Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C23 |
| commutator | C22 | C2×C42×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C23×C53 | C22×C4 |
| prime radical | C24×C53 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#97×C53 | C2 | semidirect |
| 5. C53×#336 | C2 | semidirect |
| 8. C22×C8×C53 | C4 | semidirect |
| 14. C22⋊C8×C53 | C4 | semidirect |
| 55. C8×C53 | C22⋊C4 | semidirect |
| 59. C22×C53 | #97 | semidirect |
| 69. #1752 | C53 | direct (N × H) |
| 77. C2×#97 | C2×C53 | semidirect |
| 79. #336 | C2×C53 | semidirect |
| 80. C53 | #1752 | direct (N × H) |
| 83. C22×C8 | C4×C53 | semidirect |
| 89. C22⋊C8 | C4×C53 | semidirect |
| 130. C8 | C22⋊C4×C53 | semidirect |
| 134. C22 | #97×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C8×C53 | 2 |
| C2×C22⋊C8×C53 | 2 |
| C2×#97×C53 | 2 |
| C4×C22⋊C4×C53 | 2 |
| C53×#336 | 2 |
| #1752 | 53 |