Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
264 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #121 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C2×#41×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#108×C53 | C2 | semidirect |
| 3. C2×#107×C53 | C2 | semidirect |
| 5. C53×#343 | C2 | semidirect |
| 8. #131×C53 | C4 | semidirect |
| 81. #1816 | C53 | direct (N × H) |
| 89. C2×#108 | C2×C53 | semidirect |
| 90. C2×#107 | C2×C53 | semidirect |
| 92. #343 | C2×C53 | semidirect |
| 93. C53 | #1816 | direct (N × H) |
| 96. #131 | C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#131×C53 | 2 |
| C2×#108×C53 | 2 |
| C2×#107×C53 | 2 |
| C53×#417 | 2 |
| C53×#343 | 2 |
| #1816 | 53 |