Finite Group DatabasebetaFinite Group DatabasebetaOrder 625.
25 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C25 | C5 | 1 | 5 | yes | maximal |
| 2 | C5×C25 | C5 | 1 | 1 | yes | maximal, characteristic |
| 3 | C52 | C52 | 1 | 1 | yes | characteristic |
| 4 | C25 | — | 5 | 5 | no | — |
| 5 | C25 | C25 | 1 | 5 | yes | complemented |
| 6 | C5 | C5×C25 | 1 | 1 | yes | characteristic |
| 7 | C5 | 5−1+2 | 1 | 5 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C52 | C52 |
| commutator | C5 | C5×C25 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52 ◁ G | |
| LECS | G ▷ C52 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C25 | C25 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C25 | 5 |