Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 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 | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#125 | C2 | semidirect |
| 3. C7×#479 | C2 | semidirect |
| 4. C7×#478 | C2 | semidirect |
| 5. C7×#428 | C2 | semidirect |
| 6. C7×#423 | C2 | semidirect |
| 27. #3080 | C7 | direct (N × H) |
| 53. C2×#125 | C2×C7 | semidirect |
| 55. #479 | C2×C7 | semidirect |
| 56. #478 | C2×C7 | semidirect |
| 57. #428 | C2×C7 | semidirect |
| 58. #423 | C2×C7 | semidirect |
| 163. C7 | #3080 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#125 | 2 |
| C4×C7×#41 | 2 |
| C7×#479 | 2 |
| C7×#478 | 2 |
| C7×#428 | 2 |
| C7×#423 | 2 |
| C7×#417 | 2 |
| #3080 | 7 |