Finite Group DatabasebetaFinite Group DatabasebetaOrder 24625.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C52×C197 | C5 | 1 | 1 | yes | maximal, characteristic |
| 2 | C25×C197 | C5 | 1 | 5 | yes | maximal, complemented |
| 3 | C5×C197 | — | 5 | 1 | no | — |
| 4 | C5×C197 | C52 | 1 | 1 | yes | characteristic |
| 5 | C197 | 5−1+2 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 6 | 5−1+2 | C197 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 7 | C52 | C5×C197 | 1 | 1 | yes | characteristic |
| 8 | C25 | C5×C197 | 1 | 5 | yes | complemented |
| 9 | C5 | — | 5 | 1 | no | — |
| 10 | C5 | C52×C197 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C197 | C52 |
| commutator, Frattini, cosocle | C5 | C52×C197 |
| prime radical | C52×C197 | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C197 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C25×C197 | C5 | semidirect |
| 5. C197 | 5−1+2 | direct (N × H) |
| 6. 5−1+2 | C197 | direct (N × H) |
| 8. C25 | C5×C197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×C197 | 5 |
| C25×C197 | 5 |
| 5−1+2 | 197 |