Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
360 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | D4 |
| commutator | C4 | C2×C42×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| socle | C22×C53 | C2×C22⋊C4 |
| prime radical | C2×#41×C53 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×M16×C53 | C2 | semidirect |
| 4. C43×C53 | C2 | semidirect |
| 5. C53×#339 | C2 | semidirect |
| 8. C4×Q8×C53 | C4 | semidirect |
| 10. C4×D4×C53 | C4 | semidirect |
| 17. #104×C53 | C4 | direct (N × H) |
| 31. M16×C53 | C2×C4 | semidirect |
| 42. C42×C53 | C2×C4 | semidirect |
| 48. Q8×C53 | C42 | semidirect |
| 51. D4×C53 | C42 | semidirect |
| 75. C4×C53 | #104 | direct (N × H) |
| 79. C4×#104 | C53 | direct (N × H) |
| 86. C4×M16 | C2×C53 | semidirect |
| 87. C43 | C2×C53 | semidirect |
| 88. #339 | C2×C53 | semidirect |
| 89. C53 | C4×#104 | direct (N × H) |
| 92. C4×Q8 | C4×C53 | semidirect |
| 94. C4×D4 | C4×C53 | semidirect |
| 101. #104 | C4×C53 | direct (N × H) |
| 115. M16 | C2×C4×C53 | semidirect |
| 126. C42 | C2×C4×C53 | semidirect |
| 132. Q8 | C42×C53 | semidirect |
| 135. D4 | C42×C53 | semidirect |
| 159. C4 | #104×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#41×C53 | 2 |
| C2×#104×C53 | 2 |
| C4×M16×C53 | 2 |
| C43×C53 | 2 |
| C53×#339 | 2 |
| C4×#104 | 53 |