Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
2818 conjugacy classes of subgroups; only the 350 normal subgroups are shown.
This group has 2818 conjugacy classes of subgroups — too many to draw. Only its 350 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 | #120 |
| commutator, Frattini, cosocle | C24 | C24×C7 |
| prime radical | C26×C7 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C25×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#543 | C2 | semidirect |
| 3. C7×#2029 | C2 | direct (N × H) |
| 4. C2×C7×#342 | C2 | semidirect |
| 7. C7×#543 | C22 | semidirect |
| 14. C7×#342 | C22 | semidirect |
| 15. C2×#2029 | C7 | direct (N × H) |
| 20. C25×C7 | Q8 | semidirect |
| 38. C2×#543 | C2×C7 | semidirect |
| 40. #2029 | C2×C7 | direct (N × H) |
| 41. C2×#342 | C2×C7 | semidirect |
| 60. C24×C7 | C2×Q8 | semidirect |
| 80. #543 | C22×C7 | semidirect |
| 87. #342 | C22×C7 | semidirect |
| 130. C25 | C7×Q8 | semidirect |
| 189. C24 | C2×C7×Q8 | semidirect |
| 211. C2×C7 | #2029 | direct (N × H) |
| 252. C7 | C2×#2029 | direct (N × H) |
| 280. C2 | C7×#2029 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#543 | 2 |
| C2×C7×#379 | 2 |
| C7×#2029 | 2 |
| C2×C7×#342 | 2 |
| C2×#2029 | 7 |