Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
608 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 248 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 | C24×C53 | C23 |
| commutator | C23 | C22×C4×C53 |
| Frattini, cosocle | C24 | C23×C53 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. #95×C53 | C4 | semidirect |
| 45. C2×C4×C53 | C4⋊C4 | semidirect |
| 61. #1444 | C53 | direct (N × H) |
| 72. C53 | #1444 | direct (N × H) |
| 79. #95 | C4×C53 | semidirect |
| 114. C2×C4 | C4⋊C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊C4×C53 | 2 |
| C2×#95×C53 | 2 |
| #1444 | 53 |