Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
912 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 306 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 | C23 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C2×C7×#127 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1870 | C2 | semidirect |
| 3. C7×#1839 | C2 | semidirect |
| 12. #23357 | C7 | direct (N × H) |
| 27. #1870 | C2×C7 | semidirect |
| 28. #1839 | C2×C7 | semidirect |
| 45. C42×C7 | M16 | semidirect |
| 108. C42 | C7×M16 | semidirect |
| 128. C7 | #23357 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C7×#127 | 2 |
| C7×#1870 | 2 |
| C7×#1839 | 2 |
| #23357 | 7 |