Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
336 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 208 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 | C22×C53 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C4×Q8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 20. Q16×C53 | Q8 | direct (N × H) |
| 31. Q8×C53 | Q16 | direct (N × H) |
| 47. Q8×Q16 | C53 | direct (N × H) |
| 56. C53 | Q8×Q16 | direct (N × H) |
| 71. Q16 | Q8×C53 | direct (N × H) |
| 82. Q8 | Q16×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q82×C53 | 2 |
| C53×#500 | 2 |
| C53×#477 | 2 |
| C8×Q8×C53 | 2 |
| C4×Q16×C53 | 2 |
| Q8×Q16 | 53 |