Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
912 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 130 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×C7 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#2867 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3208 | C2 | semidirect |
| 3. C7×#1994 | C2 | semidirect |
| 4. C7×#1545 | C2 | semidirect |
| 5. C7×#1276 | C2 | semidirect |
| 19. #10290 | C7 | direct (N × H) |
| 26. C7×C4⋊Q8 | D4 | semidirect |
| 42. C7×C4⋊C8 | D4 | semidirect |
| 46. #3208 | C2×C7 | semidirect |
| 48. #1994 | C2×C7 | semidirect |
| 49. #1545 | C2×C7 | semidirect |
| 50. #1276 | C2×C7 | semidirect |
| 79. C42×C7 | SD16 | semidirect |
| 128. C4⋊Q8 | C7×D4 | semidirect |
| 144. C4⋊C8 | C7×D4 | semidirect |
| 190. C42 | C7×SD16 | semidirect |
| 224. C7 | #10290 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3208 | 2 |
| C7×#2867 | 2 |
| C7×#1994 | 2 |
| C7×#1545 | 2 |
| C7×#1276 | 2 |
| #10290 | 7 |