Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
186 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | D16 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C4×D16 |
| prime radical | C7×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×C8 ◁ C4×C7×C8 ◁ C4×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2178 | C2 | semidirect |
| 3. C7×#1425 | C2 | semidirect |
| 4. C7×#1417 | C2 | semidirect |
| 5. C4×C7×C32 | C2 | semidirect |
| 8. C7×Q64 | C4 | semidirect |
| 9. C7×SD64 | C4 | semidirect |
| 10. C7×D32 | C4 | semidirect |
| 16. #13655 | C7 | direct (N × H) |
| 31. #2178 | C2×C7 | semidirect |
| 32. #1425 | C2×C7 | semidirect |
| 33. #1417 | C2×C7 | semidirect |
| 34. C4×C32 | C2×C7 | semidirect |
| 50. Q64 | C4×C7 | semidirect |
| 51. SD64 | C4×C7 | semidirect |
| 52. D32 | C4×C7 | semidirect |
| 115. C7 | #13655 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2262 | 2 |
| C7×#2178 | 2 |
| C7×#1425 | 2 |
| C7×#1417 | 2 |
| C4×C7×C32 | 2 |
| #13655 | 7 |