Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
230 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 68 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 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#135×C53 | C2 | semidirect |
| 2. C2×SD32×C53 | C2 | semidirect |
| 3. C2×D16×C53 | C2 | semidirect |
| 4. C53×#466 | C2 | semidirect |
| 5. C53×#358 | C2 | semidirect |
| 6. C53×#357 | C2 | semidirect |
| 7. C53×C22⋊C16 | C2 | semidirect |
| 33. Q16×C53 | D4 | semidirect |
| 38. D8×C53 | D4 | semidirect |
| 83. #2186 | C53 | direct (N × H) |
| 90. C2×#135 | C2×C53 | semidirect |
| 91. C2×SD32 | C2×C53 | semidirect |
| 92. C2×D16 | C2×C53 | semidirect |
| 93. #466 | C2×C53 | semidirect |
| 94. #358 | C2×C53 | semidirect |
| 95. #357 | C2×C53 | semidirect |
| 96. C22⋊C16 | C2×C53 | semidirect |
| 97. C53 | #2186 | direct (N × H) |
| 123. Q16 | D4×C53 | semidirect |
| 128. D8 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C53 | 2 |
| C2×SD32×C53 | 2 |
| C2×D16×C53 | 2 |
| C53×#466 | 2 |
| C53×#358 | 2 |
| C53×#357 | 2 |
| C53×C22⋊C16 | 2 |
| #2186 | 53 |