Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
860 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 860 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #1790 |
| commutator | C2×Q8 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| prime radical | C7×#3549 | C2 |
| LCS | G ▷ C2×Q8 ▷ 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×#2128 | C2 | semidirect |
| 3. C7×#2126 | C2 | semidirect |
| 4. C7×#1910 | C2 | semidirect |
| 5. C7×#1884 | C2 | semidirect |
| 6. C7×#568 | C22 | semidirect |
| 8. C7×H(3,𝔽4) | C4 | semidirect |
| 10. C7×#555 | C4 | semidirect |
| 20. C7×#410 | C22 | semidirect |
| 21. C7×#410 | C22 | semidirect |
| 27. #12778 | C7 | direct (N × H) |
| 78. #2128 | C2×C7 | semidirect |
| 79. #2126 | C2×C7 | semidirect |
| 80. #1910 | C2×C7 | semidirect |
| 81. #1884 | C2×C7 | semidirect |
| 142. #568 | C22×C7 | semidirect |
| 144. H(3,𝔽4) | C4×C7 | semidirect |
| 146. #555 | C4×C7 | semidirect |
| 156. #410 | C22×C7 | semidirect |
| 157. #410 | C22×C7 | semidirect |
| 386. C7 | #12778 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3549 | 2 |
| C7×#2128 | 2 |
| C7×#2126 | 2 |
| C7×#1910 | 2 |
| C7×#1884 | 2 |
| #12778 | 7 |