Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
444 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 58 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 | #11717473 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11717194 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11716943 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C5×#3937 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11716600 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×C4⋊(C5⋊D5) | — | 2 | 2 | no | — |
| 7 | C2×C4⋊(C5⋊D5) | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×C4⋊(C5⋊D5) | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C5×#709 | C22 | 1 | 2 | yes | — |
| 10 | #87886 | — | 2 | 2 | no | — |
| 11 | D5×D20 | — | 2 | 2 | no | — |
| 12 | #87874 | — | 2 | 2 | no | — |
| 13 | C42×C52 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C5×#681 | C22 | 1 | 2 | yes | complemented |
| 15 | #87774 | — | 2 | 2 | no | — |
| 16 | #87771 | — | 2 | 2 | no | — |
| 17 | #65111 | — | 5 | 2 | no | maximal |
| 18 | C22×C5⋊D5 | — | 2 | 1 | no | — |
| 19 | C22×C5⋊D5 | D4 | 1 | 2 | yes | — |
| 20 | C2×D52 | — | 2 | 2 | no | — |
| 21 | C2×C4×C52 | — | 2 | 1 | no | — |
| 22 | C2×C4×C52 | C23 | 1 | 1 | yes | characteristic |
| 23 | C4⋊(C5⋊D5) | — | 2 | 2 | no | — |
| 24 | C4⋊(C5⋊D5) | — | 2 | 2 | no | — |
| 25 | C4⋊(C5⋊D5) | — | 4 | 4 | no | — |
| 26 | C5×C22⋊D5 | — | 2 | 2 | no | — |
| 27 | C5×D20 | D4 | 1 | 2 | yes | complemented |
| 28 | C4×C5×D5 | — | 2 | 2 | no | — |
| 29 | C5×Dic10 | D4 | 1 | 2 | yes | complemented |
| 30 | #6038 | — | 2 | 4 | no | — |
| 31 | #6036 | — | 2 | 2 | no | — |
| 32 | C5×#168 | — | 2 | 2 | no | — |
| 33 | #4149 | — | 5 | 2 | no | — |
| 34 | #4095 | — | 5 | 2 | no | — |
| 35 | #4054 | — | 5 | 2 | no | — |
| 36 | #4020 | — | 5 | 2 | no | — |
| 37 | #4020 | — | 10 | 2 | no | — |
| 38 | C5×#104 | — | 5 | 2 | no | — |
| 39 | #3955 | — | 5 | 2 | no | — |
| 40 | #3937 | D5 | 1 | 2 | yes | complemented |
| 41 | C22×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 42 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 43 | C2×C5⋊D5 | — | 4 | 2 | no | — |
| 44 | C2×C5⋊D5 | — | 4 | 2 | no | — |
| 45 | C2×C5⋊D5 | — | 4 | 1 | no | — |
| 46 | C2×C5×D5 | — | 2 | 2 | no | — |
| 47 | D52 | — | 8 | 2 | no | — |
| 48 | C4×C52 | — | 2 | 2 | no | — |
| 49 | C4×C52 | C2×D4 | 1 | 2 | yes | — |
| 50 | C5×Dic5 | — | 2 | 2 | no | — |
| 51 | C5×#41 | — | 5 | 2 | no | — |
| 52 | #713 | — | 10 | 2 | no | — |
| 53 | D4×D5 | — | 10 | 2 | no | — |
| 54 | D4×D5 | — | 10 | 4 | no | — |
| 55 | #709 | — | 10 | 2 | no | — |
| 56 | #709 | C2×D5 | 1 | 2 | yes | — |
| 57 | C2×D20 | — | 5 | 2 | no | — |
| 58 | C2×D20 | — | 5 | 2 | no | — |
| 59 | C2×D20 | — | 10 | 2 | no | — |
| 60 | C2×D20 | — | 10 | 2 | no | — |
| 61 | C2×D20 | — | 10 | 2 | no | — |
| 62 | C2×D20 | — | 10 | 4 | no | — |
| 63 | C2×D20 | — | 10 | 8 | no | — |
| 64 | C5×M16 | — | 5 | 2 | no | — |
| 65 | C42×C5 | — | 2 | 2 | no | — |
| 66 | C42×C5 | C2×D5 | 1 | 2 | yes | complemented |
| 67 | #688 | — | 10 | 2 | no | — |
| 68 | #686 | — | 10 | 2 | no | — |
| 69 | #681 | C2×D5 | 1 | 2 | yes | complemented |
| 70 | D40 | — | 10 | 2 | no | — |
| 71 | #677 | — | 10 | 2 | no | — |
| 72 | #453 | — | 25 | 1 | no | Sylow |
| 73 | C2×C52 | — | 2 | 1 | no | — |
| 74 | C2×C52 | #120 | 1 | 1 | yes | characteristic |
| 75 | C5⋊D5 | — | 4 | 2 | no | — |
| 76 | C5⋊D5 | — | 8 | 1 | no | — |
| 77 | C5×D5 | — | 4 | 2 | no | — |
| 78 | C22×D5 | — | 5 | 4 | no | — |
| 79 | C22×D5 | — | 10 | 4 | no | — |
| 80 | C22×D5 | — | 10 | 4 | no | — |
| 81 | C22×D5 | — | 10 | 4 | no | — |
| 82 | C22×D5 | — | 10 | 2 | no | — |
| 83 | C5×Q8 | — | 5 | 2 | no | — |
| 84 | C5×D4 | — | 5 | 2 | no | — |
| 85 | C5×D4 | — | 10 | 2 | no | — |
| 86 | C2×C4×C5 | — | 2 | 4 | no | — |
| 87 | C2×C4×C5 | — | 2 | 2 | no | — |
| 88 | C2×C4×C5 | — | 2 | 2 | no | — |
| 89 | C2×C4×C5 | — | 10 | 2 | no | — |
| 90 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 91 | C22⋊D5 | — | 2 | 2 | no | — |
| 92 | C22⋊D5 | — | 10 | 4 | no | — |
| 93 | D20 | — | 10 | 4 | no | — |
| 94 | D20 | — | 10 | 2 | no | — |
| 95 | D20 | — | 10 | 2 | no | — |
| 96 | D20 | — | 10 | 2 | no | — |
| 97 | D20 | — | 10 | 2 | no | — |
| 98 | D20 | — | 20 | 4 | no | — |
| 99 | D20 | — | 20 | 4 | no | — |
| 100 | D20 | — | 20 | 8 | no | — |
| 101 | D20 | — | 20 | 16 | no | — |
| 102 | D20 | D20 | 1 | 2 | yes | complemented |
| 103 | C4×D5 | — | 2 | 2 | no | — |
| 104 | C4×D5 | — | 10 | 4 | no | — |
| 105 | Dic10 | D20 | 1 | 2 | yes | complemented |
| 106 | C5×C8 | — | 10 | 2 | no | — |
| 107 | #168 | — | 2 | 2 | no | — |
| 108 | H(5,𝔽2) | — | 25 | 1 | no | — |
| 109 | AGL(1,ℤ/8) | — | 25 | 2 | no | — |
| 110 | #127 | — | 25 | 1 | no | — |
| 111 | #104 | — | 5 | 2 | no | — |
| 112 | #100 | — | 25 | 1 | no | — |
| 113 | C52 | #453 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 114 | C22×C5 | — | 2 | 2 | no | — |
| 115 | C22×C5 | — | 10 | 2 | no | — |
| 116 | C22×C5 | D4×D5 | 1 | 2 | yes | — |
| 117 | C2×D5 | — | 2 | 2 | no | — |
| 118 | C2×D5 | — | 10 | 4 | no | — |
| 119 | C2×D5 | — | 20 | 4 | no | — |
| 120 | C2×D5 | — | 20 | 4 | no | — |
| 121 | C2×D5 | — | 20 | 8 | no | — |
| 122 | C2×D5 | — | 20 | 8 | no | — |
| 123 | C2×D5 | — | 20 | 4 | no | — |
| 124 | C2×D5 | — | 20 | 4 | no | — |
| 125 | C2×D5 | — | 20 | 2 | no | — |
| 126 | C2×D5 | — | 40 | 4 | no | — |
| 127 | C4×C5 | — | 2 | 8 | no | — |
| 128 | C4×C5 | — | 2 | 4 | no | — |
| 129 | C4×C5 | — | 2 | 4 | no | — |
| 130 | C4×C5 | — | 10 | 2 | no | — |
| 131 | C4×C5 | C2×D20 | 1 | 2 | yes | — |
| 132 | C4×C5 | D4×D5 | 1 | 2 | yes | — |
| 133 | Dic5 | — | 2 | 2 | no | — |
| 134 | #41 | — | 5 | 2 | no | — |
| 135 | #41 | — | 50 | 2 | no | — |
| 136 | C2×D4 | — | 25 | 1 | no | — |
| 137 | C2×D4 | — | 25 | 1 | no | — |
| 138 | C2×D4 | — | 50 | 2 | no | — |
| 139 | C2×D4 | — | 50 | 2 | no | — |
| 140 | C2×D4 | — | 50 | 2 | no | — |
| 141 | SD16 | — | 50 | 2 | no | — |
| 142 | D8 | — | 50 | 2 | no | — |
| 143 | M16 | — | 5 | 2 | no | — |
| 144 | C42 | D52 | 1 | 1 | yes | characteristic, complemented |
| 145 | C2×C5 | — | 2 | 4 | no | — |
| 146 | C2×C5 | — | 2 | 2 | no | — |
| 147 | C2×C5 | — | 2 | 2 | no | — |
| 148 | C2×C5 | — | 20 | 2 | no | — |
| 149 | C2×C5 | #4028 | 1 | 2 | yes | — |
| 150 | D5 | — | 4 | 2 | no | — |
| 151 | D5 | — | 20 | 4 | no | — |
| 152 | D5 | — | 40 | 4 | no | — |
| 153 | D5 | — | 40 | 4 | no | — |
| 154 | D5 | — | 40 | 2 | no | — |
| 155 | C23 | — | 25 | 2 | no | — |
| 156 | C23 | — | 50 | 2 | no | — |
| 157 | C23 | — | 50 | 1 | no | — |
| 158 | Q8 | — | 5 | 2 | no | — |
| 159 | D4 | — | 5 | 2 | no | — |
| 160 | D4 | — | 10 | 2 | no | — |
| 161 | D4 | — | 50 | 4 | no | — |
| 162 | D4 | — | 50 | 2 | no | — |
| 163 | D4 | — | 50 | 2 | no | — |
| 164 | D4 | — | 100 | 4 | no | — |
| 165 | C2×C4 | — | 2 | 1 | no | — |
| 166 | C2×C4 | — | 10 | 2 | no | — |
| 167 | C2×C4 | — | 50 | 2 | no | — |
| 168 | C2×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 169 | C8 | — | 10 | 2 | no | — |
| 170 | C5 | — | 2 | 2 | no | — |
| 171 | C5 | #65111 | 1 | 2 | yes | complemented |
| 172 | C22 | — | 10 | 2 | no | — |
| 173 | C22 | — | 50 | 2 | no | — |
| 174 | C22 | — | 100 | 2 | no | — |
| 175 | C22 | — | 100 | 2 | no | — |
| 176 | C22 | — | 100 | 1 | no | — |
| 177 | C22 | — | 200 | 2 | no | — |
| 178 | C22 | #87886 | 1 | 1 | yes | characteristic |
| 179 | C4 | — | 2 | 2 | no | — |
| 180 | C4 | — | 10 | 2 | no | — |
| 181 | C4 | D5×D20 | 1 | 2 | yes | — |
| 182 | C2 | — | 2 | 1 | no | — |
| 183 | C2 | — | 20 | 2 | no | — |
| 184 | C2 | — | 100 | 2 | no | — |
| 185 | C2 | — | 200 | 1 | no | — |
| 186 | C2 | #11717041 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11717041 |
| commutator, cosocle | C2×C4×C52 | C23 |
| Frattini | C2×C4 | C2×D52 |
| Fitting | C42×C52 | C22 |
| socle | C2×C52 | #120 |
| hypercenter | C42 | D52 |
| hypocenter | C52 | #453 |
| LCS | G ▷ C2×C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717473 | C2 | semidirect |
| 2. #11717194 | C2 | semidirect |
| 3. #11716943 | C2 | semidirect |
| 4. C5×#3937 | C2 | semidirect |
| 5. #11716600 | C2 | semidirect |
| 8. C2×C4⋊(C5⋊D5) | C22 | semidirect |
| 13. C42×C52 | C22 | semidirect |
| 14. C5×#681 | C22 | semidirect |
| 27. C5×D20 | D4 | semidirect |
| 29. C5×Dic10 | D4 | semidirect |
| 40. #3937 | D5 | semidirect |
| 66. C42×C5 | C2×D5 | semidirect |
| 69. #681 | C2×D5 | semidirect |
| 102. D20 | D20 | semidirect |
| 105. Dic10 | D20 | semidirect |
| 113. C52 | #453 | semidirect |
| 144. C42 | D52 | semidirect |
| 171. C5 | #65111 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717473 | 2 |
| #11717194 | 2 |
| #11716943 | 2 |
| C5×#3937 | 2 |
| #11716600 | 2 |
| #65111 | 5 |