Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
232 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #11716805 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11716795 | C2 | 1 | 3 | yes | maximal |
| 3 | #11716776 | C2 | 1 | 3 | yes | maximal |
| 4 | C22×#1044 | C22 | 1 | 3 | yes | — |
| 5 | C22×#1043 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×#6036 | C22 | 1 | 3 | yes | — |
| 7 | C22×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 8 | C2×#1044 | — | 2 | 3 | no | — |
| 9 | C2×#1044 | — | 2 | 3 | no | — |
| 10 | C2×#1044 | — | 2 | 3 | no | — |
| 11 | C2×#1043 | — | 2 | 3 | no | — |
| 12 | C2×C5×Dic5 | — | 2 | 3 | no | — |
| 13 | #6036 | — | 2 | 3 | no | — |
| 14 | #6036 | — | 2 | 3 | no | — |
| 15 | (C2×AGL(1,𝔽5))⋊C4 | — | 10 | 3 | no | — |
| 16 | C22×C52 | #41 | 1 | 1 | yes | characteristic |
| 17 | C2×C5⋊D5 | #41 | 1 | 3 | yes | — |
| 18 | C2×C5⋊D5 | #41 | 1 | 3 | yes | — |
| 19 | #1044 | — | 4 | 3 | no | — |
| 20 | #1043 | — | 4 | 1 | no | — |
| 21 | C5×Dic5 | — | 4 | 3 | no | — |
| 22 | C22×AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 23 | C22×AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 24 | C2×C4×D5 | — | 10 | 3 | no | — |
| 25 | #401 | — | 25 | 1 | no | maximal, Sylow |
| 26 | C2×C52 | #126 | 1 | 3 | yes | — |
| 27 | C5⋊D5 | #126 | 1 | 1 | yes | characteristic |
| 28 | C5⋊D5 | #126 | 1 | 3 | yes | — |
| 29 | C22×D5 | — | 10 | 3 | no | — |
| 30 | C2×AGL(1,𝔽5) | — | 10 | 6 | no | — |
| 31 | C2×AGL(1,𝔽5) | — | 10 | 6 | no | — |
| 32 | C2×AGL(1,𝔽5) | — | 20 | 3 | no | — |
| 33 | C2×AGL(1,𝔽5) | — | 20 | 3 | no | — |
| 34 | C2×AGL(1,𝔽5) | — | 20 | 3 | no | — |
| 35 | C2×AGL(1,𝔽5) | — | 20 | 3 | no | — |
| 36 | C2×C4×C5 | — | 10 | 3 | no | — |
| 37 | C2×Dic5 | — | 2 | 3 | no | — |
| 38 | C4×D5 | — | 20 | 3 | no | — |
| 39 | C4×D5 | — | 20 | 3 | no | — |
| 40 | #95 | — | 25 | 3 | no | — |
| 41 | #95 | — | 25 | 3 | no | — |
| 42 | #95 | — | 25 | 1 | no | — |
| 43 | C52 | #401 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 44 | C22×C5 | — | 2 | 3 | no | — |
| 45 | C2×D5 | — | 10 | 3 | no | — |
| 46 | C2×D5 | — | 10 | 3 | no | — |
| 47 | C2×D5 | — | 10 | 3 | no | — |
| 48 | C2×D5 | — | 10 | 3 | no | — |
| 49 | C2×D5 | — | 10 | 3 | no | — |
| 50 | C2×D5 | — | 10 | 3 | no | — |
| 51 | AGL(1,𝔽5) | — | 20 | 6 | no | — |
| 52 | AGL(1,𝔽5) | — | 20 | 6 | no | — |
| 53 | C4×C5 | — | 20 | 3 | no | — |
| 54 | Dic5 | — | 4 | 3 | no | — |
| 55 | C22×C4 | — | 25 | 3 | no | — |
| 56 | C22×C4 | — | 25 | 3 | no | — |
| 57 | C22×C4 | — | 25 | 1 | no | — |
| 58 | C2×C5 | — | 2 | 3 | no | — |
| 59 | C2×C5 | — | 2 | 3 | no | — |
| 60 | C2×C5 | — | 2 | 3 | no | — |
| 61 | D5 | — | 10 | 3 | no | — |
| 62 | D5 | — | 10 | 3 | no | — |
| 63 | D5 | — | 10 | 3 | no | — |
| 64 | D5 | — | 10 | 3 | no | — |
| 65 | C23 | — | 25 | 1 | no | — |
| 66 | C2×C4 | — | 10 | 3 | no | — |
| 67 | C2×C4 | — | 50 | 3 | no | — |
| 68 | C2×C4 | — | 50 | 3 | no | — |
| 69 | C2×C4 | — | 50 | 3 | no | — |
| 70 | C2×C4 | — | 50 | 3 | no | — |
| 71 | C2×C4 | — | 50 | 3 | no | — |
| 72 | C2×C4 | — | 50 | 3 | no | — |
| 73 | C5 | — | 2 | 3 | no | — |
| 74 | C22 | — | 25 | 3 | no | — |
| 75 | C22 | — | 25 | 3 | no | — |
| 76 | C22 | (D5⋊AGL(1,𝔽5))⋊C2 | 1 | 1 | yes | characteristic |
| 77 | C4 | — | 20 | 3 | no | — |
| 78 | C4 | — | 100 | 3 | no | — |
| 79 | C4 | — | 100 | 1 | no | — |
| 80 | C2 | — | 25 | 3 | no | — |
| 81 | C2 | — | 25 | 1 | no | — |
| 82 | C2 | #11717314 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | (D5⋊AGL(1,𝔽5))⋊C2 |
| commutator, prime radical, cosocle | C22×C5⋊D5 | C23 |
| Fitting, socle | C22×C52 | #41 |
| hypocenter | C52 | #401 |
| LCS | G ▷ C22×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 43. C52 | #401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11716805 | 2 |
| #11716795 | 2 |
| #11716776 | 2 |
| #401 | 25 |