Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
120 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 80 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×C5×C7 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23×C5×C7 |
| prime radical | C5×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C7 ◁ C2×C4×C5×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C5×C7×Q8 | C2 | semidirect |
| 3. C5×C7×Q16 | C2 | semidirect |
| 4. C5×C7×SD16 | C2 | semidirect |
| 5. C5×C7×M16 | C2 | semidirect |
| 15. C7×#137 | C5 | direct (N × H) |
| 16. C5×#137 | C7 | direct (N × H) |
| 24. C2×C7×Q8 | C2×C5 | semidirect |
| 25. C7×Q16 | C2×C5 | semidirect |
| 26. C7×SD16 | C2×C5 | semidirect |
| 27. C7×M16 | C2×C5 | semidirect |
| 29. C2×C5×Q8 | C2×C7 | semidirect |
| 30. C5×Q16 | C2×C7 | semidirect |
| 31. C5×SD16 | C2×C7 | semidirect |
| 32. C5×M16 | C2×C7 | semidirect |
| 54. C5×C7 | #137 | direct (N × H) |
| 55. #137 | C5×C7 | direct (N × H) |
| 69. C2×Q8 | C2×C5×C7 | semidirect |
| 70. Q16 | C2×C5×C7 | semidirect |
| 71. SD16 | C2×C5×C7 | semidirect |
| 72. M16 | C2×C5×C7 | semidirect |
| 88. C7 | C5×#137 | direct (N × H) |
| 89. C5 | C7×#137 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C7×#41 | 2 |
| C2×C5×C7×Q8 | 2 |
| C5×C7×Q16 | 2 |
| C5×C7×SD16 | 2 |
| C5×C7×M16 | 2 |
| C7×#137 | 5 |
| C5×#137 | 7 |