Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
424 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 424 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 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1558 | C2 | semidirect |
| 4. C7×#1556 | C2 | semidirect |
| 6. C7×#1291 | C2 | semidirect |
| 7. C7×#1291 | C2 | semidirect |
| 25. #9419 | C7 | direct (N × H) |
| 64. #1558 | C2×C7 | semidirect |
| 65. #1556 | C2×C7 | semidirect |
| 67. #1291 | C2×C7 | semidirect |
| 68. #1291 | C2×C7 | semidirect |
| 268. C7 | #9419 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3141 | 2 |
| C7×#2980 | 2 |
| C7×#1558 | 2 |
| C7×#1556 | 2 |
| C7×#1536 | 2 |
| C7×#1291 | 2 |
| #9419 | 7 |