Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #1820 |
| prime radical | C2×C7×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C42×C7 ◁ C4×C7×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C7×#424 | C4 | semidirect |
| 10. C7×#424 | C4 | semidirect |
| 15. C4×C7×M16 | C4 | semidirect |
| 16. C2×C4×C7×C8 | C4 | semidirect |
| 19. #12864 | C7 | direct (N × H) |
| 86. #424 | C4×C7 | semidirect |
| 87. #424 | C4×C7 | semidirect |
| 92. C4×M16 | C4×C7 | semidirect |
| 93. C2×C4×C8 | C4×C7 | semidirect |
| 217. C7 | #12864 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2922 | 2 |
| C7×#1827 | 2 |
| C7×#1395 | 2 |
| C7×#1392 | 2 |
| C7×#1391 | 2 |
| #12864 | 7 |