Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
418 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2044 |
| commutator, Frattini, cosocle | C2×#41 | C23×C7 |
| prime radical | C7×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2135 | C2 | semidirect |
| 4. C7×#1915 | C2 | semidirect |
| 6. C7×#1858 | C2 | semidirect |
| 7. C7×#1392 | C2 | semidirect |
| 29. #13430 | C7 | direct (N × H) |
| 76. #2135 | C2×C7 | semidirect |
| 78. #1915 | C2×C7 | semidirect |
| 80. #1858 | C2×C7 | semidirect |
| 81. #1392 | C2×C7 | semidirect |
| 359. C7 | #13430 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3200 | 2 |
| C7×#2135 | 2 |
| C7×#2134 | 2 |
| C7×#1915 | 2 |
| C7×#1914 | 2 |
| C7×#1858 | 2 |
| C7×#1392 | 2 |
| #13430 | 7 |