Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
496 conjugacy classes of subgroups; only the 328 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 328 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 | C22 | C23×C4×C53 |
| Frattini, cosocle | C23 | C24×C53 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C4⋊Q8×C53 | C4 | semidirect |
| 28. C4⋊C4×C53 | Q8 | semidirect |
| 59. #2369 | C53 | direct (N × H) |
| 70. C53 | #2369 | direct (N × H) |
| 72. C4⋊Q8 | C4×C53 | semidirect |
| 93. C4⋊C4 | 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 |
| C53×#391 | 2 |
| C53×#389 | 2 |
| C4×C4⋊C4×C53 | 2 |
| #2369 | 53 |