Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
348 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 216 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 |
|---|---|---|
| 3. C53×#501 | C2 | semidirect |
| 4. C53×#499 | C2 | semidirect |
| 27. M16×C53 | Q8 | semidirect |
| 62. #3166 | C53 | direct (N × H) |
| 68. #501 | C2×C53 | semidirect |
| 69. #499 | C2×C53 | semidirect |
| 72. C53 | #3166 | direct (N × H) |
| 93. M16 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#533 | 2 |
| C2×#125×C53 | 2 |
| C53×#501 | 2 |
| C53×#499 | 2 |
| C53×#428 | 2 |
| C4×M16×C53 | 2 |
| #3166 | 53 |