Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
448 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 232 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 | C23×C53 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C53 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 40. C42×C53 | Q8 | semidirect |
| 69. #2660 | C53 | direct (N × H) |
| 81. C53 | #2660 | direct (N × H) |
| 114. C42 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊Q8×C53 | 2 |
| C2×#125×C53 | 2 |
| C53×#398 | 2 |
| C53×#395 | 2 |
| C53×#384 | 2 |
| C53×#382 | 2 |
| C53×#376 | 2 |
| #2660 | 53 |