Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
474 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C2×#127×C53 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×SD16×C53 | C2 | semidirect |
| 3. C2×#102×C53 | C2 | semidirect |
| 4. C53×#391 | C2 | semidirect |
| 5. C53×#336 | C2 | semidirect |
| 22. C2×Q8×C53 | D4 | semidirect |
| 39. C2×C8×C53 | D4 | semidirect |
| 62. C2×C4×C53 | SD16 | semidirect |
| 83. #2014 | C53 | direct (N × H) |
| 90. C22×SD16 | C2×C53 | semidirect |
| 92. C2×#102 | C2×C53 | semidirect |
| 93. #391 | C2×C53 | semidirect |
| 94. #336 | C2×C53 | semidirect |
| 95. C53 | #2014 | direct (N × H) |
| 112. C2×Q8 | D4×C53 | semidirect |
| 129. C2×C8 | D4×C53 | semidirect |
| 152. C2×C4 | SD16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD16×C53 | 2 |
| C2×#127×C53 | 2 |
| C2×#102×C53 | 2 |
| C53×#391 | 2 |
| C53×#336 | 2 |
| #2014 | 53 |