Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
146 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C53 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C53 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 56. C8×C53 | Q16 | semidirect |
| 67. #1692 | C53 | direct (N × H) |
| 78. C53 | #1692 | direct (N × H) |
| 127. C8 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#498 | 2 |
| C53×#477 | 2 |
| C53×#462 | 2 |
| C53×#446 | 2 |
| C53×#335 | 2 |
| C53×#333 | 2 |
| C53×#332 | 2 |
| #1692 | 53 |