Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
590 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 106 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 | C2×#99 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#1351 | C2 | semidirect |
| 5. C23×C7×Q8 | C4 | semidirect |
| 10. C7×#395 | C4 | semidirect |
| 15. C7×#380 | C4 | semidirect |
| 20. #11785 | C7 | direct (N × H) |
| 54. #1351 | C2×C7 | semidirect |
| 98. C23×Q8 | C4×C7 | semidirect |
| 103. #395 | C4×C7 | semidirect |
| 108. #380 | C4×C7 | semidirect |
| 303. C7 | #11785 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2409 | 2 |
| C4×C7×#99 | 2 |
| C7×#1466 | 2 |
| C7×#1351 | 2 |
| #11785 | 7 |