Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
414 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C7 |
| prime radical | C7×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×#117 | C2 | semidirect |
| 3. C7×#485 | C2 | semidirect |
| 4. C7×#482 | C2 | semidirect |
| 5. C7×#428 | C2 | semidirect |
| 6. C7×#417 | C2 | semidirect |
| 7. C7×#407 | C2 | semidirect |
| 22. C7×#107 | C22 | semidirect |
| 23. C7×#106 | C22 | semidirect |
| 24. C7×#103 | C22 | semidirect |
| 25. C7×#102 | C22 | semidirect |
| 27. #3093 | C7 | direct (N × H) |
| 54. C2×#117 | C2×C7 | semidirect |
| 55. #485 | C2×C7 | semidirect |
| 56. #482 | C2×C7 | semidirect |
| 57. #428 | C2×C7 | semidirect |
| 58. #417 | C2×C7 | semidirect |
| 59. #407 | C2×C7 | semidirect |
| 96. #107 | C22×C7 | semidirect |
| 97. #106 | C22×C7 | semidirect |
| 98. #103 | C22×C7 | semidirect |
| 99. #102 | C22×C7 | semidirect |
| 168. C7 | #3093 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#525 | 2 |
| C2×C7×#117 | 2 |
| C7×#485 | 2 |
| C7×#482 | 2 |
| C7×#428 | 2 |
| C7×#417 | 2 |
| C7×#407 | 2 |
| #3093 | 7 |