Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
552 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 224 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| prime radical | C53×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#137×C53 | C2 | semidirect |
| 3. C2×C4⋊Q8×C53 | C2 | semidirect |
| 4. C53×#497 | C2 | semidirect |
| 5. C53×#494 | C2 | semidirect |
| 6. C53×#492 | C2 | semidirect |
| 7. C4×M16×C53 | C2 | semidirect |
| 40. M16×C53 | D4 | semidirect |
| 77. C4×C53 | #137 | semidirect |
| 79. #3152 | C53 | direct (N × H) |
| 84. C2×#137 | C2×C53 | semidirect |
| 86. C2×C4⋊Q8 | C2×C53 | semidirect |
| 87. #497 | C2×C53 | semidirect |
| 88. #494 | C2×C53 | semidirect |
| 89. #492 | C2×C53 | semidirect |
| 90. C4×M16 | C2×C53 | semidirect |
| 91. C53 | #3152 | direct (N × H) |
| 124. M16 | D4×C53 | semidirect |
| 161. C4 | #137×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137×C53 | 2 |
| C53×#532 | 2 |
| C2×C4⋊Q8×C53 | 2 |
| C53×#497 | 2 |
| C53×#494 | 2 |
| C53×#492 | 2 |
| C4×M16×C53 | 2 |
| #3152 | 53 |