Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
4778 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.
This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 1114 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 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×2−1+6 | C2 | semidirect |
| 2. C7×#3487 | C2 | semidirect |
| 3. C7×#3027 | C2 | semidirect |
| 4. C7×#3025 | C2 | semidirect |
| 5. C7×#2884 | C2 | semidirect |
| 10. C22×C7×#41 | C22 | semidirect |
| 13. C7×#535 | C22 | semidirect |
| 14. C7×#525 | C22 | semidirect |
| 15. C7×#519 | C22 | semidirect |
| 16. C7×#458 | C22 | semidirect |
| 17. C7×B(4,𝔽2) | C22 | semidirect |
| 18. C7×#410 | C22 | semidirect |
| 19. #60402 | C7 | direct (N × H) |
| 29. C2×C7×#41 | D4 | semidirect |
| 32. C22×C7×Q8 | D4 | semidirect |
| 47. 2−1+6 | C2×C7 | semidirect |
| 48. #3487 | C2×C7 | semidirect |
| 49. #3027 | C2×C7 | semidirect |
| 50. #3025 | C2×C7 | semidirect |
| 51. #2884 | C2×C7 | semidirect |
| 84. C22×#41 | C22×C7 | semidirect |
| 87. #535 | C22×C7 | semidirect |
| 88. #525 | C22×C7 | semidirect |
| 89. #519 | C22×C7 | semidirect |
| 90. #458 | C22×C7 | semidirect |
| 91. B(4,𝔽2) | C22×C7 | semidirect |
| 92. #410 | C22×C7 | semidirect |
| 122. C2×#41 | C7×D4 | semidirect |
| 125. C22×Q8 | C7×D4 | semidirect |
| 203. C7 | #60402 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×2−1+6 | 2 |
| C7×#3487 | 2 |
| C7×#3027 | 2 |
| C7×#3025 | 2 |
| C7×#2884 | 2 |
| #60402 | 7 |