Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
240 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 16 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 | C5×#5464 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#2199 | — | 3 | 1 | no | maximal, Hall |
| 3 | C5×#1020 | — | 4 | 2 | no | maximal |
| 4 | C5×#857 | — | 4 | 1 | no | maximal |
| 5 | #85559 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C5×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C5×B(4,𝔽2) | — | 3 | 2 | no | — |
| 8 | C5×#455 | — | 3 | 1 | no | — |
| 9 | C5×#354 | — | 3 | 1 | no | — |
| 10 | C5×AΣL(1,𝔽16) | — | 3 | 2 | no | — |
| 11 | C5×C22⋊A4 | — | 4 | 2 | no | — |
| 12 | C5×C42⋊C3 | — | 4 | 1 | no | — |
| 13 | C5×C42⋊C3 | — | 8 | 1 | no | — |
| 14 | #5464 | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 15 | C5×H(5,𝔽2) | — | 3 | 1 | no | — |
| 16 | C5×#137 | — | 6 | 1 | no | — |
| 17 | C5×#124 | — | 3 | 1 | no | — |
| 18 | C5×#124 | — | 6 | 1 | no | — |
| 19 | C5×#120 | — | 3 | 2 | no | — |
| 20 | C5×#120 | — | 6 | 2 | no | — |
| 21 | C5×#104 | — | 6 | 1 | no | — |
| 22 | C5×#100 | — | 3 | 1 | no | — |
| 23 | C5×#99 | — | 3 | 2 | no | — |
| 24 | C5×#99 | — | 6 | 2 | no | — |
| 25 | #2199 | — | 3 | 1 | no | Sylow |
| 26 | C5×S4 | — | 16 | 1 | no | — |
| 27 | C5×S4 | — | 16 | 4 | no | — |
| 28 | #1020 | — | 4 | 2 | no | — |
| 29 | #857 | — | 4 | 1 | no | — |
| 30 | C24×C5 | S4 | 1 | 2 | yes | complemented |
| 31 | C5×#41 | — | 6 | 1 | no | — |
| 32 | C5×#41 | — | 12 | 1 | no | — |
| 33 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 34 | C2×C5×D4 | — | 3 | 1 | no | — |
| 35 | C2×C5×D4 | — | 6 | 2 | no | — |
| 36 | C2×C5×D4 | — | 6 | 1 | no | — |
| 37 | C2×C5×D4 | — | 12 | 1 | no | — |
| 38 | C2×C5×D4 | — | 12 | 2 | no | — |
| 39 | C5×Q16 | — | 12 | 1 | no | — |
| 40 | C5×SD16 | — | 12 | 1 | no | — |
| 41 | C5×M16 | — | 6 | 1 | no | — |
| 42 | C5×C22⋊C4 | — | 3 | 2 | no | — |
| 43 | C5×C22⋊C4 | — | 6 | 2 | no | — |
| 44 | C5×C22⋊C4 | — | 6 | 2 | no | — |
| 45 | C5×C22⋊C4 | — | 12 | 2 | no | — |
| 46 | C42×C5 | — | 2 | 1 | no | — |
| 47 | C42×C5 | S4 | 1 | 1 | yes | characteristic, complemented |
| 48 | H(3,𝔽4) | C5×D3 | 1 | 1 | yes | characteristic, complemented |
| 49 | B(4,𝔽2) | — | 3 | 2 | no | — |
| 50 | #455 | — | 3 | 1 | no | — |
| 51 | #354 | — | 3 | 1 | no | — |
| 52 | AΣL(1,𝔽16) | — | 3 | 2 | no | — |
| 53 | C5×A4 | — | 16 | 4 | no | — |
| 54 | C5×A4 | — | 16 | 1 | no | — |
| 55 | C5×A4 | — | 32 | 2 | no | — |
| 56 | C22⋊A4 | — | 4 | 2 | no | — |
| 57 | C42⋊C3 | — | 4 | 1 | no | — |
| 58 | C42⋊C3 | — | 8 | 1 | no | — |
| 59 | C23×C5 | — | 3 | 2 | no | — |
| 60 | C23×C5 | — | 6 | 2 | no | — |
| 61 | C23×C5 | — | 12 | 2 | no | — |
| 62 | C5×Q8 | — | 6 | 1 | no | — |
| 63 | C5×Q8 | — | 12 | 1 | no | — |
| 64 | C5×D4 | — | 6 | 1 | no | — |
| 65 | C5×D4 | — | 12 | 1 | no | — |
| 66 | C5×D4 | — | 12 | 2 | no | — |
| 67 | C5×D4 | — | 12 | 1 | no | — |
| 68 | C5×D4 | — | 24 | 1 | no | — |
| 69 | C5×D4 | — | 24 | 4 | no | — |
| 70 | C2×C4×C5 | — | 3 | 1 | no | — |
| 71 | C2×C4×C5 | — | 6 | 2 | no | — |
| 72 | C2×C4×C5 | — | 6 | 1 | no | — |
| 73 | C2×C4×C5 | — | 12 | 1 | no | — |
| 74 | C2×C4×C5 | — | 12 | 2 | no | — |
| 75 | C5×C8 | — | 12 | 1 | no | — |
| 76 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 77 | #137 | — | 6 | 1 | no | — |
| 78 | #124 | — | 3 | 1 | no | — |
| 79 | #124 | — | 6 | 1 | no | — |
| 80 | #120 | — | 3 | 2 | no | — |
| 81 | #120 | — | 6 | 2 | no | — |
| 82 | #104 | — | 6 | 1 | no | — |
| 83 | #100 | — | 3 | 1 | no | — |
| 84 | #99 | — | 3 | 2 | no | — |
| 85 | #99 | — | 6 | 2 | no | — |
| 86 | C5×D3 | — | 64 | 1 | no | — |
| 87 | S4 | — | 16 | 1 | no | — |
| 88 | S4 | — | 16 | 4 | no | — |
| 89 | C22×C5 | — | 4 | 4 | no | — |
| 90 | C22×C5 | — | 6 | 2 | no | — |
| 91 | C22×C5 | — | 8 | 2 | no | — |
| 92 | C22×C5 | — | 12 | 2 | no | — |
| 93 | C22×C5 | — | 12 | 1 | no | — |
| 94 | C22×C5 | — | 24 | 2 | no | — |
| 95 | C22×C5 | #1020 | 1 | 1 | yes | characteristic |
| 96 | C4×C5 | — | 6 | 1 | no | — |
| 97 | C4×C5 | — | 12 | 1 | no | — |
| 98 | C4×C5 | — | 12 | 1 | no | — |
| 99 | C4×C5 | — | 24 | 2 | no | — |
| 100 | C24 | C5×S4 | 1 | 2 | yes | complemented |
| 101 | #41 | — | 6 | 1 | no | — |
| 102 | #41 | — | 12 | 1 | no | — |
| 103 | C2×Q8 | — | 3 | 1 | no | — |
| 104 | C2×D4 | — | 3 | 1 | no | — |
| 105 | C2×D4 | — | 6 | 2 | no | — |
| 106 | C2×D4 | — | 6 | 1 | no | — |
| 107 | C2×D4 | — | 12 | 1 | no | — |
| 108 | C2×D4 | — | 12 | 2 | no | — |
| 109 | Q16 | — | 12 | 1 | no | — |
| 110 | SD16 | — | 12 | 1 | no | — |
| 111 | M16 | — | 6 | 1 | no | — |
| 112 | C22⋊C4 | — | 3 | 2 | no | — |
| 113 | C22⋊C4 | — | 6 | 2 | no | — |
| 114 | C22⋊C4 | — | 6 | 2 | no | — |
| 115 | C22⋊C4 | — | 12 | 2 | no | — |
| 116 | C42 | — | 2 | 1 | no | — |
| 117 | C42 | C5×S4 | 1 | 1 | yes | characteristic, complemented |
| 118 | C3×C5 | — | 64 | 1 | no | Hall |
| 119 | A4 | — | 16 | 4 | no | — |
| 120 | A4 | — | 16 | 1 | no | — |
| 121 | A4 | — | 32 | 2 | no | — |
| 122 | C2×C5 | — | 3 | 1 | no | — |
| 123 | C2×C5 | — | 12 | 2 | no | — |
| 124 | C2×C5 | — | 24 | 1 | no | — |
| 125 | C23 | — | 3 | 2 | no | — |
| 126 | C23 | — | 6 | 2 | no | — |
| 127 | C23 | — | 12 | 2 | no | — |
| 128 | Q8 | — | 6 | 1 | no | — |
| 129 | Q8 | — | 12 | 1 | no | — |
| 130 | D4 | — | 6 | 1 | no | — |
| 131 | D4 | — | 12 | 1 | no | — |
| 132 | D4 | — | 12 | 2 | no | — |
| 133 | D4 | — | 12 | 1 | no | — |
| 134 | D4 | — | 24 | 1 | no | — |
| 135 | D4 | — | 24 | 4 | no | — |
| 136 | C2×C4 | — | 3 | 1 | no | — |
| 137 | C2×C4 | — | 6 | 2 | no | — |
| 138 | C2×C4 | — | 6 | 1 | no | — |
| 139 | C2×C4 | — | 12 | 1 | no | — |
| 140 | C2×C4 | — | 12 | 2 | no | — |
| 141 | C8 | — | 12 | 1 | no | — |
| 142 | D3 | — | 64 | 1 | no | — |
| 143 | C5 | #85559 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 144 | C22 | — | 4 | 4 | no | — |
| 145 | C22 | — | 6 | 2 | no | — |
| 146 | C22 | — | 8 | 2 | no | — |
| 147 | C22 | — | 12 | 2 | no | — |
| 148 | C22 | — | 12 | 1 | no | — |
| 149 | C22 | — | 24 | 2 | no | — |
| 150 | C22 | C5×#1020 | 1 | 1 | yes | characteristic |
| 151 | C4 | — | 6 | 1 | no | — |
| 152 | C4 | — | 12 | 1 | no | — |
| 153 | C4 | — | 12 | 1 | no | — |
| 154 | C4 | — | 24 | 2 | no | — |
| 155 | C3 | — | 64 | 1 | no | Sylow |
| 156 | C2 | — | 3 | 1 | no | — |
| 157 | C2 | — | 12 | 2 | no | — |
| 158 | C2 | — | 24 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #85559 |
| commutator, cosocle, hypocenter | #5464 | C2×C5 |
| Frattini | C22 | C5×#1020 |
| Fitting | C5×H(3,𝔽4) | D3 |
| socle | C22×C5 | #1020 |
| LCS | G ▷ #5464 | |
| UCS | C1 ◁ C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#5464 | C2 | semidirect |
| 5. #85559 | C5 | direct (N × H) |
| 6. C5×H(3,𝔽4) | D3 | semidirect |
| 14. #5464 | C2×C5 | semidirect |
| 30. C24×C5 | S4 | semidirect |
| 47. C42×C5 | S4 | semidirect |
| 48. H(3,𝔽4) | C5×D3 | semidirect |
| 100. C24 | C5×S4 | semidirect |
| 117. C42 | C5×S4 | semidirect |
| 143. C5 | #85559 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#5464 | 2 |
| C5×#2199 | 3 |
| C5×#1020 | 4 |
| C5×#857 | 4 |
| #85559 | 5 |