Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
194 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2177 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| prime radical | C7×#2263 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ C7×#346 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2179 | C2 | semidirect |
| 3. C7×#2177 | C2 | semidirect |
| 4. C7×#1426 | C2 | semidirect |
| 5. C7×#1419 | C2 | semidirect |
| 6. C7×#1418 | C2 | semidirect |
| 7. C7×#1398 | C2 | semidirect |
| 12. C7×SD64 | C4 | semidirect |
| 13. C7×D32 | C4 | semidirect |
| 21. #13658 | C7 | direct (N × H) |
| 26. C7×Q32 | C2×C4 | semidirect |
| 30. C7×D16 | C2×C4 | semidirect |
| 39. C7×C32 | C2×C4 | semidirect |
| 41. #2179 | C2×C7 | semidirect |
| 42. #2177 | C2×C7 | semidirect |
| 43. #1426 | C2×C7 | semidirect |
| 44. #1419 | C2×C7 | semidirect |
| 45. #1418 | C2×C7 | semidirect |
| 46. #1398 | C2×C7 | semidirect |
| 70. SD64 | C4×C7 | semidirect |
| 71. D32 | C4×C7 | semidirect |
| 98. Q32 | C2×C4×C7 | semidirect |
| 102. D16 | C2×C4×C7 | semidirect |
| 111. C32 | C2×C4×C7 | semidirect |
| 160. C7 | #13658 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2263 | 2 |
| C7×#2179 | 2 |
| C7×#2177 | 2 |
| C7×#1426 | 2 |
| C7×#1419 | 2 |
| C7×#1418 | 2 |
| C7×#1398 | 2 |
| #13658 | 7 |