Finite Group DatabasebetaFinite Group Databasebeta← (C2×AGL(1,𝔽5))⋊Dic7 #11764630
Order 1120.
152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D7×AGL(1,𝔽5) |
| commutator | C2×C5×C7 | C42 |
| Fitting, socle | C22×C5×C7 | C2×C4 |
| prime radical, cosocle | C22×C7×D5 | C22 |
| hypocenter | C5×C7 | #95 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C2×C7⋊AGL(1,𝔽5) | C4 | semidirect |
| 10. C2×C7×AGL(1,𝔽5) | C4 | semidirect |
| 11. C2×Dic35 | C4 | semidirect |
| 12. C2×C5×Dic7 | C4 | semidirect |
| 51. C2×Dic7 | AGL(1,𝔽5) | semidirect |
| 58. C2×AGL(1,𝔽5) | Dic7 | semidirect |
| 63. C5×C7 | #95 | semidirect |
| 115. C7 | (C2×AGL(1,𝔽5))⋊C4 | semidirect |
| 116. C5 | C7⋊#95 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7⋊AGL(1,𝔽5) | 2 |
| C22×C7×AGL(1,𝔽5) | 2 |
| C2×D5×Dic7 | 2 |
| C7⋊#95 | 5 |
| (C2×AGL(1,𝔽5))⋊C4 | 7 |