Finite Group DatabasebetaFinite Group DatabasebetaOrder 2400.
239 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×D5×A5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | A5⋊Dic5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×A5⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×C5×A5 | C22 | 1 | 1 | yes | characteristic |
| 5 | D5×A5 | C4 | 1 | 2 | yes | complemented |
| 6 | D5×SL(2,ℤ/4) | — | 5 | 1 | no | maximal |
| 7 | C2×A5⋊C4 | — | 5 | 1 | no | maximal |
| 8 | C2×D5×AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 9 | C5×A5 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×D5×A4 | — | 5 | 1 | no | — |
| 11 | C22×A5 | — | 5 | 1 | no | — |
| 12 | C4×D3×D5 | — | 10 | 1 | no | maximal |
| 13 | A4⋊Dic5 | — | 5 | 1 | no | — |
| 14 | C5×SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 15 | A5⋊C4 | — | 5 | 1 | no | — |
| 16 | A5⋊C4 | D5 | 1 | 1 | yes | characteristic, direct product factor |
| 17 | C2×D52 | — | 6 | 1 | no | — |
| 18 | C2×#1042 | — | 6 | 1 | no | — |
| 19 | C2×C5×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 20 | D5×AGL(1,𝔽5) | — | 6 | 4 | no | — |
| 21 | D5×C22⋊C4 | — | 15 | 1 | no | — |
| 22 | C2×C5×A4 | — | 5 | 1 | no | — |
| 23 | C2×D3×D5 | — | 10 | 1 | no | — |
| 24 | D5×A4 | — | 5 | 2 | no | — |
| 25 | C2×A5 | — | 5 | 2 | no | — |
| 26 | C2×A5 | C2×D5 | 1 | 1 | yes | characteristic |
| 27 | C4×D15 | — | 10 | 1 | no | — |
| 28 | C4×C5×D3 | — | 10 | 1 | no | — |
| 29 | C3×C4×D5 | — | 10 | 1 | no | — |
| 30 | #1200 | — | 10 | 1 | no | — |
| 31 | D3×Dic5 | — | 10 | 1 | no | — |
| 32 | D5×Dic3 | — | 10 | 1 | no | — |
| 33 | C2×C5⋊D5 | — | 6 | 1 | no | — |
| 34 | C2×C5×D5 | — | 6 | 1 | no | — |
| 35 | C2×C5×D5 | — | 6 | 1 | no | — |
| 36 | D52 | — | 6 | 2 | no | — |
| 37 | D52 | — | 6 | 2 | no | — |
| 38 | #1042 | — | 6 | 2 | no | — |
| 39 | C5×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 40 | C2×SL(2,ℤ/4) | — | 25 | 1 | no | Hall |
| 41 | C23×D5 | — | 5 | 1 | no | — |
| 42 | C22×AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 43 | C2×C4×D5 | — | 15 | 1 | no | — |
| 44 | C2×C4×D5 | — | 15 | 1 | no | — |
| 45 | C5×C22⋊C4 | — | 15 | 1 | no | — |
| 46 | C22⋊Dic5 | — | 15 | 1 | no | — |
| 47 | #685 | — | 15 | 1 | no | — |
| 48 | #685 | — | 15 | 1 | no | — |
| 49 | C2×D15 | — | 10 | 1 | no | — |
| 50 | C2×C5×D3 | — | 10 | 1 | no | — |
| 51 | C2×C3×D5 | — | 10 | 1 | no | — |
| 52 | C5×A4 | — | 5 | 1 | no | — |
| 53 | D3×D5 | — | 10 | 2 | no | — |
| 54 | D3×D5 | — | 10 | 2 | no | — |
| 55 | A5 | C4×D5 | 1 | 1 | yes | characteristic, complemented |
| 56 | C3×C4×C5 | — | 10 | 1 | no | — |
| 57 | Dic15 | — | 10 | 1 | no | — |
| 58 | C3×Dic5 | — | 10 | 1 | no | — |
| 59 | C5×Dic3 | — | 10 | 1 | no | — |
| 60 | C2×C52 | — | 6 | 1 | no | — |
| 61 | C5⋊D5 | — | 6 | 2 | no | — |
| 62 | C5×D5 | — | 6 | 1 | no | — |
| 63 | C5×D5 | — | 6 | 1 | no | — |
| 64 | C5×D5 | — | 6 | 2 | no | — |
| 65 | C22×A4 | — | 25 | 1 | no | — |
| 66 | C2×C4×D3 | — | 50 | 1 | no | — |
| 67 | SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 68 | SL(2,ℤ/4) | — | 25 | 1 | no | — |
| 69 | C23×C5 | — | 5 | 1 | no | — |
| 70 | C22×D5 | — | 5 | 2 | no | — |
| 71 | C22×D5 | — | 15 | 1 | no | — |
| 72 | C22×D5 | — | 15 | 1 | no | — |
| 73 | C22×D5 | — | 15 | 2 | no | — |
| 74 | C22×D5 | — | 30 | 1 | no | — |
| 75 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 76 | C2×AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 77 | C2×AGL(1,𝔽5) | — | 30 | 4 | no | — |
| 78 | C2×C4×C5 | — | 15 | 1 | no | — |
| 79 | C2×C4×C5 | — | 15 | 1 | no | — |
| 80 | C2×Dic5 | — | 15 | 1 | no | — |
| 81 | C2×Dic5 | — | 15 | 1 | no | — |
| 82 | C4×D5 | — | 10 | 1 | no | — |
| 83 | C4×D5 | — | 30 | 1 | no | — |
| 84 | C4×D5 | — | 30 | 2 | no | — |
| 85 | C2×C22⋊C4 | — | 75 | 1 | no | Sylow |
| 86 | C2×C3×C5 | — | 10 | 1 | no | — |
| 87 | D15 | — | 10 | 2 | no | — |
| 88 | C3×D5 | — | 10 | 2 | no | — |
| 89 | C5×D3 | — | 10 | 1 | no | — |
| 90 | C5×D3 | — | 10 | 1 | no | — |
| 91 | C52 | — | 6 | 1 | no | Sylow |
| 92 | C22×D3 | — | 50 | 1 | no | — |
| 93 | C2×A4 | — | 5 | 1 | no | — |
| 94 | C2×A4 | — | 25 | 2 | no | — |
| 95 | C2×C3×C4 | — | 50 | 1 | no | — |
| 96 | C2×Dic3 | — | 50 | 1 | no | — |
| 97 | C4×D3 | — | 10 | 1 | no | — |
| 98 | C4×D3 | — | 50 | 1 | no | — |
| 99 | C4×D3 | — | 50 | 1 | no | — |
| 100 | C4×D3 | — | 50 | 1 | no | — |
| 101 | C22×C5 | — | 5 | 1 | no | — |
| 102 | C22×C5 | — | 15 | 1 | no | — |
| 103 | C22×C5 | — | 15 | 1 | no | — |
| 104 | C22×C5 | — | 30 | 1 | no | — |
| 105 | C2×D5 | — | 6 | 1 | no | — |
| 106 | C2×D5 | — | 15 | 1 | no | — |
| 107 | C2×D5 | — | 15 | 2 | no | — |
| 108 | C2×D5 | — | 15 | 2 | no | — |
| 109 | C2×D5 | — | 15 | 2 | no | — |
| 110 | C2×D5 | — | 30 | 2 | no | — |
| 111 | C2×D5 | — | 30 | 2 | no | — |
| 112 | C2×D5 | — | 30 | 1 | no | — |
| 113 | C2×D5 | — | 30 | 1 | no | — |
| 114 | C2×D5 | — | 120 | 1 | no | — |
| 115 | C2×D5 | S5 | 1 | 1 | yes | characteristic |
| 116 | AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 117 | AGL(1,𝔽5) | — | 30 | 2 | no | — |
| 118 | C4×C5 | — | 10 | 1 | no | — |
| 119 | C4×C5 | — | 30 | 1 | no | — |
| 120 | Dic5 | — | 10 | 1 | no | — |
| 121 | Dic5 | — | 30 | 1 | no | — |
| 122 | C24 | — | 25 | 1 | no | — |
| 123 | C22×C4 | — | 75 | 1 | no | — |
| 124 | C22×C4 | — | 75 | 1 | no | — |
| 125 | C22⋊C4 | — | 15 | 1 | no | — |
| 126 | C22⋊C4 | — | 75 | 1 | no | — |
| 127 | C22⋊C4 | — | 75 | 1 | no | — |
| 128 | C22⋊C4 | — | 75 | 1 | no | — |
| 129 | C3×C5 | — | 10 | 1 | no | — |
| 130 | C22×C3 | — | 50 | 1 | no | — |
| 131 | C2×D3 | — | 10 | 1 | no | — |
| 132 | C2×D3 | — | 50 | 2 | no | — |
| 133 | C2×D3 | — | 50 | 2 | no | — |
| 134 | C2×D3 | — | 50 | 1 | no | — |
| 135 | A4 | — | 5 | 1 | no | — |
| 136 | C3×C4 | — | 10 | 1 | no | — |
| 137 | C3×C4 | — | 50 | 1 | no | — |
| 138 | Dic3 | — | 10 | 1 | no | — |
| 139 | Dic3 | — | 50 | 1 | no | — |
| 140 | C2×C5 | — | 6 | 1 | no | — |
| 141 | C2×C5 | — | 15 | 1 | no | — |
| 142 | C2×C5 | — | 15 | 1 | no | — |
| 143 | C2×C5 | — | 24 | 1 | no | — |
| 144 | C2×C5 | — | 30 | 2 | no | — |
| 145 | C2×C5 | C2×S5 | 1 | 1 | yes | characteristic |
| 146 | D5 | — | 6 | 1 | no | — |
| 147 | D5 | — | 6 | 1 | no | — |
| 148 | D5 | — | 15 | 2 | no | — |
| 149 | D5 | — | 30 | 2 | no | — |
| 150 | D5 | — | 120 | 2 | no | — |
| 151 | D5 | A5⋊C4 | 1 | 2 | yes | direct product factor |
| 152 | C23 | — | 5 | 1 | no | — |
| 153 | C23 | — | 25 | 2 | no | — |
| 154 | C23 | — | 75 | 1 | no | — |
| 155 | C23 | — | 75 | 1 | no | — |
| 156 | C23 | — | 75 | 2 | no | — |
| 157 | C2×C4 | — | 15 | 1 | no | — |
| 158 | C2×C4 | — | 15 | 1 | no | — |
| 159 | C2×C4 | — | 50 | 1 | no | — |
| 160 | C2×C4 | — | 75 | 1 | no | — |
| 161 | C2×C4 | — | 75 | 1 | no | — |
| 162 | C2×C4 | — | 150 | 1 | no | — |
| 163 | C2×C4 | — | 150 | 2 | no | — |
| 164 | C2×C3 | — | 10 | 1 | no | — |
| 165 | C2×C3 | — | 50 | 2 | no | — |
| 166 | D3 | — | 10 | 1 | no | — |
| 167 | D3 | — | 10 | 1 | no | — |
| 168 | D3 | — | 50 | 2 | no | — |
| 169 | C5 | — | 6 | 1 | no | — |
| 170 | C5 | — | 24 | 1 | no | — |
| 171 | C5 | C2×A5⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 172 | C22 | — | 5 | 1 | no | — |
| 173 | C22 | — | 5 | 1 | no | — |
| 174 | C22 | — | 15 | 1 | no | — |
| 175 | C22 | — | 15 | 1 | no | — |
| 176 | C22 | — | 75 | 2 | no | — |
| 177 | C22 | — | 75 | 2 | no | — |
| 178 | C22 | — | 75 | 2 | no | — |
| 179 | C22 | — | 75 | 1 | no | — |
| 180 | C22 | — | 150 | 1 | no | — |
| 181 | C4 | — | 10 | 1 | no | — |
| 182 | C4 | — | 30 | 1 | no | — |
| 183 | C4 | — | 50 | 1 | no | — |
| 184 | C4 | — | 150 | 1 | no | — |
| 185 | C3 | — | 10 | 1 | no | Sylow |
| 186 | C2 | — | 5 | 2 | no | — |
| 187 | C2 | — | 15 | 1 | no | — |
| 188 | C2 | — | 15 | 1 | no | — |
| 189 | C2 | — | 75 | 2 | no | — |
| 190 | C2 | D5×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×S5 |
| commutator, hypocenter | C5×A5 | C2×C4 |
| Fitting | C2×C5 | C2×S5 |
| socle, cosocle | C2×C5×A5 | C22 |
| prime radical | C2×D5×A5 | C2 |
| LCS | G ▷ C5×A5 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A5⋊Dic5 | C2 | semidirect |
| 3. C5×A5⋊C4 | C2 | semidirect |
| 5. D5×A5 | C4 | semidirect |
| 9. C5×A5 | C2×C4 | semidirect |
| 16. A5⋊C4 | D5 | direct (N × H) |
| 55. A5 | C4×D5 | semidirect |
| 151. D5 | A5⋊C4 | direct (N × H) |
| 171. C5 | C2×A5⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D5×A5 | 2 |
| A5⋊Dic5 | 2 |
| C5×A5⋊C4 | 2 |
| D5×SL(2,ℤ/4) | 5 |
| C2×A5⋊C4 | 5 |
| C2×D5×AGL(1,𝔽5) | 6 |
| C4×D3×D5 | 10 |