Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
216 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| prime radical | C23×C53 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q32×C53 | C2 | semidirect |
| 4. C53×#358 | C2 | semidirect |
| 53. C22×C53 | Q32 | semidirect |
| 61. #2188 | C53 | direct (N × H) |
| 67. C2×Q32 | C2×C53 | semidirect |
| 69. #358 | C2×C53 | semidirect |
| 71. C53 | #2188 | direct (N × H) |
| 119. C22 | Q32×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×Q32×C53 | 2 |
| C53×#467 | 2 |
| C53×#358 | 2 |
| C53×C22⋊C16 | 2 |
| #2188 | 53 |