Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
734 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 186 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×C7 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C2×C7×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#498 | C2 | semidirect |
| 3. C7×#1990 | C2 | semidirect |
| 5. C7×#1273 | C2 | semidirect |
| 18. #10315 | C7 | direct (N × H) |
| 44. C2×#498 | C2×C7 | semidirect |
| 46. #1990 | C2×C7 | semidirect |
| 48. #1273 | C2×C7 | semidirect |
| 71. C2×C7×C8 | SD16 | semidirect |
| 72. C2×C7×C8 | SD16 | semidirect |
| 184. C2×C8 | C7×SD16 | semidirect |
| 185. C2×C8 | C7×SD16 | semidirect |
| 228. C7 | #10315 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#498 | 2 |
| C2×C7×#492 | 2 |
| C7×#1990 | 2 |
| C7×#1957 | 2 |
| C7×#1273 | 2 |
| #10315 | 7 |