Finite Group DatabasebetaFinite Group DatabasebetaOrder 26900.
48 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5⋊D5×C269 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C52×C269 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | AGL(1,𝔽5)×C269 | — | 5 | 6 | no | maximal |
| 4 | D5×C269 | — | 5 | 6 | no | — |
| 5 | C5×C269 | AGL(1,𝔽5) | 1 | 6 | yes | complemented |
| 6 | C4×C269 | — | 25 | 1 | no | Hall |
| 7 | C2×C269 | — | 25 | 1 | no | — |
| 8 | C269 | #1043 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | #1043 | C269 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 10 | C5⋊D5 | C2×C269 | 1 | 1 | yes | characteristic |
| 11 | C52 | C4×C269 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 13 | D5 | — | 5 | 6 | no | — |
| 14 | C5 | AGL(1,𝔽5)×C269 | 1 | 6 | yes | complemented |
| 15 | C4 | — | 25 | 1 | no | Sylow |
| 16 | C2 | — | 25 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C269 | #1043 |
| commutator, hypocenter | C52 | C4×C269 |
| Frattini | C1 | G |
| Fitting, socle | C52×C269 | C4 |
| prime radical | C5⋊D5×C269 | C2 |
| cosocle | C5⋊D5 | C2×C269 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C269 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C52×C269 | C4 | semidirect |
| 5. C5×C269 | AGL(1,𝔽5) | semidirect |
| 8. C269 | #1043 | direct (N × H) |
| 9. #1043 | C269 | direct (N × H) |
| 11. C52 | C4×C269 | semidirect |
| 14. C5 | AGL(1,𝔽5)×C269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5⋊D5×C269 | 2 |
| AGL(1,𝔽5)×C269 | 5 |
| #1043 | 269 |