Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
348 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 90 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 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 2. C7×#1838 | C2 | semidirect |
| 3. C7×#1559 | C2 | semidirect |
| 5. C7×#1531 | C2 | semidirect |
| 7. C7×#1307 | C2 | semidirect |
| 27. #10527 | C7 | direct (N × H) |
| 65. #1838 | C2×C7 | semidirect |
| 66. #1559 | C2×C7 | semidirect |
| 68. #1531 | C2×C7 | semidirect |
| 70. #1307 | C2×C7 | semidirect |
| 284. C7 | #10527 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3254 | 2 |
| C7×#1838 | 2 |
| C7×#1559 | 2 |
| C7×#1557 | 2 |
| C7×#1531 | 2 |
| C7×#1530 | 2 |
| C7×#1307 | 2 |
| #10527 | 7 |