Finite Group DatabasebetaFinite Group DatabasebetaOrder 200.
34 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D52 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1042 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×AGL(1,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5⋊D5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C5×D5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C5×D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×AGL(1,𝔽5) | — | 5 | 1 | no | maximal |
| 8 | C4×D5 | — | 5 | 1 | no | maximal |
| 9 | C52 | C2×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C2×D5 | — | 5 | 1 | no | — |
| 11 | C2×D5 | — | 5 | 1 | no | — |
| 12 | AGL(1,𝔽5) | — | 5 | 1 | no | — |
| 13 | AGL(1,𝔽5) | D5 | 1 | 1 | yes | characteristic, direct product factor |
| 14 | C4×C5 | — | 5 | 1 | no | — |
| 15 | Dic5 | — | 5 | 1 | no | — |
| 16 | C2×C5 | — | 5 | 1 | no | — |
| 17 | C2×C5 | — | 5 | 1 | no | — |
| 18 | D5 | — | 5 | 1 | no | — |
| 19 | D5 | — | 5 | 1 | no | — |
| 20 | D5 | — | 20 | 1 | no | — |
| 21 | D5 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, direct product factor |
| 22 | D5 | C2×D5 | 1 | 1 | yes | characteristic |
| 23 | C2×C4 | — | 25 | 1 | no | Sylow |
| 24 | C5 | — | 4 | 1 | no | — |
| 25 | C5 | C4×D5 | 1 | 1 | yes | characteristic, complemented |
| 26 | C5 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 27 | C22 | — | 25 | 1 | no | — |
| 28 | C4 | — | 5 | 1 | no | — |
| 29 | C4 | — | 25 | 1 | no | — |
| 30 | C2 | — | 5 | 1 | no | — |
| 31 | C2 | — | 5 | 1 | no | — |
| 32 | C2 | — | 25 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52 | C2×C4 |
| prime radical | D52 | C2 |
| cosocle | C5×D5 | C22 |
| LCS | G ▷ C52 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1042 | C2 | semidirect |
| 3. C5×AGL(1,𝔽5) | C2 | semidirect |
| 4. C5⋊D5 | C4 | semidirect |
| 5. C5×D5 | C4 | semidirect |
| 9. C52 | C2×C4 | semidirect |
| 13. AGL(1,𝔽5) | D5 | direct (N × H) |
| 21. D5 | AGL(1,𝔽5) | direct (N × H) |
| 25. C5 | C4×D5 | semidirect |
| 26. C5 | C2×AGL(1,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D52 | 2 |
| #1042 | 2 |
| C5×AGL(1,𝔽5) | 2 |
| C2×AGL(1,𝔽5) | 5 |
| C4×D5 | 5 |