Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 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 | C2×D4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C42 | C23×C53 |
| socle | C22×C53 | C4×D4 |
| prime radical | C2×D8×C53 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#436 | C2 | semidirect |
| 3. C53×#334 | C2 | semidirect |
| 4. C53×D4⋊C8 | C2 | semidirect |
| 5. C53×#322 | C2 | semidirect |
| 41. C8×C53 | M16 | semidirect |
| 51. #1594 | C53 | direct (N × H) |
| 57. #436 | C2×C53 | semidirect |
| 58. #334 | C2×C53 | semidirect |
| 59. D4⋊C8 | C2×C53 | semidirect |
| 60. #322 | C2×C53 | semidirect |
| 61. C53 | #1594 | direct (N × H) |
| 97. C8 | M16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D8×C53 | 2 |
| C53×#436 | 2 |
| C53×#334 | 2 |
| C53×D4⋊C8 | 2 |
| C53×#322 | 2 |
| #1594 | 53 |