Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
626 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 78 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 | D5×#994 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11756726 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11756725 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10617559 | — | 3 | 1 | no | maximal, Hall |
| 5 | C5×#994 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | Q8×ΓL(1,𝔽16) | — | 4 | 1 | no | maximal |
| 7 | D5×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 8 | C5⋊#994 | C4 | 1 | 3 | yes | complemented |
| 9 | #91886 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #91886 | C22 | 1 | 3 | yes | complemented |
| 11 | #85528 | — | 5 | 1 | no | maximal, Hall |
| 12 | D5×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 13 | #66266 | — | 3 | 1 | no | — |
| 14 | D5⋊#131 | — | 3 | 1 | no | — |
| 15 | #65794 | — | 3 | 3 | no | — |
| 16 | #65793 | — | 3 | 3 | no | — |
| 17 | #65767 | — | 3 | 3 | no | — |
| 18 | C2×#4007 | — | 3 | 3 | no | — |
| 19 | C3×Q8×D5 | — | 4 | 1 | no | — |
| 20 | C5×#41⋊C3 | C2×C4 | 1 | 3 | yes | complemented |
| 21 | C4⋊ΓL(1,𝔽16) | — | 4 | 3 | no | — |
| 22 | C4×ΓL(1,𝔽16) | — | 4 | 3 | no | — |
| 23 | D5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 24 | C5⋊(#41⋊C3) | C2×C4 | 1 | 3 | yes | complemented |
| 25 | C2×#994 | — | 5 | 1 | no | — |
| 26 | #5426 | — | 5 | 3 | no | — |
| 27 | #5424 | — | 5 | 3 | no | — |
| 28 | C5×H(5,𝔽2) | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 29 | #4149 | — | 3 | 1 | no | — |
| 30 | #4149 | Dic3 | 1 | 3 | yes | complemented |
| 31 | D5×#41 | — | 3 | 1 | no | — |
| 32 | D5×#41 | C2×D3 | 1 | 3 | yes | — |
| 33 | #4144 | — | 3 | 3 | no | — |
| 34 | C2×D4×D5 | — | 3 | 3 | no | — |
| 35 | Q8×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 36 | #4133 | — | 6 | 1 | no | — |
| 37 | D4×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 38 | #4131 | — | 6 | 3 | no | — |
| 39 | #4130 | — | 3 | 3 | no | — |
| 40 | C2×C4⋊AGL(1,𝔽5) | — | 3 | 3 | no | — |
| 41 | D5⋊M16 | — | 3 | 3 | no | — |
| 42 | C2×D5⋊C8 | — | 3 | 3 | no | — |
| 43 | #4009 | — | 3 | 3 | no | — |
| 44 | #4009 | — | 3 | 1 | no | — |
| 45 | #4007 | — | 3 | 6 | no | — |
| 46 | #4007 | — | 3 | 3 | no | — |
| 47 | #4007 | — | 3 | 3 | no | — |
| 48 | #2897 | — | 15 | 1 | no | Sylow |
| 49 | C2×ΓL(1,𝔽16) | — | 4 | 1 | no | — |
| 50 | C2×ΓL(1,𝔽16) | — | 4 | 3 | no | — |
| 51 | C3×C5×Q8 | — | 4 | 1 | no | — |
| 52 | C3×C4×D5 | — | 4 | 3 | no | — |
| 53 | C3×Dic10 | — | 4 | 3 | no | — |
| 54 | C5×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | #994 | — | 5 | 3 | no | — |
| 56 | #994 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 57 | C2×#41⋊C3 | — | 5 | 3 | no | — |
| 58 | Q8×Dic3 | — | 20 | 1 | no | — |
| 59 | #859 | — | 5 | 1 | no | — |
| 60 | #859 | — | 5 | 3 | no | — |
| 61 | C23×D5 | — | 6 | 1 | no | — |
| 62 | C22×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 63 | C5×#41 | — | 3 | 1 | no | — |
| 64 | C5×#41 | C2×Dic3 | 1 | 3 | yes | — |
| 65 | C2×C5×D4 | — | 3 | 3 | no | — |
| 66 | C2×C22⋊D5 | — | 6 | 3 | no | — |
| 67 | #713 | — | 3 | 1 | no | — |
| 68 | #713 | C2×Dic3 | 1 | 3 | yes | — |
| 69 | Q8×D5 | S4 | 1 | 1 | yes | characteristic |
| 70 | Q8×D5 | C22×D3 | 1 | 1 | yes | characteristic |
| 71 | #711 | — | 3 | 3 | no | — |
| 72 | #711 | — | 3 | 3 | no | — |
| 73 | D4×D5 | — | 3 | 6 | no | — |
| 74 | D4×D5 | — | 3 | 3 | no | — |
| 75 | D4×D5 | — | 3 | 3 | no | — |
| 76 | D4×D5 | — | 6 | 3 | no | — |
| 77 | #709 | — | 3 | 3 | no | — |
| 78 | #709 | — | 3 | 3 | no | — |
| 79 | C2×D20 | — | 3 | 3 | no | — |
| 80 | C2×C4×D5 | — | 3 | 3 | no | — |
| 81 | C22⋊AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 82 | #704 | — | 6 | 3 | no | — |
| 83 | C2×#170 | — | 6 | 3 | no | — |
| 84 | C4⋊AGL(1,𝔽5) | — | 3 | 3 | no | — |
| 85 | C4⋊AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 86 | C4⋊AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 87 | C4×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 88 | #700 | — | 6 | 3 | no | — |
| 89 | D5⋊C8 | — | 3 | 1 | no | — |
| 90 | D5⋊C8 | — | 3 | 3 | no | — |
| 91 | D5⋊C8 | — | 6 | 3 | no | — |
| 92 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 93 | C2×#131 | — | 15 | 1 | no | — |
| 94 | #520 | — | 15 | 1 | no | — |
| 95 | #418 | — | 15 | 3 | no | — |
| 96 | #417 | — | 15 | 3 | no | — |
| 97 | #416 | — | 15 | 3 | no | — |
| 98 | C2×#102 | — | 15 | 3 | no | — |
| 99 | C2×C3×D5 | — | 4 | 1 | no | — |
| 100 | ΓL(1,𝔽16) | — | 4 | 2 | no | — |
| 101 | ΓL(1,𝔽16) | — | 8 | 3 | no | — |
| 102 | C3×C4×C5 | — | 4 | 3 | no | — |
| 103 | C3×Dic5 | — | 4 | 3 | no | — |
| 104 | C2×C3×Q8 | — | 20 | 1 | no | — |
| 105 | #41⋊C3 | — | 5 | 3 | no | — |
| 106 | #41⋊C3 | C2×AGL(1,𝔽5) | 1 | 3 | yes | complemented |
| 107 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 108 | C4⋊Dic3 | — | 20 | 3 | no | — |
| 109 | C4×Dic3 | — | 20 | 3 | no | — |
| 110 | C23×C5 | — | 6 | 1 | no | — |
| 111 | C22×D5 | — | 3 | 3 | no | — |
| 112 | C22×D5 | — | 6 | 3 | no | — |
| 113 | C22×D5 | — | 24 | 1 | no | — |
| 114 | C22×D5 | — | 24 | 1 | no | — |
| 115 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 116 | C2×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 117 | C2×AGL(1,𝔽5) | — | 24 | 3 | no | — |
| 118 | C5×Q8 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 119 | C5×Q8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 120 | C5×D4 | — | 3 | 3 | no | — |
| 121 | C5×D4 | — | 3 | 3 | no | — |
| 122 | C2×C4×C5 | — | 3 | 3 | no | — |
| 123 | C22⋊D5 | — | 3 | 6 | no | — |
| 124 | C22⋊D5 | — | 6 | 3 | no | — |
| 125 | C2×Dic5 | — | 3 | 3 | no | — |
| 126 | D20 | — | 3 | 3 | no | — |
| 127 | D20 | — | 3 | 3 | no | — |
| 128 | C4×D5 | — | 3 | 1 | no | — |
| 129 | C4×D5 | — | 3 | 3 | no | — |
| 130 | C4×D5 | — | 3 | 3 | no | — |
| 131 | C4×D5 | C2×S4 | 1 | 3 | yes | — |
| 132 | Dic10 | — | 3 | 1 | no | — |
| 133 | Dic10 | SL(2,ℤ/4) | 1 | 3 | yes | — |
| 134 | #170 | — | 6 | 1 | no | — |
| 135 | #170 | — | 6 | 3 | no | — |
| 136 | H(5,𝔽2) | — | 5 | 3 | no | — |
| 137 | H(5,𝔽2) | — | 15 | 3 | no | — |
| 138 | H(5,𝔽2) | — | 15 | 1 | no | — |
| 139 | H(5,𝔽2) | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, complemented |
| 140 | C2×#41 | — | 5 | 3 | no | — |
| 141 | C2×#41 | — | 15 | 1 | no | — |
| 142 | C22×D4 | — | 15 | 3 | no | — |
| 143 | #131 | — | 30 | 3 | no | — |
| 144 | #131 | — | 30 | 1 | no | — |
| 145 | C2×M16 | — | 15 | 3 | no | — |
| 146 | C22×C8 | — | 15 | 3 | no | — |
| 147 | C4×Q8 | — | 30 | 1 | no | — |
| 148 | C4×D4 | — | 30 | 3 | no | — |
| 149 | #117 | — | 15 | 3 | no | — |
| 150 | C2×C4⋊C4 | — | 15 | 3 | no | — |
| 151 | #103 | — | 15 | 1 | no | — |
| 152 | #103 | — | 15 | 3 | no | — |
| 153 | #102 | — | 15 | 3 | no | — |
| 154 | #102 | — | 15 | 6 | no | — |
| 155 | #102 | — | 15 | 3 | no | — |
| 156 | C2×C3×C5 | — | 4 | 1 | no | — |
| 157 | C3×D5 | — | 4 | 1 | no | — |
| 158 | C3×D5 | — | 4 | 1 | no | — |
| 159 | C3×Q8 | — | 4 | 1 | no | — |
| 160 | C3×Q8 | — | 20 | 3 | no | — |
| 161 | C2×C3×C4 | — | 20 | 3 | no | — |
| 162 | C2×Dic3 | — | 20 | 1 | no | — |
| 163 | C2×Dic3 | — | 20 | 3 | no | — |
| 164 | SL(2,𝔽3) | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 165 | C22×C5 | — | 3 | 3 | no | — |
| 166 | C22×C5 | — | 24 | 1 | no | — |
| 167 | C2×D5 | — | 3 | 3 | no | — |
| 168 | C2×D5 | — | 6 | 3 | no | — |
| 169 | C2×D5 | — | 6 | 3 | no | — |
| 170 | C2×D5 | — | 24 | 3 | no | — |
| 171 | C2×D5 | C22×S4 | 1 | 1 | yes | characteristic |
| 172 | AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 173 | AGL(1,𝔽5) | — | 12 | 3 | no | — |
| 174 | C4×C5 | — | 3 | 1 | no | — |
| 175 | C4×C5 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 176 | Dic5 | — | 3 | 1 | no | — |
| 177 | Dic5 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 178 | C24 | — | 30 | 1 | no | — |
| 179 | #41 | — | 3 | 1 | no | — |
| 180 | #41 | — | 5 | 3 | no | — |
| 181 | #41 | — | 15 | 3 | no | — |
| 182 | #41 | — | 15 | 3 | no | — |
| 183 | #41 | — | 15 | 3 | no | — |
| 184 | #41 | — | 15 | 1 | no | — |
| 185 | #41 | — | 15 | 3 | no | — |
| 186 | #41 | C2×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 187 | C2×Q8 | — | 5 | 1 | no | — |
| 188 | C2×Q8 | — | 5 | 1 | no | — |
| 189 | C2×D4 | — | 3 | 3 | no | — |
| 190 | C2×D4 | — | 15 | 3 | no | — |
| 191 | C2×D4 | — | 15 | 3 | no | — |
| 192 | C2×D4 | — | 15 | 6 | no | — |
| 193 | C2×D4 | — | 15 | 3 | no | — |
| 194 | C2×D4 | — | 30 | 3 | no | — |
| 195 | C2×D4 | — | 30 | 3 | no | — |
| 196 | C22×C4 | — | 15 | 3 | no | — |
| 197 | C22×C4 | — | 30 | 3 | no | — |
| 198 | M16 | — | 30 | 3 | no | — |
| 199 | M16 | — | 30 | 3 | no | — |
| 200 | C2×C8 | — | 15 | 1 | no | — |
| 201 | C2×C8 | — | 15 | 3 | no | — |
| 202 | C2×C8 | — | 30 | 3 | no | — |
| 203 | C2×C8 | — | 30 | 3 | no | — |
| 204 | C4⋊C4 | — | 15 | 3 | no | — |
| 205 | C4⋊C4 | — | 15 | 1 | no | — |
| 206 | C4⋊C4 | — | 30 | 3 | no | — |
| 207 | C22⋊C4 | — | 30 | 3 | no | — |
| 208 | C42 | — | 30 | 3 | no | — |
| 209 | C3×C5 | — | 4 | 1 | no | Hall |
| 210 | C22×C3 | — | 20 | 1 | no | — |
| 211 | C3×C4 | — | 4 | 3 | no | — |
| 212 | C3×C4 | — | 20 | 3 | no | — |
| 213 | Dic3 | — | 20 | 2 | no | — |
| 214 | Dic3 | — | 40 | 3 | no | — |
| 215 | C2×C5 | — | 6 | 3 | no | — |
| 216 | C2×C5 | C22×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 217 | D5 | — | 6 | 3 | no | — |
| 218 | D5 | #5924 | 1 | 1 | yes | characteristic |
| 219 | D5 | #5925 | 1 | 1 | yes | characteristic |
| 220 | C23 | — | 6 | 1 | no | — |
| 221 | C23 | — | 15 | 3 | no | — |
| 222 | C23 | — | 30 | 3 | no | — |
| 223 | C23 | — | 120 | 1 | no | — |
| 224 | C23 | — | 120 | 1 | no | — |
| 225 | Q8 | — | 5 | 3 | no | — |
| 226 | Q8 | — | 15 | 1 | no | — |
| 227 | Q8 | A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 228 | Q8 | C22×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 229 | D4 | — | 3 | 3 | no | — |
| 230 | D4 | — | 3 | 3 | no | — |
| 231 | D4 | — | 15 | 3 | no | — |
| 232 | D4 | — | 15 | 6 | no | — |
| 233 | D4 | — | 15 | 3 | no | — |
| 234 | D4 | — | 30 | 3 | no | — |
| 235 | C2×C4 | — | 3 | 3 | no | — |
| 236 | C2×C4 | — | 5 | 3 | no | — |
| 237 | C2×C4 | — | 15 | 3 | no | — |
| 238 | C2×C4 | — | 15 | 1 | no | — |
| 239 | C2×C4 | — | 15 | 3 | no | — |
| 240 | C2×C4 | — | 15 | 3 | no | — |
| 241 | C2×C4 | — | 30 | 1 | no | — |
| 242 | C2×C4 | — | 30 | 3 | no | — |
| 243 | C2×C4 | — | 120 | 3 | no | — |
| 244 | C8 | — | 30 | 1 | no | — |
| 245 | C8 | — | 30 | 3 | no | — |
| 246 | C2×C3 | — | 4 | 1 | no | — |
| 247 | C2×C3 | — | 20 | 1 | no | — |
| 248 | C2×C3 | — | 20 | 1 | no | — |
| 249 | C5 | #85528 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 250 | C22 | — | 3 | 3 | no | — |
| 251 | C22 | — | 5 | 1 | no | — |
| 252 | C22 | — | 15 | 3 | no | — |
| 253 | C22 | — | 24 | 1 | no | — |
| 254 | C22 | — | 30 | 3 | no | — |
| 255 | C22 | — | 30 | 3 | no | — |
| 256 | C22 | — | 120 | 3 | no | — |
| 257 | C4 | — | 3 | 1 | no | — |
| 258 | C4 | — | 5 | 3 | no | — |
| 259 | C4 | — | 15 | 1 | no | — |
| 260 | C4 | — | 60 | 3 | no | — |
| 261 | C4 | — | 60 | 1 | no | — |
| 262 | C4 | C2×A4⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 263 | C3 | — | 4 | 1 | no | Sylow |
| 264 | C2 | — | 5 | 1 | no | — |
| 265 | C2 | — | 5 | 1 | no | — |
| 266 | C2 | — | 6 | 3 | no | — |
| 267 | C2 | — | 30 | 3 | no | — |
| 268 | C2 | C22×A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11756726 | C2 | semidirect |
| 3. #11756725 | C2 | semidirect |
| 5. C5×#994 | C4 | semidirect |
| 8. C5⋊#994 | C4 | semidirect |
| 9. #91886 | C22 | semidirect |
| 10. #91886 | C22 | semidirect |
| 20. C5×#41⋊C3 | C2×C4 | semidirect |
| 24. C5⋊(#41⋊C3) | C2×C4 | semidirect |
| 28. C5×H(5,𝔽2) | Dic3 | semidirect |
| 30. #4149 | Dic3 | semidirect |
| 56. #994 | AGL(1,𝔽5) | semidirect |
| 106. #41⋊C3 | C2×AGL(1,𝔽5) | semidirect |
| 139. H(5,𝔽2) | ΓL(1,𝔽16) | semidirect |
| 249. C5 | #85528 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×#994 | 2 |
| #11756726 | 2 |
| #11756725 | 2 |
| #10617559 | 3 |
| Q8×ΓL(1,𝔽16) | 4 |
| #85528 | 5 |