Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
256 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 256 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 | C22×C53 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C24×C53 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#451 | C2 | semidirect |
| 4. C53×#328 | C2 | semidirect |
| 44. C23×C53 | Q16 | semidirect |
| 81. #1602 | C53 | direct (N × H) |
| 89. #451 | C2×C53 | semidirect |
| 91. #328 | C2×C53 | semidirect |
| 93. C53 | #1602 | direct (N × H) |
| 132. C23 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#543 | 2 |
| C53×#451 | 2 |
| C53×#393 | 2 |
| C53×#328 | 2 |
| C53×#323 | 2 |
| #1602 | 53 |