Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
392 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 94 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 | C22×C7×D4 | D4 |
| 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 |
|---|---|---|
| 3. C7×#1573 | C2 | semidirect |
| 4. C7×#1552 | C2 | semidirect |
| 5. C7×#1503 | C2 | semidirect |
| 7. C7×#1285 | C2 | semidirect |
| 29. #10510 | C7 | direct (N × H) |
| 75. #1573 | C2×C7 | semidirect |
| 76. #1552 | C2×C7 | semidirect |
| 77. #1503 | C2×C7 | semidirect |
| 79. #1285 | C2×C7 | semidirect |
| 339. C7 | #10510 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3215 | 2 |
| C7×#1983 | 2 |
| C7×#1573 | 2 |
| C7×#1552 | 2 |
| C7×#1503 | 2 |
| C7×#1469 | 2 |
| C7×#1285 | 2 |
| #10510 | 7 |