Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
260 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 28 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×C5⋊(#127⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11751554 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×#64853 | — | 3 | 1 | no | maximal, Hall |
| 4 | C2×A4⋊AGL(1,𝔽5) | — | 4 | 1 | no | maximal |
| 5 | C2×C5×C42⋊C3 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C5⋊(#127⋊C3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C5⋊(#127⋊C3) | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#4626 | — | 5 | 1 | no | maximal, Hall |
| 9 | C2×#4020 | D3 | 1 | 1 | yes | characteristic |
| 10 | C2×#3999 | — | 3 | 1 | no | — |
| 11 | C2×#3999 | — | 3 | 1 | no | — |
| 12 | #64853 | — | 3 | 2 | no | — |
| 13 | #64853 | — | 3 | 2 | no | — |
| 14 | C2×D5×A4 | — | 4 | 1 | no | — |
| 15 | A4⋊AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 16 | C5×C42⋊C3 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×#127⋊C3 | — | 5 | 1 | no | — |
| 18 | #4626 | — | 5 | 2 | no | — |
| 19 | C22×D20 | — | 3 | 1 | no | — |
| 20 | C22×D20 | — | 3 | 1 | no | — |
| 21 | C2×C22⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 22 | C2×C22⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 23 | C2×C42×C5 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 24 | #4020 | — | 3 | 1 | no | — |
| 25 | #4020 | — | 3 | 1 | no | — |
| 26 | #4020 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 27 | #4020 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | #3999 | — | 3 | 2 | no | — |
| 29 | #3999 | — | 3 | 2 | no | — |
| 30 | #3999 | — | 6 | 1 | no | — |
| 31 | #3999 | — | 6 | 1 | no | — |
| 32 | C2×#353 | — | 15 | 1 | no | Sylow |
| 33 | C2×C5×A4 | — | 4 | 1 | no | — |
| 34 | C2×ΓL(1,𝔽16) | — | 16 | 1 | no | — |
| 35 | D5×A4 | — | 4 | 1 | no | — |
| 36 | D5×A4 | — | 4 | 1 | no | — |
| 37 | C2×SL(2,ℤ/4) | — | 20 | 1 | no | — |
| 38 | #127⋊C3 | — | 5 | 1 | no | — |
| 39 | #127⋊C3 | — | 5 | 1 | no | — |
| 40 | C2×C42⋊C3 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 41 | C23×D5 | — | 3 | 1 | no | — |
| 42 | C23×D5 | S4 | 1 | 1 | yes | characteristic |
| 43 | C22×AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 44 | C22×AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 45 | C22×C4×C5 | — | 3 | 1 | no | — |
| 46 | C2×D20 | — | 3 | 1 | no | — |
| 47 | C2×D20 | — | 3 | 1 | no | — |
| 48 | C2×D20 | — | 3 | 1 | no | — |
| 49 | C2×D20 | — | 3 | 1 | no | — |
| 50 | C2×D20 | — | 3 | 1 | no | — |
| 51 | C2×D20 | — | 3 | 1 | no | — |
| 52 | C2×D20 | — | 6 | 1 | no | — |
| 53 | C2×D20 | — | 12 | 1 | no | — |
| 54 | C2×D20 | — | 12 | 1 | no | — |
| 55 | C2×D20 | — | 12 | 1 | no | — |
| 56 | C2×D20 | — | 12 | 1 | no | — |
| 57 | C22⋊AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 58 | C22⋊AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 59 | C22⋊AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 60 | C22⋊AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 61 | C42×C5 | — | 3 | 1 | no | — |
| 62 | C42×C5 | C2×Dic3 | 1 | 1 | yes | characteristic, complemented |
| 63 | C2×#127 | — | 5 | 1 | no | — |
| 64 | C2×#99 | — | 15 | 1 | no | — |
| 65 | C2×#99 | — | 15 | 1 | no | — |
| 66 | #353 | — | 15 | 2 | no | — |
| 67 | #353 | — | 15 | 2 | no | — |
| 68 | C2×C3×D5 | — | 16 | 1 | no | — |
| 69 | C5×A4 | — | 4 | 1 | no | — |
| 70 | ΓL(1,𝔽16) | — | 16 | 2 | no | — |
| 71 | C22×A4 | — | 20 | 1 | no | — |
| 72 | SL(2,ℤ/4) | — | 20 | 2 | no | — |
| 73 | C42⋊C3 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 74 | C23×C5 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 75 | C22×D5 | — | 3 | 1 | no | — |
| 76 | C22×D5 | — | 3 | 1 | no | — |
| 77 | C22×D5 | — | 6 | 1 | no | — |
| 78 | C22×D5 | — | 6 | 1 | no | — |
| 79 | C22×D5 | — | 6 | 1 | no | — |
| 80 | C22×D5 | — | 6 | 1 | no | — |
| 81 | C22×D5 | — | 12 | 1 | no | — |
| 82 | C22×D5 | — | 12 | 1 | no | — |
| 83 | C22×D5 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 84 | C22×D5 | C2×S4 | 1 | 1 | yes | characteristic |
| 85 | C2×AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 86 | C2×AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 87 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 88 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 89 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 90 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 91 | C2×C4×C5 | — | 3 | 1 | no | — |
| 92 | C2×C4×C5 | — | 3 | 1 | no | — |
| 93 | C2×C4×C5 | — | 6 | 1 | no | — |
| 94 | C2×C4×C5 | — | 6 | 1 | no | — |
| 95 | D20 | — | 12 | 1 | no | — |
| 96 | D20 | — | 12 | 1 | no | — |
| 97 | D20 | — | 12 | 1 | no | — |
| 98 | D20 | — | 12 | 1 | no | — |
| 99 | D20 | — | 12 | 1 | no | — |
| 100 | D20 | — | 12 | 1 | no | — |
| 101 | D20 | — | 12 | 2 | no | — |
| 102 | C22×D4 | — | 15 | 1 | no | — |
| 103 | C22×D4 | — | 15 | 1 | no | — |
| 104 | #127 | — | 5 | 1 | no | — |
| 105 | #127 | — | 5 | 1 | no | — |
| 106 | #127 | — | 15 | 1 | no | — |
| 107 | #127 | — | 15 | 1 | no | — |
| 108 | C2×C22⋊C4 | — | 30 | 1 | no | — |
| 109 | C2×C22⋊C4 | — | 30 | 1 | no | — |
| 110 | C2×C42 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, complemented |
| 111 | #99 | — | 15 | 2 | no | — |
| 112 | #99 | — | 15 | 2 | no | — |
| 113 | #99 | — | 30 | 1 | no | — |
| 114 | #99 | — | 30 | 1 | no | — |
| 115 | C2×C3×C5 | — | 16 | 1 | no | — |
| 116 | C3×D5 | — | 16 | 1 | no | — |
| 117 | C3×D5 | — | 16 | 1 | no | — |
| 118 | C2×A4 | — | 4 | 1 | no | — |
| 119 | C2×A4 | — | 20 | 1 | no | — |
| 120 | C2×A4 | — | 20 | 1 | no | — |
| 121 | C2×Dic3 | — | 80 | 1 | no | — |
| 122 | C22×C5 | — | 3 | 1 | no | — |
| 123 | C22×C5 | — | 3 | 1 | no | — |
| 124 | C22×C5 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 125 | C2×D5 | — | 4 | 1 | no | — |
| 126 | C2×D5 | — | 6 | 1 | no | — |
| 127 | C2×D5 | — | 6 | 1 | no | — |
| 128 | C2×D5 | — | 6 | 1 | no | — |
| 129 | C2×D5 | — | 6 | 1 | no | — |
| 130 | C2×D5 | — | 12 | 1 | no | — |
| 131 | C2×D5 | — | 12 | 1 | no | — |
| 132 | C2×D5 | — | 12 | 1 | no | — |
| 133 | C2×D5 | — | 12 | 1 | no | — |
| 134 | C2×D5 | — | 12 | 1 | no | — |
| 135 | C2×D5 | — | 24 | 1 | no | — |
| 136 | AGL(1,𝔽5) | — | 24 | 2 | no | — |
| 137 | AGL(1,𝔽5) | — | 24 | 2 | no | — |
| 138 | C4×C5 | — | 6 | 1 | no | — |
| 139 | C4×C5 | — | 6 | 1 | no | — |
| 140 | C24 | — | 5 | 1 | no | — |
| 141 | C24 | — | 15 | 1 | no | — |
| 142 | C2×D4 | — | 15 | 1 | no | — |
| 143 | C2×D4 | — | 15 | 1 | no | — |
| 144 | C2×D4 | — | 15 | 1 | no | — |
| 145 | C2×D4 | — | 15 | 1 | no | — |
| 146 | C2×D4 | — | 15 | 1 | no | — |
| 147 | C2×D4 | — | 15 | 1 | no | — |
| 148 | C2×D4 | — | 30 | 1 | no | — |
| 149 | C2×D4 | — | 60 | 1 | no | — |
| 150 | C2×D4 | — | 60 | 1 | no | — |
| 151 | C2×D4 | — | 60 | 1 | no | — |
| 152 | C2×D4 | — | 60 | 1 | no | — |
| 153 | C22×C4 | — | 3 | 1 | no | — |
| 154 | C22×C4 | — | 60 | 1 | no | — |
| 155 | C22×C4 | — | 60 | 1 | no | — |
| 156 | C22⋊C4 | — | 30 | 2 | no | — |
| 157 | C22⋊C4 | — | 30 | 2 | no | — |
| 158 | C22⋊C4 | — | 60 | 1 | no | — |
| 159 | C22⋊C4 | — | 60 | 1 | no | — |
| 160 | C42 | — | 3 | 1 | no | — |
| 161 | C42 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, complemented |
| 162 | C3×C5 | — | 16 | 1 | no | Hall |
| 163 | C22×C3 | — | 80 | 1 | no | — |
| 164 | A4 | — | 4 | 1 | no | — |
| 165 | Dic3 | — | 80 | 2 | no | — |
| 166 | C2×C5 | — | 3 | 1 | no | — |
| 167 | C2×C5 | — | 3 | 1 | no | — |
| 168 | C2×C5 | #4626 | 1 | 1 | yes | characteristic, complemented |
| 169 | D5 | — | 4 | 1 | no | — |
| 170 | D5 | — | 4 | 1 | no | — |
| 171 | D5 | — | 12 | 1 | no | — |
| 172 | D5 | — | 12 | 1 | no | — |
| 173 | C23 | — | 5 | 1 | no | — |
| 174 | C23 | — | 5 | 1 | no | — |
| 175 | C23 | — | 15 | 1 | no | — |
| 176 | C23 | — | 15 | 1 | no | — |
| 177 | C23 | — | 30 | 1 | no | — |
| 178 | C23 | — | 30 | 1 | no | — |
| 179 | C23 | — | 30 | 1 | no | — |
| 180 | C23 | — | 30 | 1 | no | — |
| 181 | C23 | — | 60 | 1 | no | — |
| 182 | C23 | — | 60 | 1 | no | — |
| 183 | C23 | A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 184 | D4 | — | 60 | 1 | no | — |
| 185 | D4 | — | 60 | 1 | no | — |
| 186 | D4 | — | 60 | 1 | no | — |
| 187 | D4 | — | 60 | 1 | no | — |
| 188 | D4 | — | 60 | 1 | no | — |
| 189 | D4 | — | 60 | 1 | no | — |
| 190 | D4 | — | 60 | 2 | no | — |
| 191 | C2×C4 | — | 3 | 1 | no | — |
| 192 | C2×C4 | — | 3 | 1 | no | — |
| 193 | C2×C4 | — | 6 | 1 | no | — |
| 194 | C2×C4 | — | 6 | 1 | no | — |
| 195 | C2×C4 | — | 60 | 2 | no | — |
| 196 | C2×C4 | — | 60 | 2 | no | — |
| 197 | C2×C4 | — | 120 | 1 | no | — |
| 198 | C2×C4 | — | 120 | 1 | no | — |
| 199 | C2×C4 | — | 120 | 1 | no | — |
| 200 | C2×C4 | — | 120 | 1 | no | — |
| 201 | C2×C3 | — | 16 | 1 | no | — |
| 202 | C2×C3 | — | 80 | 1 | no | — |
| 203 | C2×C3 | — | 80 | 1 | no | — |
| 204 | C5 | C2×#4626 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 205 | C22 | — | 3 | 1 | no | — |
| 206 | C22 | — | 3 | 1 | no | — |
| 207 | C22 | — | 20 | 1 | no | — |
| 208 | C22 | — | 30 | 1 | no | — |
| 209 | C22 | — | 30 | 1 | no | — |
| 210 | C22 | — | 30 | 1 | no | — |
| 211 | C22 | — | 30 | 1 | no | — |
| 212 | C22 | — | 60 | 1 | no | — |
| 213 | C22 | — | 60 | 1 | no | — |
| 214 | C22 | — | 60 | 1 | no | — |
| 215 | C22 | — | 60 | 1 | no | — |
| 216 | C22 | — | 60 | 1 | no | — |
| 217 | C22 | — | 120 | 1 | no | — |
| 218 | C22 | C2×A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 219 | C4 | — | 6 | 1 | no | — |
| 220 | C4 | — | 6 | 1 | no | — |
| 221 | C4 | — | 120 | 2 | no | — |
| 222 | C4 | — | 120 | 2 | no | — |
| 223 | C3 | — | 16 | 1 | no | Sylow |
| 224 | C2 | — | 3 | 1 | no | — |
| 225 | C2 | — | 3 | 1 | no | — |
| 226 | C2 | — | 20 | 1 | no | — |
| 227 | C2 | — | 20 | 1 | no | — |
| 228 | C2 | — | 60 | 1 | no | — |
| 229 | C2 | — | 60 | 1 | no | — |
| 230 | C2 | #11751554 | 1 | 1 | yes | characteristic, direct product factor |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11751554 | C2 | direct (N × H) |
| 5. C2×C5×C42⋊C3 | C4 | semidirect |
| 6. C5⋊(#127⋊C3) | C4 | semidirect |
| 16. C5×C42⋊C3 | C2×C4 | semidirect |
| 23. C2×C42×C5 | Dic3 | semidirect |
| 26. #4020 | Dic3 | semidirect |
| 40. C2×C42⋊C3 | AGL(1,𝔽5) | semidirect |
| 62. C42×C5 | C2×Dic3 | semidirect |
| 73. C42⋊C3 | C2×AGL(1,𝔽5) | semidirect |
| 110. C2×C42 | ΓL(1,𝔽16) | semidirect |
| 161. C42 | C2×ΓL(1,𝔽16) | semidirect |
| 168. C2×C5 | #4626 | semidirect |
| 204. C5 | C2×#4626 | semidirect |
| 230. C2 | #11751554 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5⋊(#127⋊C3) | 2 |
| #11751554 | 2 |
| C2×#64853 | 3 |
| C2×A4⋊AGL(1,𝔽5) | 4 |
| C2×#4626 | 5 |