Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
550 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 90 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 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | #127 | C23×C7 |
| prime radical | C2×C7×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×#127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×#353 | C2 | semidirect |
| 3. C7×#1555 | C2 | semidirect |
| 4. C7×#1402 | C2 | semidirect |
| 5. C2×C7×H(5,𝔽2) | C4 | semidirect |
| 10. C7×#453 | C4 | semidirect |
| 14. C2×C7×#100 | C4 | semidirect |
| 17. C7×#353 | C22 | semidirect |
| 19. C7×#331 | C22 | semidirect |
| 20. #13561 | C7 | direct (N × H) |
| 24. C7×H(5,𝔽2) | C2×C4 | semidirect |
| 49. C7×#100 | C2×C4 | semidirect |
| 55. C2×#353 | C2×C7 | semidirect |
| 56. #1555 | C2×C7 | semidirect |
| 57. #1402 | C2×C7 | semidirect |
| 112. C2×H(5,𝔽2) | C4×C7 | semidirect |
| 117. #453 | C4×C7 | semidirect |
| 121. C2×#100 | C4×C7 | semidirect |
| 124. #353 | C22×C7 | semidirect |
| 126. #331 | C22×C7 | semidirect |
| 179. H(5,𝔽2) | C2×C4×C7 | semidirect |
| 204. #100 | C2×C4×C7 | semidirect |
| 347. C7 | #13561 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#453 | 2 |
| C2×C7×#353 | 2 |
| C7×#1555 | 2 |
| C7×#1402 | 2 |
| #13561 | 7 |