Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
348 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 188 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| prime radical | C23×C53 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#477 | C2 | semidirect |
| 4. C53×#462 | C2 | semidirect |
| 47. C2×C4×C53 | Q16 | semidirect |
| 64. #3249 | C53 | direct (N × H) |
| 70. #477 | C2×C53 | semidirect |
| 72. #462 | C2×C53 | semidirect |
| 74. C53 | #3249 | direct (N × H) |
| 116. C2×C4 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#533 | 2 |
| C53×#477 | 2 |
| C53×#467 | 2 |
| C53×#462 | 2 |
| C2×C4⋊C8×C53 | 2 |
| #3249 | 53 |