Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
498 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 498 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 | C4×C7 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C7 |
| Frattini, cosocle | #124 | C23×C7 |
| socle | C2×C7 | C2×AΣL(1,𝔽16) |
| prime radical | C7×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C7×#532 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2126 | C2 | semidirect |
| 3. C7×#1525 | C2 | semidirect |
| 4. C7×#1404 | C2 | semidirect |
| 5. C7×#585 | C4 | semidirect |
| 8. C7×#456 | C4 | semidirect |
| 13. C7×#413 | C4 | semidirect |
| 16. C7×#354 | C22 | semidirect |
| 19. C7×#329 | C22 | semidirect |
| 20. #13568 | C7 | direct (N × H) |
| 55. #2126 | C2×C7 | semidirect |
| 56. #1525 | C2×C7 | semidirect |
| 57. #1404 | C2×C7 | semidirect |
| 112. #585 | C4×C7 | semidirect |
| 115. #456 | C4×C7 | semidirect |
| 120. #413 | C4×C7 | semidirect |
| 123. #354 | C22×C7 | semidirect |
| 126. #329 | C22×C7 | semidirect |
| 323. C7 | #13568 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3018 | 2 |
| C7×#2126 | 2 |
| C7×#1525 | 2 |
| C7×#1404 | 2 |
| #13568 | 7 |