Finite Group DatabasebetaFinite Group DatabasebetaOrder 400.
64 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #6057 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D52⋊C2 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | D5⋊AGL(1,𝔽5) | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | D52 | C22 | 1 | 3 | yes | — |
| 5 | #1044 | C22 | 1 | 3 | yes | — |
| 6 | #1043 | C22 | 1 | 1 | yes | characteristic |
| 7 | C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 8 | C5×D5 | — | 2 | 3 | no | — |
| 9 | C2×AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 10 | C52 | #41 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C2×D5 | — | 10 | 3 | no | — |
| 12 | AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 13 | AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 14 | #41 | — | 25 | 1 | no | maximal, Sylow |
| 15 | C2×C5 | — | 10 | 3 | no | — |
| 16 | D5 | — | 2 | 3 | no | — |
| 17 | D5 | — | 10 | 3 | no | — |
| 18 | Q8 | — | 25 | 1 | no | — |
| 19 | D4 | — | 25 | 3 | no | — |
| 20 | C2×C4 | — | 25 | 3 | no | — |
| 21 | C5 | — | 2 | 3 | no | — |
| 22 | C22 | — | 25 | 3 | no | — |
| 23 | C4 | — | 25 | 1 | no | — |
| 24 | C4 | — | 25 | 3 | no | — |
| 25 | C2 | — | 10 | 3 | no | — |
| 26 | C2 | — | 25 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C5⋊D5 | C23 |
| Fitting, socle, hypocenter | C52 | #41 |
| LCS | G ▷ C5⋊D5 ▷ C52 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #6057 | C2 | semidirect |
| 2. D52⋊C2 | C2 | semidirect |
| 3. D5⋊AGL(1,𝔽5) | C2 | semidirect |
| 10. C52 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6057 | 2 |
| D52⋊C2 | 2 |
| D5⋊AGL(1,𝔽5) | 2 |
| #41 | 25 |