Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1502 conjugacy classes of subgroups; only the 1294 normal subgroups are shown.
This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 1294 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C2 | C2×C43×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1726 | C2 | semidirect |
| 6. C4×C7×M16 | C4 | semidirect |
| 12. #20329 | C7 | direct (N × H) |
| 20. C7×#97 | C2×C4 | semidirect |
| 21. C4×C7×C8 | C2×C4 | semidirect |
| 25. #1726 | C2×C7 | semidirect |
| 40. C4×M16 | C4×C7 | semidirect |
| 63. #97 | C2×C4×C7 | semidirect |
| 64. C4×C8 | C2×C4×C7 | semidirect |
| 98. C7 | #20329 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C7×M16 | 2 |
| C2×C7×#376 | 2 |
| C7×#1726 | 2 |
| #20329 | 7 |