Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
364 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #486 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #2023 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3162 | C2 | semidirect |
| 4. C7×#1389 | C2 | semidirect |
| 5. C7×#1375 | C2 | semidirect |
| 17. #13373 | C7 | direct (N × H) |
| 21. C2×C7×Q16 | Q8 | semidirect |
| 23. C2×C7×D8 | Q8 | semidirect |
| 44. #3162 | C2×C7 | semidirect |
| 47. #1389 | C2×C7 | semidirect |
| 48. #1375 | C2×C7 | semidirect |
| 120. C2×Q16 | C7×Q8 | semidirect |
| 122. C2×D8 | C7×Q8 | semidirect |
| 218. C7 | #13373 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3162 | 2 |
| C7×#2145 | 2 |
| C7×#1975 | 2 |
| C7×#1389 | 2 |
| C7×#1375 | 2 |
| #13373 | 7 |