Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
340 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C7×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2163 | C2 | semidirect |
| 3. C7×#1564 | C2 | semidirect |
| 4. C7×#1364 | C2 | semidirect |
| 5. C7×#1361 | C2 | semidirect |
| 21. #11951 | C7 | direct (N × H) |
| 48. #2163 | C2×C7 | semidirect |
| 49. #1564 | C2×C7 | semidirect |
| 50. #1364 | C2×C7 | semidirect |
| 51. #1361 | C2×C7 | semidirect |
| 218. C7 | #11951 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3169 | 2 |
| C7×#2163 | 2 |
| C7×#1564 | 2 |
| C7×#1364 | 2 |
| C7×#1361 | 2 |
| #11951 | 7 |