Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
690 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 134 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 | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C7×#2617 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2312 | C2 | semidirect |
| 3. C7×#1519 | C2 | semidirect |
| 4. C2×C7×#323 | C2 | semidirect |
| 5. C7×#1289 | C2 | semidirect |
| 10. C7×#397 | C4 | semidirect |
| 12. C7×#392 | C4 | semidirect |
| 21. #9205 | C7 | direct (N × H) |
| 52. C7×#95 | D4 | semidirect |
| 54. #2312 | C2×C7 | semidirect |
| 55. #1519 | C2×C7 | semidirect |
| 56. C2×#323 | C2×C7 | semidirect |
| 57. #1289 | C2×C7 | semidirect |
| 108. #397 | C4×C7 | semidirect |
| 110. #392 | C4×C7 | semidirect |
| 198. #95 | C7×D4 | semidirect |
| 334. C7 | #9205 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2617 | 2 |
| C7×#2312 | 2 |
| C7×#1519 | 2 |
| C2×C7×#323 | 2 |
| C7×#1289 | 2 |
| #9205 | 7 |