Finite Group DatabasebetaFinite Group DatabasebetaOrder 2400.
314 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 34 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×D5×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C5⋊(C5⋊S4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×A4⋊Dic5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11717095 | — | 3 | 1 | no | maximal, Hall |
| 5 | C2×C52×A4 | C22 | 1 | 1 | yes | characteristic |
| 6 | C5×D5×A4 | — | 2 | 1 | no | — |
| 7 | C5⋊(C5⋊S4) | — | 2 | 1 | no | — |
| 8 | #10597123 | — | 4 | 1 | no | maximal |
| 9 | #91958 | — | 5 | 1 | no | maximal |
| 10 | #91905 | — | 5 | 1 | no | maximal |
| 11 | C23×C5×D5 | D3 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×C22⋊(C5⋊D5) | — | 3 | 1 | no | — |
| 13 | C2×#6038 | — | 3 | 1 | no | — |
| 14 | C2×#6038 | — | 3 | 1 | no | — |
| 15 | C5×C22⋊Dic5 | — | 3 | 1 | no | — |
| 16 | #87778 | — | 3 | 1 | no | — |
| 17 | #87778 | — | 3 | 1 | no | — |
| 18 | C2×C3⋊(C5⋊D5) | — | 4 | 1 | no | — |
| 19 | C2×C3×C5×D5 | — | 4 | 1 | no | — |
| 20 | C52×A4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C5×Dic15 | — | 4 | 1 | no | — |
| 22 | C22×C5×A4 | — | 5 | 1 | no | — |
| 23 | C2×D5×A4 | D5 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×C5⋊S4 | — | 5 | 1 | no | — |
| 25 | C2×C5⋊S4 | — | 5 | 1 | no | — |
| 26 | C2×C5⋊S4 | — | 10 | 2 | no | — |
| 27 | A4⋊Dic5 | D5 | 1 | 1 | yes | characteristic, complemented |
| 28 | C5×SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 29 | C23×C52 | C2×D3 | 1 | 1 | yes | characteristic |
| 30 | C22×C5⋊D5 | — | 3 | 1 | no | — |
| 31 | C22×C5×D5 | — | 2 | 1 | no | — |
| 32 | C22×C5×D5 | — | 3 | 1 | no | — |
| 33 | C22×C5×D5 | — | 3 | 1 | no | — |
| 34 | C22×C5×D5 | — | 6 | 1 | no | — |
| 35 | C22⋊(C5⋊D5) | — | 6 | 1 | no | — |
| 36 | C22⋊(C5⋊D5) | — | 6 | 1 | no | — |
| 37 | C2×#1039 | — | 3 | 1 | no | — |
| 38 | C2×C5×Dic5 | — | 3 | 1 | no | — |
| 39 | C2×C5×Dic5 | — | 3 | 1 | no | — |
| 40 | #6038 | — | 6 | 2 | no | — |
| 41 | #6038 | — | 6 | 1 | no | — |
| 42 | #6038 | — | 6 | 1 | no | — |
| 43 | #4099 | — | 15 | 1 | no | — |
| 44 | #4028 | — | 15 | 1 | no | — |
| 45 | C2×C3×C52 | — | 4 | 1 | no | — |
| 46 | C3⋊(C5⋊D5) | — | 8 | 1 | no | — |
| 47 | C3×C5×D5 | — | 8 | 1 | no | — |
| 48 | C2×C5×A4 | — | 2 | 2 | no | — |
| 49 | C2×C5×A4 | — | 10 | 1 | no | — |
| 50 | C2×C5×A4 | C2×D5 | 1 | 1 | yes | characteristic |
| 51 | C2×C5×A4 | C2×D5 | 1 | 1 | yes | characteristic |
| 52 | D5×A4 | — | 2 | 1 | no | — |
| 53 | C5⋊S4 | — | 10 | 4 | no | — |
| 54 | C5⋊S4 | — | 10 | 1 | no | — |
| 55 | C5⋊S4 | — | 10 | 1 | no | — |
| 56 | #1220 | — | 20 | 1 | no | — |
| 57 | #1202 | — | 20 | 1 | no | — |
| 58 | C22×C52 | — | 3 | 1 | no | — |
| 59 | C22×C52 | — | 3 | 1 | no | — |
| 60 | C22×C52 | #67 | 1 | 1 | yes | characteristic, complemented |
| 61 | C2×C5⋊D5 | — | 6 | 1 | no | — |
| 62 | C2×C5⋊D5 | — | 6 | 1 | no | — |
| 63 | C2×C5⋊D5 | — | 6 | 1 | no | — |
| 64 | C2×C5×D5 | — | 3 | 2 | no | — |
| 65 | C2×C5×D5 | — | 3 | 1 | no | — |
| 66 | C2×C5×D5 | — | 6 | 1 | no | — |
| 67 | C2×C5×D5 | — | 6 | 1 | no | — |
| 68 | C2×C5×D5 | — | 6 | 1 | no | — |
| 69 | C2×C5×D5 | S4 | 1 | 1 | yes | characteristic, complemented |
| 70 | #1039 | — | 6 | 1 | no | — |
| 71 | C5×Dic5 | — | 6 | 1 | no | — |
| 72 | C5×Dic5 | — | 6 | 1 | no | — |
| 73 | GL(2,ℤ/4) | — | 25 | 1 | no | Hall |
| 74 | C24×C5 | — | 5 | 1 | no | — |
| 75 | C23×D5 | D15 | 1 | 1 | yes | characteristic, complemented |
| 76 | C2×C22⋊D5 | — | 15 | 1 | no | — |
| 77 | C2×C22⋊D5 | — | 15 | 1 | no | — |
| 78 | C2×C22⋊D5 | — | 15 | 1 | no | — |
| 79 | C2×C22⋊D5 | — | 15 | 1 | no | — |
| 80 | C2×C22⋊D5 | — | 30 | 2 | no | — |
| 81 | C2×D20 | — | 15 | 1 | no | — |
| 82 | C2×D20 | — | 15 | 1 | no | — |
| 83 | C5×C22⋊C4 | — | 15 | 1 | no | — |
| 84 | C22⋊Dic5 | — | 3 | 1 | no | — |
| 85 | C22⋊Dic5 | — | 15 | 1 | no | — |
| 86 | C22⋊Dic5 | — | 15 | 1 | no | — |
| 87 | #685 | — | 15 | 1 | no | — |
| 88 | #685 | — | 15 | 1 | no | — |
| 89 | C3×C52 | — | 4 | 1 | no | Hall |
| 90 | C22×C3×C5 | — | 20 | 1 | no | — |
| 91 | C2×D15 | — | 20 | 1 | no | — |
| 92 | C2×D15 | — | 20 | 1 | no | — |
| 93 | C2×D15 | — | 40 | 2 | no | — |
| 94 | C2×C3×D5 | — | 4 | 1 | no | — |
| 95 | C5×A4 | — | 2 | 2 | no | — |
| 96 | C5×A4 | D20 | 1 | 1 | yes | characteristic, complemented |
| 97 | C5×A4 | C22⋊D5 | 1 | 1 | yes | characteristic, complemented |
| 98 | Dic15 | — | 4 | 1 | no | — |
| 99 | C5×Dic3 | — | 20 | 1 | no | — |
| 100 | C2×C52 | — | 3 | 1 | no | — |
| 101 | C2×C52 | — | 3 | 1 | no | — |
| 102 | C2×C52 | C2×S4 | 1 | 1 | yes | characteristic |
| 103 | C5⋊D5 | — | 12 | 1 | no | — |
| 104 | C5×D5 | — | 2 | 1 | no | — |
| 105 | C5×D5 | — | 6 | 1 | no | — |
| 106 | C22×A4 | — | 5 | 1 | no | — |
| 107 | C2×S4 | — | 25 | 1 | no | — |
| 108 | SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 109 | C23×C5 | — | 2 | 2 | no | — |
| 110 | C23×C5 | — | 10 | 1 | no | — |
| 111 | C23×C5 | — | 15 | 1 | no | — |
| 112 | C23×C5 | — | 15 | 1 | no | — |
| 113 | C23×C5 | — | 30 | 1 | no | — |
| 114 | C23×C5 | D3×D5 | 1 | 1 | yes | characteristic |
| 115 | C23×C5 | C2×D15 | 1 | 1 | yes | characteristic |
| 116 | C22×D5 | — | 2 | 1 | no | — |
| 117 | C22×D5 | — | 3 | 1 | no | — |
| 118 | C22×D5 | — | 3 | 1 | no | — |
| 119 | C22×D5 | — | 6 | 1 | no | — |
| 120 | C22×D5 | — | 15 | 1 | no | — |
| 121 | C22×D5 | — | 15 | 1 | no | — |
| 122 | C22×D5 | — | 30 | 2 | no | — |
| 123 | C2×C4×C5 | — | 15 | 1 | no | — |
| 124 | C2×C4×C5 | — | 15 | 1 | no | — |
| 125 | C22⋊D5 | — | 30 | 1 | no | — |
| 126 | C22⋊D5 | — | 30 | 2 | no | — |
| 127 | C22⋊D5 | — | 30 | 1 | no | — |
| 128 | C22⋊D5 | — | 30 | 4 | no | — |
| 129 | C22⋊D5 | — | 30 | 1 | no | — |
| 130 | C22⋊D5 | — | 30 | 1 | no | — |
| 131 | C22⋊D5 | — | 30 | 4 | no | — |
| 132 | C22⋊D5 | — | 30 | 1 | no | — |
| 133 | C22⋊D5 | — | 30 | 1 | no | — |
| 134 | C2×Dic5 | — | 3 | 1 | no | — |
| 135 | C2×Dic5 | — | 3 | 1 | no | — |
| 136 | C2×Dic5 | — | 15 | 1 | no | — |
| 137 | C2×Dic5 | — | 15 | 1 | no | — |
| 138 | C2×Dic5 | — | 30 | 2 | no | — |
| 139 | D20 | — | 30 | 2 | no | — |
| 140 | D20 | — | 30 | 1 | no | — |
| 141 | D20 | — | 30 | 1 | no | — |
| 142 | #120 | — | 75 | 1 | no | Sylow |
| 143 | C2×C3×C5 | — | 4 | 1 | no | — |
| 144 | C2×C3×C5 | — | 4 | 1 | no | — |
| 145 | C2×C3×C5 | — | 8 | 2 | no | — |
| 146 | C2×C3×C5 | — | 40 | 1 | no | — |
| 147 | D15 | — | 40 | 1 | no | — |
| 148 | D15 | — | 40 | 1 | no | — |
| 149 | D15 | — | 40 | 4 | no | — |
| 150 | C3×D5 | — | 8 | 1 | no | — |
| 151 | C52 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 152 | C2×A4 | — | 10 | 1 | no | — |
| 153 | C2×A4 | D52 | 1 | 1 | yes | characteristic |
| 154 | S4 | — | 50 | 1 | no | — |
| 155 | #67 | — | 100 | 1 | no | — |
| 156 | C22×C5 | — | 2 | 2 | no | — |
| 157 | C22×C5 | — | 3 | 1 | no | — |
| 158 | C22×C5 | — | 3 | 1 | no | — |
| 159 | C22×C5 | — | 3 | 1 | no | — |
| 160 | C22×C5 | — | 3 | 1 | no | — |
| 161 | C22×C5 | — | 5 | 1 | no | — |
| 162 | C22×C5 | — | 6 | 2 | no | — |
| 163 | C22×C5 | — | 6 | 2 | no | — |
| 164 | C22×C5 | — | 15 | 2 | no | — |
| 165 | C22×C5 | — | 15 | 1 | no | — |
| 166 | C22×C5 | — | 30 | 1 | no | — |
| 167 | C22×C5 | — | 30 | 1 | no | — |
| 168 | C22×C5 | — | 30 | 1 | no | — |
| 169 | C22×C5 | #1202 | 1 | 1 | yes | characteristic, complemented |
| 170 | C22×C5 | #1220 | 1 | 1 | yes | characteristic, complemented |
| 171 | C2×D5 | — | 3 | 2 | no | — |
| 172 | C2×D5 | — | 3 | 1 | no | — |
| 173 | C2×D5 | — | 6 | 1 | no | — |
| 174 | C2×D5 | — | 6 | 1 | no | — |
| 175 | C2×D5 | — | 6 | 1 | no | — |
| 176 | C2×D5 | — | 30 | 1 | no | — |
| 177 | C2×D5 | — | 30 | 1 | no | — |
| 178 | C2×D5 | — | 30 | 1 | no | — |
| 179 | C2×D5 | — | 30 | 1 | no | — |
| 180 | C2×D5 | — | 30 | 4 | no | — |
| 181 | C2×D5 | — | 30 | 1 | no | — |
| 182 | C2×D5 | — | 30 | 1 | no | — |
| 183 | C2×D5 | — | 60 | 2 | no | — |
| 184 | C2×D5 | — | 60 | 2 | no | — |
| 185 | C2×D5 | C5⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 186 | C4×C5 | — | 30 | 1 | no | — |
| 187 | C4×C5 | — | 30 | 1 | no | — |
| 188 | Dic5 | — | 6 | 1 | no | — |
| 189 | Dic5 | — | 6 | 1 | no | — |
| 190 | Dic5 | — | 30 | 1 | no | — |
| 191 | Dic5 | — | 30 | 1 | no | — |
| 192 | Dic5 | — | 30 | 4 | no | — |
| 193 | C24 | — | 5 | 1 | no | — |
| 194 | C2×D4 | — | 75 | 1 | no | — |
| 195 | C2×D4 | — | 75 | 1 | no | — |
| 196 | C2×D4 | — | 75 | 1 | no | — |
| 197 | C22⋊C4 | — | 15 | 1 | no | — |
| 198 | C22⋊C4 | — | 75 | 1 | no | — |
| 199 | C22⋊C4 | — | 75 | 1 | no | — |
| 200 | C3×C5 | — | 4 | 1 | no | — |
| 201 | C3×C5 | — | 4 | 1 | no | — |
| 202 | C3×C5 | — | 8 | 2 | no | — |
| 203 | C22×C3 | — | 20 | 1 | no | — |
| 204 | C2×D3 | — | 100 | 1 | no | — |
| 205 | A4 | #6038 | 1 | 1 | yes | characteristic, complemented |
| 206 | Dic3 | — | 20 | 1 | no | — |
| 207 | C2×C5 | — | 2 | 2 | no | — |
| 208 | C2×C5 | — | 3 | 1 | no | — |
| 209 | C2×C5 | — | 3 | 1 | no | — |
| 210 | C2×C5 | — | 3 | 1 | no | — |
| 211 | C2×C5 | — | 3 | 1 | no | — |
| 212 | C2×C5 | — | 6 | 2 | no | — |
| 213 | C2×C5 | — | 6 | 2 | no | — |
| 214 | C2×C5 | — | 10 | 1 | no | — |
| 215 | C2×C5 | — | 30 | 1 | no | — |
| 216 | C2×C5 | D5×S4 | 1 | 1 | yes | characteristic |
| 217 | C2×C5 | C2×C5⋊S4 | 1 | 1 | yes | characteristic |
| 218 | D5 | — | 2 | 1 | no | — |
| 219 | D5 | — | 6 | 1 | no | — |
| 220 | D5 | — | 60 | 4 | no | — |
| 221 | D5 | — | 60 | 1 | no | — |
| 222 | D5 | — | 60 | 1 | no | — |
| 223 | C23 | — | 10 | 1 | no | — |
| 224 | C23 | — | 15 | 1 | no | — |
| 225 | C23 | — | 15 | 1 | no | — |
| 226 | C23 | — | 30 | 1 | no | — |
| 227 | C23 | — | 75 | 1 | no | — |
| 228 | C23 | D5×D15 | 1 | 1 | yes | characteristic |
| 229 | D4 | — | 150 | 1 | no | — |
| 230 | D4 | — | 150 | 2 | no | — |
| 231 | D4 | — | 150 | 1 | no | — |
| 232 | D4 | — | 150 | 1 | no | — |
| 233 | D4 | — | 150 | 1 | no | — |
| 234 | C2×C4 | — | 15 | 1 | no | — |
| 235 | C2×C4 | — | 15 | 1 | no | — |
| 236 | C2×C4 | — | 75 | 1 | no | — |
| 237 | C2×C3 | — | 4 | 1 | no | — |
| 238 | C2×C3 | — | 40 | 1 | no | — |
| 239 | D3 | — | 200 | 1 | no | — |
| 240 | C5 | — | 2 | 2 | no | — |
| 241 | C5 | #91905 | 1 | 1 | yes | characteristic, complemented |
| 242 | C5 | #91958 | 1 | 1 | yes | characteristic, complemented |
| 243 | C22 | — | 3 | 1 | no | — |
| 244 | C22 | — | 3 | 1 | no | — |
| 245 | C22 | — | 5 | 1 | no | — |
| 246 | C22 | — | 15 | 2 | no | — |
| 247 | C22 | — | 15 | 1 | no | — |
| 248 | C22 | — | 30 | 1 | no | — |
| 249 | C22 | — | 30 | 1 | no | — |
| 250 | C22 | — | 30 | 1 | no | — |
| 251 | C22 | — | 150 | 1 | no | — |
| 252 | C22 | — | 150 | 1 | no | — |
| 253 | C22 | — | 150 | 1 | no | — |
| 254 | C22 | #10597123 | 1 | 1 | yes | characteristic, complemented |
| 255 | C4 | — | 30 | 1 | no | — |
| 256 | C4 | — | 30 | 1 | no | — |
| 257 | C4 | — | 150 | 1 | no | — |
| 258 | C3 | — | 4 | 1 | no | Sylow |
| 259 | C2 | — | 3 | 1 | no | — |
| 260 | C2 | — | 3 | 1 | no | — |
| 261 | C2 | — | 10 | 1 | no | — |
| 262 | C2 | — | 30 | 1 | no | — |
| 263 | C2 | — | 300 | 1 | no | — |
| 264 | C2 | D5×C5⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C5⋊S4 |
| commutator, cosocle | C2×C52×A4 | C22 |
| Fitting, socle | C23×C52 | C2×D3 |
| hypocenter | C52×A4 | D4 |
| LCS | G ▷ C2×C52×A4 ▷ C52×A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×D5×A4 | C2 | semidirect |
| 2. C2×C5⋊(C5⋊S4) | C2 | semidirect |
| 3. C5×A4⋊Dic5 | C2 | semidirect |
| 11. C23×C5×D5 | D3 | semidirect |
| 20. C52×A4 | D4 | semidirect |
| 23. C2×D5×A4 | D5 | semidirect |
| 27. A4⋊Dic5 | D5 | semidirect |
| 60. C22×C52 | #67 | semidirect |
| 69. C2×C5×D5 | S4 | semidirect |
| 75. C23×D5 | D15 | semidirect |
| 96. C5×A4 | D20 | semidirect |
| 97. C5×A4 | C22⋊D5 | semidirect |
| 151. C52 | GL(2,ℤ/4) | semidirect |
| 169. C22×C5 | #1202 | semidirect |
| 170. C22×C5 | #1220 | semidirect |
| 185. C2×D5 | C5⋊S4 | semidirect |
| 205. A4 | #6038 | semidirect |
| 241. C5 | #91905 | semidirect |
| 242. C5 | #91958 | semidirect |
| 254. C22 | #10597123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×D5×A4 | 2 |
| C2×C5⋊(C5⋊S4) | 2 |
| C5×A4⋊Dic5 | 2 |
| #11717095 | 3 |
| #10597123 | 4 |
| #91958 | 5 |
| #91905 | 5 |