Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
402 conjugacy classes of subgroups; only the 276 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 276 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 | C2×#41×C53 | 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. C8×Q8×C53 | C2 | semidirect |
| 3. C53×#436 | C2 | semidirect |
| 4. C53×#432 | C2 | semidirect |
| 5. C4×M16×C53 | C2 | semidirect |
| 36. Q8×C53 | M16 | semidirect |
| 48. C8×C53 | #41 | semidirect |
| 59. #2971 | C53 | direct (N × H) |
| 65. C8×Q8 | C2×C53 | semidirect |
| 66. #436 | C2×C53 | semidirect |
| 67. #432 | C2×C53 | semidirect |
| 68. C4×M16 | C2×C53 | semidirect |
| 69. C53 | #2971 | direct (N × H) |
| 100. Q8 | M16×C53 | semidirect |
| 112. C8 | #41×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#41×C53 | 2 |
| C8×Q8×C53 | 2 |
| C53×#436 | 2 |
| C53×#432 | 2 |
| C4×M16×C53 | 2 |
| #2971 | 53 |