Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
558 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 128 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 | #11717462 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #11717227 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#4055 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#87775 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11716940 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11716929 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #11716927 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11716920 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #87900 | C22 | 1 | 1 | yes | characteristic |
| 10 | C5×#709 | C22 | 1 | 2 | yes | — |
| 11 | C2×C5×Dic10 | C22 | 1 | 2 | yes | — |
| 12 | #87879 | — | 2 | 4 | no | — |
| 13 | #87874 | — | 2 | 2 | no | — |
| 14 | #87872 | C22 | 1 | 4 | yes | — |
| 15 | #87871 | C22 | 1 | 4 | yes | — |
| 16 | D5×Dic10 | — | 2 | 2 | no | — |
| 17 | C52×M16 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#6029 | C22 | 1 | 1 | yes | characteristic |
| 19 | #87799 | C22 | 1 | 2 | yes | — |
| 20 | C8×C5⋊D5 | C22 | 1 | 2 | yes | — |
| 21 | C5×Dic20 | C22 | 1 | 4 | yes | — |
| 22 | C5×#677 | C22 | 1 | 4 | yes | — |
| 23 | #87775 | C22 | 1 | 1 | yes | characteristic |
| 24 | #87775 | C22 | 1 | 2 | yes | — |
| 25 | #87775 | C22 | 1 | 2 | yes | — |
| 26 | #87772 | C22 | 1 | 2 | yes | — |
| 27 | #87770 | C22 | 1 | 1 | yes | characteristic |
| 28 | #66113 | — | 5 | 2 | no | maximal |
| 29 | C2×C4×C52 | C23 | 1 | 1 | yes | characteristic |
| 30 | C22⋊(C5⋊D5) | D4 | 1 | 2 | yes | — |
| 31 | C4⋊(C5⋊D5) | D4 | 1 | 1 | yes | characteristic |
| 32 | C4×C5⋊D5 | C23 | 1 | 2 | yes | — |
| 33 | #6045 | D4 | 1 | 1 | yes | characteristic |
| 34 | C5×C22⋊D5 | — | 2 | 2 | no | — |
| 35 | C2×C5×Dic5 | — | 2 | 2 | no | — |
| 36 | C5×D20 | C23 | 1 | 2 | yes | — |
| 37 | C4×C5×D5 | — | 2 | 2 | no | — |
| 38 | C5×Dic10 | — | 2 | 2 | no | — |
| 39 | C5×Dic10 | C23 | 1 | 4 | yes | — |
| 40 | C5×Dic10 | C23 | 1 | 2 | yes | — |
| 41 | #6039 | — | 2 | 4 | no | — |
| 42 | #6038 | — | 2 | 4 | no | — |
| 43 | #6036 | — | 2 | 4 | no | — |
| 44 | D5×Dic5 | — | 2 | 4 | no | — |
| 45 | C52×C8 | C23 | 1 | 2 | yes | — |
| 46 | #6029 | C23 | 1 | 1 | yes | characteristic |
| 47 | #6029 | C23 | 1 | 1 | yes | characteristic |
| 48 | #4150 | — | 5 | 2 | no | — |
| 49 | #4147 | — | 5 | 2 | no | — |
| 50 | C5×#137 | — | 5 | 2 | no | — |
| 51 | #4096 | — | 5 | 2 | no | — |
| 52 | C2×#689 | — | 5 | 2 | no | — |
| 53 | #4064 | — | 5 | 4 | no | — |
| 54 | D5×Q16 | — | 5 | 4 | no | — |
| 55 | #4062 | — | 5 | 4 | no | — |
| 56 | #4061 | — | 5 | 4 | no | — |
| 57 | #4055 | D5 | 1 | 2 | yes | complemented |
| 58 | #4053 | — | 5 | 2 | no | — |
| 59 | #4053 | — | 10 | 2 | no | — |
| 60 | C22×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C5⋊D5 | C2×D4 | 1 | 2 | yes | — |
| 62 | C2×C5×D5 | — | 2 | 2 | no | — |
| 63 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C4×C52 | C24 | 1 | 1 | yes | characteristic |
| 65 | #1039 | C2×D4 | 1 | 2 | yes | — |
| 66 | C5×Dic5 | — | 2 | 4 | no | — |
| 67 | C5×Dic5 | — | 2 | 2 | no | — |
| 68 | C5×#41 | — | 5 | 2 | no | — |
| 69 | C2×C5×Q8 | — | 5 | 2 | no | — |
| 70 | #713 | — | 5 | 4 | no | — |
| 71 | #713 | — | 10 | 2 | no | — |
| 72 | Q8×D5 | — | 5 | 4 | no | — |
| 73 | Q8×D5 | — | 5 | 4 | no | — |
| 74 | Q8×D5 | — | 10 | 2 | no | — |
| 75 | #711 | — | 5 | 4 | no | — |
| 76 | #711 | — | 10 | 4 | no | — |
| 77 | #709 | — | 5 | 2 | no | — |
| 78 | #709 | — | 10 | 2 | no | — |
| 79 | #709 | — | 10 | 2 | no | — |
| 80 | #709 | — | 10 | 4 | no | — |
| 81 | #709 | C2×D5 | 1 | 2 | yes | — |
| 82 | C2×Dic10 | — | 10 | 2 | no | — |
| 83 | C2×Dic10 | C2×D5 | 1 | 2 | yes | — |
| 84 | C5×Q16 | — | 5 | 4 | no | — |
| 85 | C5×SD16 | — | 5 | 4 | no | — |
| 86 | C5×M16 | — | 2 | 2 | no | — |
| 87 | C5×M16 | C2×D5 | 1 | 2 | yes | — |
| 88 | #689 | — | 5 | 2 | no | — |
| 89 | #689 | — | 5 | 4 | no | — |
| 90 | #689 | — | 5 | 4 | no | — |
| 91 | #688 | — | 5 | 2 | no | — |
| 92 | #687 | — | 5 | 2 | no | — |
| 93 | #686 | — | 5 | 2 | no | — |
| 94 | C2×#168 | — | 5 | 2 | no | — |
| 95 | C2×#168 | — | 10 | 2 | no | — |
| 96 | Dic20 | C2×D5 | 1 | 4 | yes | — |
| 97 | #677 | C2×D5 | 1 | 4 | yes | — |
| 98 | #676 | — | 5 | 4 | no | — |
| 99 | #676 | — | 10 | 4 | no | — |
| 100 | C8×D5 | — | 5 | 4 | no | — |
| 101 | C8×D5 | — | 10 | 4 | no | — |
| 102 | #578 | — | 25 | 1 | no | Sylow |
| 103 | C2×C52 | — | 2 | 1 | no | — |
| 104 | C2×C52 | C22×D4 | 1 | 1 | yes | characteristic |
| 105 | C5⋊D5 | — | 2 | 2 | no | — |
| 106 | C5×D5 | — | 4 | 2 | no | — |
| 107 | C5×Q8 | — | 5 | 2 | no | — |
| 108 | C5×Q8 | — | 5 | 4 | no | — |
| 109 | C5×Q8 | — | 10 | 2 | no | — |
| 110 | C5×D4 | — | 5 | 2 | no | — |
| 111 | C5×D4 | — | 10 | 2 | no | — |
| 112 | C2×C4×C5 | — | 2 | 2 | no | — |
| 113 | C2×C4×C5 | — | 10 | 2 | no | — |
| 114 | C2×C4×C5 | — | 10 | 2 | no | — |
| 115 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 116 | C22⋊D5 | — | 2 | 2 | no | — |
| 117 | C22⋊D5 | — | 5 | 4 | no | — |
| 118 | C22⋊D5 | — | 10 | 4 | no | — |
| 119 | C22⋊D5 | — | 10 | 4 | no | — |
| 120 | C2×Dic5 | — | 2 | 2 | no | — |
| 121 | C2×Dic5 | — | 10 | 4 | no | — |
| 122 | D20 | — | 5 | 2 | no | — |
| 123 | D20 | — | 10 | 2 | no | — |
| 124 | D20 | — | 10 | 4 | no | — |
| 125 | D20 | C22×D5 | 1 | 2 | yes | — |
| 126 | C4×D5 | — | 2 | 2 | no | — |
| 127 | C4×D5 | — | 5 | 4 | no | — |
| 128 | C4×D5 | — | 10 | 4 | no | — |
| 129 | C4×D5 | — | 10 | 4 | no | — |
| 130 | C4×D5 | — | 10 | 4 | no | — |
| 131 | C4×D5 | — | 10 | 4 | no | — |
| 132 | Dic10 | — | 2 | 2 | no | — |
| 133 | Dic10 | — | 5 | 2 | no | — |
| 134 | Dic10 | — | 10 | 2 | no | — |
| 135 | Dic10 | — | 10 | 4 | no | — |
| 136 | Dic10 | — | 10 | 4 | no | — |
| 137 | Dic10 | C22×D5 | 1 | 4 | yes | — |
| 138 | Dic10 | C22×D5 | 1 | 2 | yes | — |
| 139 | C5×C8 | — | 2 | 4 | no | — |
| 140 | C5×C8 | C22×D5 | 1 | 4 | yes | — |
| 141 | #168 | — | 5 | 2 | no | — |
| 142 | #168 | — | 5 | 2 | no | — |
| 143 | #168 | — | 10 | 2 | no | — |
| 144 | #168 | — | 10 | 2 | no | — |
| 145 | 2−1+4 | — | 25 | 2 | no | — |
| 146 | #137 | — | 5 | 2 | no | — |
| 147 | #137 | — | 25 | 4 | no | — |
| 148 | #135 | — | 25 | 2 | no | — |
| 149 | #135 | — | 25 | 1 | no | — |
| 150 | C2×Q16 | — | 25 | 2 | no | — |
| 151 | C2×Q16 | — | 25 | 1 | no | — |
| 152 | #131 | — | 25 | 1 | no | — |
| 153 | C52 | #578 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 154 | C22×C5 | — | 2 | 2 | no | — |
| 155 | C22×C5 | — | 10 | 2 | no | — |
| 156 | C22×C5 | D4×D5 | 1 | 2 | yes | — |
| 157 | C2×D5 | — | 2 | 2 | no | — |
| 158 | C2×D5 | — | 5 | 4 | no | — |
| 159 | C2×D5 | — | 10 | 4 | no | — |
| 160 | C4×C5 | — | 2 | 2 | no | — |
| 161 | C4×C5 | — | 2 | 2 | no | — |
| 162 | C4×C5 | — | 10 | 2 | no | — |
| 163 | C4×C5 | — | 10 | 4 | no | — |
| 164 | C4×C5 | D4×D5 | 1 | 2 | yes | — |
| 165 | C4×C5 | C23×D5 | 1 | 2 | yes | — |
| 166 | Dic5 | — | 2 | 4 | no | — |
| 167 | Dic5 | — | 2 | 2 | no | — |
| 168 | Dic5 | — | 5 | 4 | no | — |
| 169 | Dic5 | — | 10 | 4 | no | — |
| 170 | #41 | — | 5 | 2 | no | — |
| 171 | #41 | — | 25 | 1 | no | — |
| 172 | #41 | — | 25 | 4 | no | — |
| 173 | #41 | — | 50 | 4 | no | — |
| 174 | #41 | — | 50 | 2 | no | — |
| 175 | C2×Q8 | — | 5 | 2 | no | — |
| 176 | C2×Q8 | — | 25 | 4 | no | — |
| 177 | C2×Q8 | — | 50 | 2 | no | — |
| 178 | Q16 | — | 5 | 4 | no | — |
| 179 | Q16 | — | 25 | 2 | no | — |
| 180 | Q16 | — | 25 | 1 | no | — |
| 181 | Q16 | — | 25 | 2 | no | — |
| 182 | SD16 | — | 5 | 4 | no | — |
| 183 | SD16 | — | 25 | 2 | no | — |
| 184 | D8 | — | 25 | 1 | no | — |
| 185 | M16 | — | 25 | 2 | no | — |
| 186 | M16 | D52 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | — | 25 | 1 | no | — |
| 188 | C2×C8 | — | 25 | 2 | no | — |
| 189 | C2×C5 | — | 2 | 2 | no | — |
| 190 | C2×C5 | — | 2 | 2 | no | — |
| 191 | C2×C5 | — | 4 | 2 | no | — |
| 192 | C2×C5 | — | 20 | 2 | no | — |
| 193 | C2×C5 | C2×D4×D5 | 1 | 2 | yes | — |
| 194 | D5 | — | 4 | 2 | no | — |
| 195 | D5 | — | 10 | 4 | no | — |
| 196 | D5 | — | 20 | 4 | no | — |
| 197 | Q8 | — | 5 | 4 | no | — |
| 198 | Q8 | — | 5 | 2 | no | — |
| 199 | Q8 | — | 10 | 2 | no | — |
| 200 | Q8 | — | 25 | 1 | no | — |
| 201 | Q8 | — | 50 | 4 | no | — |
| 202 | D4 | — | 5 | 2 | no | — |
| 203 | D4 | — | 10 | 2 | no | — |
| 204 | D4 | — | 25 | 2 | no | — |
| 205 | D4 | — | 25 | 1 | no | — |
| 206 | D4 | — | 50 | 4 | no | — |
| 207 | C2×C4 | — | 10 | 2 | no | — |
| 208 | C2×C4 | — | 10 | 2 | no | — |
| 209 | C2×C4 | — | 25 | 2 | no | — |
| 210 | C2×C4 | — | 50 | 4 | no | — |
| 211 | C2×C4 | — | 50 | 4 | no | — |
| 212 | C2×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 213 | C8 | — | 25 | 1 | no | — |
| 214 | C8 | — | 25 | 1 | no | — |
| 215 | C8 | C2×D52 | 1 | 2 | yes | — |
| 216 | C5 | — | 2 | 2 | no | — |
| 217 | C5 | #66113 | 1 | 2 | yes | complemented |
| 218 | C22 | — | 10 | 2 | no | — |
| 219 | C22 | — | 25 | 2 | no | — |
| 220 | C22 | #87877 | 1 | 1 | yes | characteristic |
| 221 | C4 | — | 10 | 2 | no | — |
| 222 | C4 | — | 10 | 4 | no | — |
| 223 | C4 | — | 25 | 2 | no | — |
| 224 | C4 | #87877 | 1 | 1 | yes | characteristic |
| 225 | C4 | C22×D52 | 1 | 1 | yes | characteristic |
| 226 | C2 | — | 2 | 1 | no | — |
| 227 | C2 | — | 20 | 2 | no | — |
| 228 | C2 | — | 50 | 2 | no | — |
| 229 | C2 | C2×#87877 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#87877 |
| commutator, cosocle | C4×C52 | C24 |
| Frattini | C4 | C22×D52 |
| Fitting | C52×M16 | C22 |
| socle | C2×C52 | C22×D4 |
| hypercenter | M16 | D52 |
| hypocenter | C52 | #578 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717462 | C2 | semidirect |
| 2. #11717227 | C2 | semidirect |
| 3. C5×#4055 | C2 | semidirect |
| 4. C2×#87775 | C2 | semidirect |
| 5. #11716940 | C2 | semidirect |
| 6. #11716929 | C2 | semidirect |
| 7. #11716927 | C2 | semidirect |
| 8. #11716920 | C2 | semidirect |
| 57. #4055 | D5 | semidirect |
| 153. C52 | #578 | semidirect |
| 217. C5 | #66113 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717462 | 2 |
| #11717227 | 2 |
| C5×#4055 | 2 |
| C2×#87775 | 2 |
| #11716940 | 2 |
| #11716929 | 2 |
| #11716927 | 2 |
| #11716920 | 2 |
| #66113 | 5 |