C8 | C22×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#117×C53 | 2 |
| C53×#433 | 2 |
| C53×#423 | 2 |
| C53×#407 | 2 |
| C4×M16×C53 | 2 |
| #2921 | 53 |