Finite Group DatabasebetaFinite Group DatabasebetaOrder 400.
23 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×#168 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×C52 | C4 | 1 | 1 | yes | characteristic |
| 3 | #674 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 4 | C5×C16 | — | 5 | 1 | no | maximal |
| 5 | C2×C52 | C8 | 1 | 1 | yes | characteristic |
| 6 | C5×C8 | — | 5 | 1 | no | — |
| 7 | #168 | C2×C5 | 1 | 1 | yes | characteristic |
| 8 | C52 | C16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C4×C5 | — | 4 | 1 | no | — |
| 10 | C4×C5 | C4×C5 | 1 | 1 | yes | characteristic |
| 11 | C4×C5 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 12 | C16 | — | 5 | 1 | no | Sylow |
| 13 | C2×C5 | — | 4 | 1 | no | — |
| 14 | C2×C5 | C5×C8 | 1 | 1 | yes | characteristic |
| 15 | C2×C5 | #170 | 1 | 1 | yes | characteristic |
| 16 | C8 | — | 5 | 1 | no | — |
| 17 | C5 | — | 4 | 1 | no | — |
| 18 | C5 | C5×C16 | 1 | 1 | yes | characteristic, complemented |
| 19 | C5 | #674 | 1 | 1 | yes | characteristic, direct product factor |
| 20 | C4 | C5×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 21 | C2 | C5×#170 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C5×C16 |
| Frattini | C4 | C5×AGL(1,𝔽5) |
| Fitting | C4×C52 | C4 |
| socle, prime radical | C2×C52 | C8 |
| cosocle | #168 | C2×C5 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C4×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #674 | C5 | direct (N × H) |
| 8. C52 | C16 | semidirect |
| 18. C5 | C5×C16 | semidirect |
| 19. C5 | #674 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#168 | 2 |
| #674 | 5 |
| C5×C16 | 5 |