Finite Group DatabasebetaFinite Group DatabasebetaOrder 200.
21 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #1044 | C2 | 1 | 3 | yes | maximal |
| 2 | C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 3 | C52 | Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 5 | D5 | — | 10 | 3 | no | — |
| 6 | Q8 | — | 25 | 1 | no | maximal, Sylow |
| 7 | C5 | — | 2 | 3 | no | — |
| 8 | C4 | — | 25 | 3 | no | — |
| 9 | C2 | — | 25 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, prime radical, cosocle | C5⋊D5 | C22 |
| Fitting, socle, hypocenter | C52 | Q8 |
| LCS | G ▷ C5⋊D5 ▷ C52 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C52 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #1044 | 2 |
| Q8 | 25 |