Finite Group DatabasebetaFinite Group DatabasebetaOrder 2000.
41 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #6029⋊C5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×H(3,𝔽5) | C4 | 1 | 1 | yes | characteristic |
| 3 | #87764 | C5 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×#674 | — | 5 | 1 | no | maximal |
| 5 | C2×H(3,𝔽5) | C8 | 1 | 1 | yes | characteristic |
| 6 | #6029 | C2×C5 | 1 | 1 | yes | characteristic |
| 7 | C5×#168 | — | 5 | 1 | no | — |
| 8 | H(3,𝔽5) | C16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C4×C52 | — | 4 | 1 | no | — |
| 10 | C4×C52 | C4×C5 | 1 | 1 | yes | characteristic |
| 11 | C4×C52 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 12 | #674 | — | 5 | 1 | no | — |
| 13 | #674 | — | 25 | 1 | no | — |
| 14 | C5×C16 | — | 25 | 1 | no | — |
| 15 | C2×C52 | — | 4 | 1 | no | — |
| 16 | C2×C52 | C5×C8 | 1 | 1 | yes | characteristic |
| 17 | C2×C52 | #170 | 1 | 1 | yes | characteristic |
| 18 | C5×C8 | — | 25 | 1 | no | — |
| 19 | #168 | — | 5 | 1 | no | — |
| 20 | #168 | — | 25 | 1 | no | — |
| 21 | C52 | — | 4 | 1 | no | — |
| 22 | C52 | C5×C16 | 1 | 1 | yes | characteristic, complemented |
| 23 | C52 | #674 | 1 | 1 | yes | characteristic, complemented |
| 24 | C4×C5 | — | 5 | 1 | no | — |
| 25 | C4×C5 | — | 5 | 1 | no | — |
| 26 | C4×C5 | — | 20 | 1 | no | — |
| 27 | C4×C5 | C5×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 28 | C16 | — | 25 | 1 | no | Sylow |
| 29 | C2×C5 | — | 5 | 1 | no | — |
| 30 | C2×C5 | — | 5 | 1 | no | — |
| 31 | C2×C5 | — | 20 | 1 | no | — |
| 32 | C2×C5 | C5×#170 | 1 | 1 | yes | characteristic |
| 33 | C8 | — | 25 | 1 | no | — |
| 34 | C5 | — | 5 | 1 | no | — |
| 35 | C5 | — | 5 | 1 | no | — |
| 36 | C5 | — | 20 | 1 | no | — |
| 37 | C5 | C5×#674 | 1 | 1 | yes | characteristic |
| 38 | C4 | #1043⋊C5 | 1 | 1 | yes | characteristic |
| 39 | C2 | #6033⋊C5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #1043⋊C5 |
| commutator, hypocenter | C52 | C5×C16 |
| Frattini | C4×C5 | C5×AGL(1,𝔽5) |
| Fitting | C4×H(3,𝔽5) | C4 |
| socle | C2×C5 | C5×#170 |
| prime radical | C2×H(3,𝔽5) | C8 |
| cosocle | #6029 | C2×C5 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #87764 | C5 | semidirect |
| 8. H(3,𝔽5) | C16 | semidirect |
| 22. C52 | C5×C16 | semidirect |
| 23. C52 | #674 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6029⋊C5 | 2 |
| #87764 | 5 |
| C5×#674 | 5 |