Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1952 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 282 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 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3484 | C2 | semidirect |
| 2. C7×#3019 | C2 | semidirect |
| 3. C7×#2010 | C2 | semidirect |
| 4. C7×#1911 | C2 | semidirect |
| 5. C7×#1783 | C2 | semidirect |
| 6. C2×C7×H(5,𝔽2) | C22 | semidirect |
| 8. C7×#575 | C22 | semidirect |
| 13. C7×#525 | C22 | semidirect |
| 15. C7×#518 | C22 | semidirect |
| 18. C7×#421 | C22 | semidirect |
| 19. C2×C7×#100 | C22 | semidirect |
| 20. C7×#407 | C22 | semidirect |
| 21. C7×H(3,ℤ/4) | C22 | semidirect |
| 22. #23897 | C7 | direct (N × H) |
| 40. C7×#127 | D4 | semidirect |
| 41. C7×#124 | D4 | semidirect |
| 60. #3484 | C2×C7 | semidirect |
| 61. #3019 | C2×C7 | semidirect |
| 62. #2010 | C2×C7 | semidirect |
| 63. #1911 | C2×C7 | semidirect |
| 64. #1783 | C2×C7 | semidirect |
| 74. C2×C7×Q8 | C2×D4 | semidirect |
| 89. C2×C7×D4 | C2×D4 | semidirect |
| 109. C42×C7 | C2×D4 | semidirect |
| 110. C2×H(5,𝔽2) | C22×C7 | semidirect |
| 112. #575 | C22×C7 | semidirect |
| 117. #525 | C22×C7 | semidirect |
| 119. #518 | C22×C7 | semidirect |
| 122. #421 | C22×C7 | semidirect |
| 123. C2×#100 | C22×C7 | semidirect |
| 124. #407 | C22×C7 | semidirect |
| 125. H(3,ℤ/4) | C22×C7 | semidirect |
| 179. #127 | C7×D4 | semidirect |
| 180. #124 | C7×D4 | semidirect |
| 228. C2×Q8 | C2×C7×D4 | semidirect |
| 243. C2×D4 | C2×C7×D4 | semidirect |
| 263. C42 | C2×C7×D4 | semidirect |
| 306. C7 | #23897 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3484 | 2 |
| C7×#3019 | 2 |
| C7×#2010 | 2 |
| C7×#1911 | 2 |
| C7×#1783 | 2 |
| #23897 | 7 |