Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
854 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 854 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 | C2×C7 | #2840 |
| commutator, Frattini, cosocle | C24 | C24×C7 |
| prime radical | C23×C7×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C24×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2032 | C2 | semidirect |
| 2. C7×#2029 | C2 | semidirect |
| 5. C7×#543 | C22 | semidirect |
| 8. C7×#342 | C22 | semidirect |
| 9. #32873 | C7 | direct (N × H) |
| 10. C25×C7 | Q8 | semidirect |
| 13. C22×C7×D4 | Q8 | semidirect |
| 15. C23×C4×C7 | Q8 | semidirect |
| 20. #2032 | C2×C7 | semidirect |
| 21. #2029 | C2×C7 | semidirect |
| 42. #543 | C22×C7 | semidirect |
| 45. #342 | C22×C7 | semidirect |
| 68. C25 | C7×Q8 | semidirect |
| 71. C22×D4 | C7×Q8 | semidirect |
| 73. C23×C4 | C7×Q8 | semidirect |
| 139. C7 | #32873 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2032 | 2 |
| C7×#2029 | 2 |
| C7×#1781 | 2 |
| #32873 | 7 |