Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
380 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×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×#575 | C2 | semidirect |
| 2. C2×C7×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C7×#535 | C2 | semidirect |
| 4. C7×#421 | C2 | semidirect |
| 5. C7×#418 | C2 | semidirect |
| 6. C7×#410 | C2 | semidirect |
| 7. C7×#344 | C2 | semidirect |
| 31. #2008 | C7 | direct (N × H) |
| 42. C2×C7×Q8 | D4 | semidirect |
| 53. C2×C7×D4 | D4 | semidirect |
| 66. C7×M16 | D4 | semidirect |
| 68. C2×C7×C8 | D4 | semidirect |
| 76. #575 | C2×C7 | semidirect |
| 77. C2×AGL(1,ℤ/8) | C2×C7 | semidirect |
| 78. #535 | C2×C7 | semidirect |
| 79. #421 | C2×C7 | semidirect |
| 80. #418 | C2×C7 | semidirect |
| 81. #410 | C2×C7 | semidirect |
| 82. #344 | C2×C7 | semidirect |
| 176. C2×Q8 | C7×D4 | semidirect |
| 187. C2×D4 | C7×D4 | semidirect |
| 200. M16 | C7×D4 | semidirect |
| 202. C2×C8 | C7×D4 | semidirect |
| 257. C7 | #2008 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#575 | 2 |
| C2×C7×AGL(1,ℤ/8) | 2 |
| C7×#535 | 2 |
| C7×#421 | 2 |
| C7×#418 | 2 |
| C7×#410 | 2 |
| C7×#344 | 2 |
| #2008 | 7 |