Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
210 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 210 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 | C2×C53 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ #131×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#578 | C2 | semidirect |
| 2. C53×#509 | C2 | semidirect |
| 3. C2×SD32×C53 | C2 | semidirect |
| 4. C53×#472 | C2 | semidirect |
| 5. C53×#362 | C2 | semidirect |
| 6. C53×#359 | C2 | semidirect |
| 7. C53×#350 | C2 | semidirect |
| 85. #2193 | C53 | direct (N × H) |
| 91. #578 | C2×C53 | semidirect |
| 92. #509 | C2×C53 | semidirect |
| 93. C2×SD32 | C2×C53 | semidirect |
| 94. #472 | C2×C53 | semidirect |
| 95. #362 | C2×C53 | semidirect |
| 96. #359 | C2×C53 | semidirect |
| 97. #350 | C2×C53 | semidirect |
| 98. C53 | #2193 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#578 | 2 |
| C53×#509 | 2 |
| C2×SD32×C53 | 2 |
| C53×#472 | 2 |
| C53×#362 | 2 |
| C53×#359 | 2 |
| C53×#350 | 2 |
| #2193 | 53 |