Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
472 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #41×C5⋊D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11716992 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11716936 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×#3969 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11716611 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C52×#41 | C22 | 1 | 1 | yes | characteristic |
| 7 | #87904 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | Q8×C5⋊D5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | #87902 | — | 2 | 1 | no | — |
| 10 | #87902 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | D4×C5⋊D5 | — | 2 | 1 | no | — |
| 12 | D4×C5⋊D5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | #87900 | — | 2 | 1 | no | — |
| 14 | #87900 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C4×C5⋊D5 | — | 2 | 1 | no | — |
| 16 | C2×C4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4×C5×Dic5 | C22 | 1 | 2 | yes | complemented |
| 18 | C5×#681 | C22 | 1 | 2 | yes | complemented |
| 19 | #87781 | — | 2 | 1 | no | — |
| 20 | #87779 | — | 2 | 1 | no | — |
| 21 | #87769 | — | 2 | 1 | no | — |
| 22 | #87767 | — | 2 | 1 | no | — |
| 23 | #65110 | — | 5 | 2 | no | maximal |
| 24 | C22×C5⋊D5 | — | 2 | 1 | no | — |
| 25 | C22×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 26 | C52×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C52×D4 | — | 2 | 1 | no | — |
| 28 | C52×D4 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×C4×C52 | — | 2 | 1 | no | — |
| 30 | C2×C4×C52 | C23 | 1 | 1 | yes | characteristic |
| 31 | C22⋊(C5⋊D5) | — | 2 | 1 | no | — |
| 32 | C22⋊(C5⋊D5) | — | 2 | 1 | no | — |
| 33 | C22⋊(C5⋊D5) | — | 2 | 1 | no | — |
| 34 | C2×#1039 | — | 2 | 1 | no | — |
| 35 | C2×#1039 | D4 | 1 | 1 | yes | characteristic |
| 36 | C4⋊(C5⋊D5) | — | 2 | 1 | no | — |
| 37 | C4⋊(C5⋊D5) | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C4×C5⋊D5 | — | 2 | 1 | no | — |
| 39 | C4×C5⋊D5 | — | 2 | 1 | no | — |
| 40 | C4×C5⋊D5 | — | 2 | 1 | no | — |
| 41 | C4×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C4×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 44 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 45 | #6045 | — | 2 | 1 | no | — |
| 46 | #6045 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C2×C5×Dic5 | — | 2 | 2 | no | — |
| 48 | C5×#168 | — | 2 | 2 | no | — |
| 49 | D5×#41 | — | 5 | 2 | no | — |
| 50 | D5×#41 | — | 10 | 2 | no | — |
| 51 | D5×M16 | — | 5 | 2 | no | — |
| 52 | #4018 | — | 5 | 2 | no | — |
| 53 | C5×#104 | — | 5 | 2 | no | — |
| 54 | #3969 | D5 | 1 | 2 | yes | complemented |
| 55 | #3957 | — | 5 | 2 | no | — |
| 56 | #3937 | — | 5 | 2 | no | — |
| 57 | C22×C52 | — | 2 | 1 | no | — |
| 58 | C22×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | C2×C5⋊D5 | — | 2 | 1 | no | — |
| 60 | C2×C5⋊D5 | — | 4 | 1 | no | — |
| 61 | C2×C5⋊D5 | — | 4 | 2 | no | — |
| 62 | C2×C5⋊D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C5⋊D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C4×C52 | — | 2 | 1 | no | — |
| 65 | C4×C52 | C22×C4 | 1 | 1 | yes | characteristic |
| 66 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | #1039 | — | 2 | 1 | no | — |
| 68 | #1039 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | #1039 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C5×Dic5 | — | 4 | 2 | no | — |
| 71 | C5×#41 | — | 2 | 2 | no | — |
| 72 | C5×#41 | C2×D5 | 1 | 2 | yes | — |
| 73 | #713 | — | 5 | 2 | no | — |
| 74 | #713 | — | 10 | 2 | no | — |
| 75 | Q8×D5 | — | 5 | 2 | no | — |
| 76 | Q8×D5 | — | 10 | 2 | no | — |
| 77 | #711 | — | 5 | 2 | no | — |
| 78 | #711 | — | 10 | 2 | no | — |
| 79 | #711 | — | 10 | 2 | no | — |
| 80 | #711 | — | 10 | 4 | no | — |
| 81 | D4×D5 | — | 5 | 2 | no | — |
| 82 | D4×D5 | — | 10 | 2 | no | — |
| 83 | D4×D5 | — | 10 | 2 | no | — |
| 84 | D4×D5 | — | 10 | 4 | no | — |
| 85 | #709 | — | 5 | 2 | no | — |
| 86 | #709 | — | 10 | 2 | no | — |
| 87 | #709 | — | 10 | 2 | no | — |
| 88 | #709 | — | 10 | 4 | no | — |
| 89 | C2×C4×D5 | — | 5 | 2 | no | — |
| 90 | C2×C4×D5 | — | 10 | 2 | no | — |
| 91 | C2×C4×D5 | — | 10 | 2 | no | — |
| 92 | C2×C4×D5 | — | 10 | 4 | no | — |
| 93 | C5×M16 | — | 5 | 2 | no | — |
| 94 | C42×C5 | — | 5 | 2 | no | — |
| 95 | #685 | — | 10 | 2 | no | — |
| 96 | #683 | — | 10 | 2 | no | — |
| 97 | C4×Dic5 | C2×D5 | 1 | 2 | yes | complemented |
| 98 | #681 | C2×D5 | 1 | 2 | yes | complemented |
| 99 | #676 | — | 10 | 2 | no | — |
| 100 | C8×D5 | — | 10 | 2 | no | — |
| 101 | #421 | — | 25 | 1 | no | Sylow |
| 102 | C2×C52 | — | 2 | 1 | no | — |
| 103 | C2×C52 | — | 4 | 1 | no | — |
| 104 | C2×C52 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 105 | C5⋊D5 | — | 2 | 1 | no | — |
| 106 | C5⋊D5 | — | 2 | 1 | no | — |
| 107 | C5⋊D5 | — | 4 | 1 | no | — |
| 108 | C22×D5 | — | 5 | 2 | no | — |
| 109 | C22×D5 | — | 10 | 2 | no | — |
| 110 | C22×D5 | — | 10 | 2 | no | — |
| 111 | C22×D5 | — | 10 | 4 | no | — |
| 112 | C5×Q8 | — | 2 | 2 | no | — |
| 113 | C5×Q8 | C4×D5 | 1 | 2 | yes | — |
| 114 | C5×D4 | — | 2 | 4 | no | — |
| 115 | C5×D4 | — | 2 | 2 | no | — |
| 116 | C5×D4 | — | 2 | 2 | no | — |
| 117 | C5×D4 | C4×D5 | 1 | 2 | yes | — |
| 118 | C2×C4×C5 | — | 2 | 4 | no | — |
| 119 | C2×C4×C5 | — | 2 | 2 | no | — |
| 120 | C2×C4×C5 | — | 2 | 2 | no | — |
| 121 | C2×C4×C5 | — | 10 | 2 | no | — |
| 122 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 123 | C22⋊D5 | — | 10 | 2 | no | — |
| 124 | C22⋊D5 | — | 10 | 2 | no | — |
| 125 | C22⋊D5 | — | 10 | 4 | no | — |
| 126 | C22⋊D5 | — | 10 | 4 | no | — |
| 127 | C22⋊D5 | — | 10 | 4 | no | — |
| 128 | C22⋊D5 | — | 10 | 2 | no | — |
| 129 | C2×Dic5 | — | 2 | 2 | no | — |
| 130 | C2×Dic5 | — | 5 | 2 | no | — |
| 131 | C2×Dic5 | — | 10 | 2 | no | — |
| 132 | C2×Dic5 | — | 10 | 2 | no | — |
| 133 | C2×Dic5 | — | 10 | 4 | no | — |
| 134 | D20 | — | 5 | 2 | no | — |
| 135 | D20 | — | 10 | 2 | no | — |
| 136 | D20 | — | 10 | 2 | no | — |
| 137 | D20 | — | 10 | 4 | no | — |
| 138 | C4×D5 | — | 5 | 2 | no | — |
| 139 | C4×D5 | — | 5 | 2 | no | — |
| 140 | C4×D5 | — | 5 | 2 | no | — |
| 141 | C4×D5 | — | 5 | 2 | no | — |
| 142 | C4×D5 | — | 10 | 2 | no | — |
| 143 | C4×D5 | — | 10 | 2 | no | — |
| 144 | C4×D5 | — | 10 | 2 | no | — |
| 145 | C4×D5 | — | 10 | 2 | no | — |
| 146 | C4×D5 | — | 10 | 2 | no | — |
| 147 | C4×D5 | — | 10 | 2 | no | — |
| 148 | C4×D5 | — | 10 | 4 | no | — |
| 149 | C4×D5 | — | 10 | 4 | no | — |
| 150 | C4×D5 | — | 10 | 4 | no | — |
| 151 | C4×D5 | — | 10 | 2 | no | — |
| 152 | Dic10 | — | 5 | 2 | no | — |
| 153 | Dic10 | — | 10 | 2 | no | — |
| 154 | Dic10 | — | 10 | 2 | no | — |
| 155 | Dic10 | — | 10 | 4 | no | — |
| 156 | C5×C8 | — | 10 | 2 | no | — |
| 157 | #168 | — | 2 | 2 | no | — |
| 158 | C2×#41 | — | 25 | 1 | no | — |
| 159 | C2×M16 | — | 25 | 1 | no | — |
| 160 | #117 | — | 25 | 1 | no | — |
| 161 | #104 | — | 5 | 2 | no | — |
| 162 | #104 | — | 25 | 2 | no | — |
| 163 | C52 | #421 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 164 | C22×C5 | — | 2 | 4 | no | — |
| 165 | C22×C5 | — | 2 | 2 | no | — |
| 166 | C22×C5 | — | 2 | 2 | no | — |
| 167 | C22×C5 | D4×D5 | 1 | 2 | yes | — |
| 168 | C2×D5 | — | 5 | 2 | no | — |
| 169 | C2×D5 | — | 5 | 2 | no | — |
| 170 | C2×D5 | — | 10 | 2 | no | — |
| 171 | C2×D5 | — | 10 | 2 | no | — |
| 172 | C2×D5 | — | 10 | 4 | no | — |
| 173 | C2×D5 | — | 10 | 2 | no | — |
| 174 | C2×D5 | — | 20 | 2 | no | — |
| 175 | C2×D5 | — | 20 | 4 | no | — |
| 176 | C2×D5 | — | 20 | 8 | no | — |
| 177 | C2×D5 | — | 20 | 4 | no | — |
| 178 | C4×C5 | — | 2 | 4 | no | — |
| 179 | C4×C5 | — | 2 | 2 | no | — |
| 180 | C4×C5 | — | 2 | 2 | no | — |
| 181 | C4×C5 | — | 2 | 2 | no | — |
| 182 | C4×C5 | — | 20 | 2 | no | — |
| 183 | C4×C5 | C2×C4×D5 | 1 | 2 | yes | — |
| 184 | C4×C5 | D4×D5 | 1 | 2 | yes | — |
| 185 | Dic5 | — | 4 | 2 | no | — |
| 186 | Dic5 | — | 5 | 2 | no | — |
| 187 | Dic5 | — | 5 | 2 | no | — |
| 188 | Dic5 | — | 10 | 2 | no | — |
| 189 | Dic5 | — | 10 | 2 | no | — |
| 190 | Dic5 | — | 10 | 4 | no | — |
| 191 | Dic5 | — | 10 | 2 | no | — |
| 192 | #41 | — | 25 | 1 | no | — |
| 193 | #41 | — | 25 | 1 | no | — |
| 194 | #41 | — | 25 | 1 | no | — |
| 195 | #41 | — | 50 | 1 | no | — |
| 196 | #41 | — | 50 | 1 | no | — |
| 197 | #41 | D52 | 1 | 1 | yes | characteristic |
| 198 | C2×Q8 | — | 25 | 1 | no | — |
| 199 | C2×D4 | — | 25 | 1 | no | — |
| 200 | C2×D4 | — | 50 | 1 | no | — |
| 201 | C22×C4 | — | 25 | 1 | no | — |
| 202 | C22×C4 | — | 50 | 1 | no | — |
| 203 | M16 | — | 5 | 2 | no | — |
| 204 | M16 | — | 50 | 1 | no | — |
| 205 | C2×C8 | — | 50 | 1 | no | — |
| 206 | C4⋊C4 | — | 50 | 1 | no | — |
| 207 | C22⋊C4 | — | 50 | 1 | no | — |
| 208 | C42 | — | 5 | 2 | no | — |
| 209 | C2×C5 | — | 2 | 2 | no | — |
| 210 | C2×C5 | — | 2 | 2 | no | — |
| 211 | C2×C5 | — | 4 | 2 | no | — |
| 212 | C2×C5 | — | 4 | 4 | no | — |
| 213 | C2×C5 | — | 4 | 2 | no | — |
| 214 | C2×C5 | D5×C22⋊C4 | 1 | 2 | yes | — |
| 215 | D5 | — | 10 | 4 | no | — |
| 216 | D5 | — | 10 | 2 | no | — |
| 217 | D5 | — | 10 | 2 | no | — |
| 218 | D5 | — | 20 | 2 | no | — |
| 219 | D5 | — | 20 | 4 | no | — |
| 220 | D5 | — | 20 | 2 | no | — |
| 221 | C23 | — | 25 | 1 | no | — |
| 222 | C23 | — | 50 | 1 | no | — |
| 223 | Q8 | — | 25 | 1 | no | — |
| 224 | Q8 | — | 50 | 1 | no | — |
| 225 | Q8 | #6036 | 1 | 1 | yes | characteristic |
| 226 | D4 | — | 2 | 1 | no | — |
| 227 | D4 | — | 25 | 1 | no | — |
| 228 | D4 | — | 50 | 1 | no | — |
| 229 | D4 | — | 50 | 1 | no | — |
| 230 | D4 | — | 50 | 1 | no | — |
| 231 | D4 | — | 50 | 1 | no | — |
| 232 | D4 | #6036 | 1 | 1 | yes | characteristic |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 10 | 2 | no | — |
| 235 | C2×C4 | — | 25 | 1 | no | — |
| 236 | C2×C4 | — | 25 | 1 | no | — |
| 237 | C2×C4 | — | 25 | 1 | no | — |
| 238 | C2×C4 | — | 25 | 1 | no | — |
| 239 | C2×C4 | — | 25 | 1 | no | — |
| 240 | C2×C4 | — | 50 | 1 | no | — |
| 241 | C2×C4 | — | 50 | 1 | no | — |
| 242 | C2×C4 | — | 50 | 1 | no | — |
| 243 | C2×C4 | — | 50 | 1 | no | — |
| 244 | C2×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 245 | C8 | — | 10 | 2 | no | — |
| 246 | C5 | — | 2 | 2 | no | — |
| 247 | C5 | #65110 | 1 | 2 | yes | complemented |
| 248 | C22 | — | 2 | 1 | no | — |
| 249 | C22 | — | 25 | 1 | no | — |
| 250 | C22 | — | 25 | 1 | no | — |
| 251 | C22 | — | 50 | 1 | no | — |
| 252 | C22 | — | 100 | 1 | no | — |
| 253 | C22 | — | 100 | 2 | no | — |
| 254 | C22 | #87886 | 1 | 1 | yes | characteristic |
| 255 | C4 | — | 2 | 1 | no | — |
| 256 | C4 | — | 20 | 2 | no | — |
| 257 | C4 | — | 25 | 1 | no | — |
| 258 | C4 | — | 25 | 1 | no | — |
| 259 | C4 | — | 50 | 1 | no | — |
| 260 | C4 | C2×#6036 | 1 | 1 | yes | characteristic |
| 261 | C4 | #87886 | 1 | 1 | yes | characteristic |
| 262 | C2 | — | 2 | 1 | no | — |
| 263 | C2 | — | 4 | 1 | no | — |
| 264 | C2 | — | 50 | 1 | no | — |
| 265 | C2 | — | 50 | 1 | no | — |
| 266 | C2 | — | 100 | 1 | no | — |
| 267 | C2 | #11717092 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11716992 | C2 | semidirect |
| 3. #11716936 | C2 | semidirect |
| 4. C5×#3969 | C2 | semidirect |
| 5. #11716611 | C2 | semidirect |
| 7. #87904 | C4 | semidirect |
| 8. Q8×C5⋊D5 | C4 | semidirect |
| 10. #87902 | C4 | semidirect |
| 12. D4×C5⋊D5 | C4 | semidirect |
| 17. C4×C5×Dic5 | C22 | semidirect |
| 18. C5×#681 | C22 | semidirect |
| 54. #3969 | D5 | semidirect |
| 97. C4×Dic5 | C2×D5 | semidirect |
| 98. #681 | C2×D5 | semidirect |
| 163. C52 | #421 | semidirect |
| 247. C5 | #65110 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×C5⋊D5 | 2 |
| #11716992 | 2 |
| #11716936 | 2 |
| C5×#3969 | 2 |
| #11716611 | 2 |
| #65110 | 5 |