Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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 | #410 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| socle | C2×C7 | #1799 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×C8 ◁ C2×C7×#131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1812 | C2 | semidirect |
| 4. C7×#1391 | C2 | semidirect |
| 5. C2×C7×#131 | C4 | semidirect |
| 11. C7×#423 | C4 | semidirect |
| 13. C7×#408 | C4 | semidirect |
| 15. C2×C7×#97 | C4 | semidirect |
| 19. #12808 | C7 | direct (N × H) |
| 51. #1812 | C2×C7 | semidirect |
| 52. #1391 | C2×C7 | semidirect |
| 91. C2×#131 | C4×C7 | semidirect |
| 97. #423 | C4×C7 | semidirect |
| 99. #408 | C4×C7 | semidirect |
| 101. C2×#97 | C4×C7 | semidirect |
| 250. C7 | #12808 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2932 | 2 |
| C7×#2126 | 2 |
| C7×#1812 | 2 |
| C7×#1391 | 2 |
| #12808 | 7 |