Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #354 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | #124 | C23×C7 |
| socle | C2×C7 | C2×#354 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C7×#532 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#1414 | C2 | semidirect |
| 5. C7×#533 | C4 | semidirect |
| 7. C7×#488 | C4 | semidirect |
| 15. C2×C4×C7×C8 | C4 | semidirect |
| 20. #13602 | C7 | direct (N × H) |
| 51. #1414 | C2×C7 | semidirect |
| 88. #533 | C4×C7 | semidirect |
| 90. #488 | C4×C7 | semidirect |
| 98. C2×C4×C8 | C4×C7 | semidirect |
| 243. C7 | #13602 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3131 | 2 |
| C7×#2132 | 2 |
| C7×#2126 | 2 |
| C7×#1414 | 2 |
| #13602 | 7 |