Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
170 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 64 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, Frattini, cosocle | C42 | C23×C53 |
| prime radical | C53×#459 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#486 | C2 | semidirect |
| 2. C53×#476 | C2 | semidirect |
| 4. C53×#436 | C2 | semidirect |
| 5. C53×#335 | C2 | semidirect |
| 6. C53×#333 | C2 | semidirect |
| 7. C53×#331 | C2 | semidirect |
| 77. #1697 | C53 | direct (N × H) |
| 83. #486 | C2×C53 | semidirect |
| 84. #476 | C2×C53 | semidirect |
| 86. #436 | C2×C53 | semidirect |
| 87. #335 | C2×C53 | semidirect |
| 88. #333 | C2×C53 | semidirect |
| 89. #331 | C2×C53 | semidirect |
| 90. C53 | #1697 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#486 | 2 |
| C53×#476 | 2 |
| C53×#459 | 2 |
| C53×#436 | 2 |
| C53×#335 | 2 |
| C53×#333 | 2 |
| C53×#331 | 2 |
| #1697 | 53 |