Finite Group DatabasebetaFinite Group Databasebeta← (C2×AGL(1,𝔽5))⋊(C5⋊D5) #14522722
Order 2000.
332 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 70 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 | C2 | AGL(1,𝔽5)×C5⋊D5 |
| commutator, Fitting, socle | C2×C53 | C2×C4 |
| prime radical | C2×D5×C5⋊D5 | C2 |
| cosocle | C2×C52×D5 | C22 |
| hypocenter | C53 | C22⋊C4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#92559 | C2 | semidirect |
| 3. C2×C52×AGL(1,𝔽5) | C2 | semidirect |
| 4. C2×C5⋊(C5⋊D5) | C4 | semidirect |
| 5. C2×C5×C5⋊D5 | C4 | semidirect |
| 21. C2×C5×AGL(1,𝔽5) | D5 | semidirect |
| 24. C53 | C22⋊C4 | semidirect |
| 31. C2×C5⋊D5 | AGL(1,𝔽5) | semidirect |
| 57. C2×AGL(1,𝔽5) | C5⋊D5 | semidirect |
| 61. C52 | #685 | semidirect |
| 62. C52 | C22⋊AGL(1,𝔽5) | semidirect |
| 95. C5 | #87808 | semidirect |
| 96. C5 | (C2×AGL(1,𝔽5))⋊D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D5×C5⋊D5 | 2 |
| C2×#92559 | 2 |
| C2×C52×AGL(1,𝔽5) | 2 |
| (C2×AGL(1,𝔽5))⋊D5 | 5 |
| #87808 | 5 |