Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
250 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #99 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×Q8 | C23×C53 |
| socle | C2×C53 | C2×#99 |
| prime radical | C53×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#413 | C2 | semidirect |
| 3. C53×#356 | C2 | semidirect |
| 4. C53×#355 | C2 | semidirect |
| 79. #2132 | C53 | direct (N × H) |
| 86. #413 | C2×C53 | semidirect |
| 87. #356 | C2×C53 | semidirect |
| 88. #355 | C2×C53 | semidirect |
| 89. C53 | #2132 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#532 | 2 |
| C53×#413 | 2 |
| C53×#356 | 2 |
| C53×#355 | 2 |
| #2132 | 53 |