Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
510 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C7 |
| Frattini, cosocle | #124 | C23×C7 |
| prime radical | C2×C7×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×#124 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×#354 | C2 | semidirect |
| 3. C2×C7×#329 | C2 | semidirect |
| 4. C7×#1404 | C2 | direct (N × H) |
| 5. C2×C7×2−1+4 | C4 | semidirect |
| 10. C7×#454 | C4 | semidirect |
| 13. C2×C7×#101 | C4 | semidirect |
| 17. C7×#354 | C22 | semidirect |
| 19. C7×#329 | C22 | semidirect |
| 20. C2×#1404 | C7 | direct (N × H) |
| 24. C7×2−1+4 | C2×C4 | semidirect |
| 49. C7×#101 | C2×C4 | semidirect |
| 55. C2×#354 | C2×C7 | semidirect |
| 56. C2×#329 | C2×C7 | semidirect |
| 57. #1404 | C2×C7 | direct (N × H) |
| 112. C2×2−1+4 | C4×C7 | semidirect |
| 117. #454 | C4×C7 | semidirect |
| 120. C2×#101 | C4×C7 | semidirect |
| 124. #354 | C22×C7 | semidirect |
| 126. #329 | C22×C7 | semidirect |
| 173. 2−1+4 | C2×C4×C7 | semidirect |
| 198. #101 | C2×C4×C7 | semidirect |
| 282. C2×C7 | #1404 | direct (N × H) |
| 327. C7 | C2×#1404 | direct (N × H) |
| 353. C2 | C7×#1404 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#454 | 2 |
| C2×C7×#354 | 2 |
| C2×C7×#329 | 2 |
| C7×#1404 | 2 |
| C2×#1404 | 7 |