Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
280 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23×C7 |
| prime radical | C7×#469 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×M16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2239 | C2 | semidirect |
| 4. C7×#2102 | C2 | semidirect |
| 5. C7×#1340 | C2 | semidirect |
| 7. C7×#1327 | C2 | semidirect |
| 27. #12309 | C7 | direct (N × H) |
| 60. #2239 | C2×C7 | semidirect |
| 62. #2102 | C2×C7 | semidirect |
| 63. #1340 | C2×C7 | semidirect |
| 65. #1327 | C2×C7 | semidirect |
| 255. C7 | #12309 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3256 | 2 |
| C7×#2239 | 2 |
| C7×#2238 | 2 |
| C7×#2102 | 2 |
| C7×#1340 | 2 |
| C7×#1339 | 2 |
| C7×#1327 | 2 |
| #12309 | 7 |