Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1112 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 218 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C7 |
| socle | C2×C7 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3529 | C2 | semidirect |
| 2. C7×#3018 | C2 | semidirect |
| 3. C7×#2203 | C2 | semidirect |
| 4. C7×#2202 | C2 | semidirect |
| 5. C7×#2132 | C2 | semidirect |
| 6. C7×#585 | C22 | semidirect |
| 8. C7×#563 | C22 | semidirect |
| 10. C7×H(3,𝔽4) | C22 | semidirect |
| 15. C7×#456 | C22 | semidirect |
| 20. C7×#413 | C22 | semidirect |
| 21. C7×#356 | C22 | semidirect |
| 22. C7×#355 | C22 | semidirect |
| 23. #33564 | C7 | direct (N × H) |
| 65. #3529 | C2×C7 | semidirect |
| 66. #3018 | C2×C7 | semidirect |
| 67. #2203 | C2×C7 | semidirect |
| 68. #2202 | C2×C7 | semidirect |
| 69. #2132 | C2×C7 | semidirect |
| 120. #585 | C22×C7 | semidirect |
| 122. #563 | C22×C7 | semidirect |
| 124. H(3,𝔽4) | C22×C7 | semidirect |
| 129. #456 | C22×C7 | semidirect |
| 134. #413 | C22×C7 | semidirect |
| 135. #356 | C22×C7 | semidirect |
| 136. #355 | C22×C7 | semidirect |
| 316. C7 | #33564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3529 | 2 |
| C7×#3018 | 2 |
| C7×#2203 | 2 |
| C7×#2202 | 2 |
| C7×#2132 | 2 |
| #33564 | 7 |