Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
392 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C24×C53 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q16×C53 | C2 | semidirect |
| 3. C2×#103×C53 | C2 | semidirect |
| 5. C53×#340 | C2 | semidirect |
| 47. C23×C53 | Q16 | semidirect |
| 86. #2001 | C53 | direct (N × H) |
| 93. C22×Q16 | C2×C53 | semidirect |
| 95. C2×#103 | C2×C53 | semidirect |
| 97. #340 | C2×C53 | semidirect |
| 98. C53 | #2001 | direct (N × H) |
| 140. C23 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q16×C53 | 2 |
| C2×C22⋊Q8×C53 | 2 |
| C2×#103×C53 | 2 |
| C2×C22⋊C8×C53 | 2 |
| C53×#340 | 2 |
| #2001 | 53 |