Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
416 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 416 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#477 | C2 | direct (N × H) |
| 27. C2×Q8×C53 | Q8 | semidirect |
| 50. Q8×C53 | C2×Q8 | semidirect |
| 78. C2×#477 | C53 | direct (N × H) |
| 79. C2×C53 | #477 | direct (N × H) |
| 85. #477 | C2×C53 | direct (N × H) |
| 89. C53 | C2×#477 | direct (N × H) |
| 110. C2×Q8 | Q8×C53 | semidirect |
| 133. Q8 | C2×Q8×C53 | semidirect |
| 161. C2 | C53×#477 | direct (N × H) |
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×C4×Q8×C53 | 2 |
| C53×#477 | 2 |
| C2×#107×C53 | 2 |
| C2×C4⋊C8×C53 | 2 |
| C2×#103×C53 | 2 |
| C2×#477 | 53 |