Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
278 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C22×C53 | C2×C4⋊C4 |
| prime radical | C24×C53 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#108×C53 | C2 | semidirect |
| 5. C53×#341 | C2 | semidirect |
| 71. C22×C53 | #108 | semidirect |
| 79. #1855 | C53 | direct (N × H) |
| 87. C2×#108 | C2×C53 | semidirect |
| 90. #341 | C2×C53 | semidirect |
| 91. C53 | #1855 | direct (N × H) |
| 157. C22 | #108×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C53 | 2 |
| C2×#108×C53 | 2 |
| C53×#407 | 2 |
| C2×C22⋊C8×C53 | 2 |
| C53×#341 | 2 |
| #1855 | 53 |