Finite Group Databasebeta

← C5×#5683 #11756068

Subgroups — C5×#5683 #11756068

Order 960.

520 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#124C211yesmaximal, characteristic, complemented
2C5×#947C212yesmaximal, complemented
3C5×#937C212yesmaximal, complemented
4C5×#885C214yesmaximal, complemented
5C5×#882C214yesmaximal, complemented
6C5×#875C212yesmaximal, complemented
7C5×H(3,𝔽4)—31nomaximal, Hall
8C23×C5×D3C2212yescomplemented
9C2×C3×C5×Q8C2211yescharacteristic, complemented
10C2×C3×C5×D4C2211yescharacteristic
11C2×C5×#67C2214yes—
12C2×C5×D12C2212yes—
13C2×C5×D12C2212yes—
14C2×C5×Dic6C2212yescomplemented
15C3×C5×C22⋊C4C2214yescomplemented
16C3×C42×C5C2211yescharacteristic, complemented
17C5×#217C2212yescomplemented
18C5×(C2×D3)⋊C4C2214yescomplemented
19C5×(C2×D3)⋊C4C2214yescomplemented
20C5×(C2×D3)⋊C4C2214yescomplemented
21C4×C5×Dic3C2212yescomplemented
22#5683C511yesmaximal, characteristic, direct product factor, Hall
23C5×#124—32no—
24C5×#124—34no—
25C5×#124—32no—
26C5×#124D311yescharacteristic, complemented
27C5×#120—32no—
28C5×#120—34no—
29C23×C3×C5C2312yes—
30C22×C5×D3—42no—
31C22×C5×D3—44no—
32C22×C5×D3C2314yes—
33C3×C5×Q8—41no—
34C3×C5×D4—41no—
35C2×C3×C4×C5C2312yes—
36C2×C3×C4×C5C2311yescharacteristic
37C2×C3×C4×C5C2312yes—
38C5×#67—44no—
39C2×C5×Dic3C2314yes—
40C5×D12—42no—
41C5×D12—42no—
42C5×Dic6—42no—
43C3×#124C2×C511yescharacteristic, complemented
44#947C2×C512yescomplemented
45#937C2×C512yescomplemented
46#885C2×C514yescomplemented
47#882C2×C514yescomplemented
48#875C2×C512yescomplemented
49C24×C5—32no—
50C2×C5×Q8—32no—
51C2×C5×Q8C2×D311yescharacteristic, complemented
52C2×C5×D4—34no—
53C2×C5×D4—32no—
54C2×C5×D4—32no—
55C2×C5×D4C2×D311yescharacteristic
56C5×C22⋊C4—32no—
57C5×C22⋊C4—34no—
58C5×C22⋊C4—34no—
59C5×C22⋊C4—34no—
60C5×C22⋊C4C2×D314yescomplemented
61C42×C5—32no—
62C42×C5C2×D311yescharacteristic, complemented
63H(3,𝔽4)—31noSylow
64C22×C3×C5—24no—
65C22×C3×C5—22no—
66C22×C3×C5C2411yescharacteristic
67C2×C5×D3—24no—
68C2×C5×D3—24no—
69C2×C5×D3—24no—
70C2×C5×D3—48no—
71C3×C4×C5—22no—
72C3×C4×C5—22no—
73C3×C4×C5—21no—
74C5×Dic3—24no—
75C23×D3C22×C512yescomplemented
76C2×C3×Q8C22×C511yescharacteristic, complemented
77C2×C3×D4C22×C511yescharacteristic
78C2×#67C22×C514yes—
79C2×D12C22×C512yes—
80C2×D12C22×C512yes—
81C2×Dic6C22×C512yescomplemented
82C3×C22⋊C4C22×C514yescomplemented
83C3×C42C22×C511yescharacteristic, complemented
84#217C22×C512yescomplemented
85(C2×D3)⋊C4C22×C514yescomplemented
86(C2×D3)⋊C4C22×C514yescomplemented
87(C2×D3)⋊C4C22×C514yescomplemented
88C4×Dic3C22×C512yescomplemented
89C23×C5—34no—
90C23×C5—122no—
91C23×C5—124no—
92C23×C5C22×D312yes—
93C5×Q8—41no—
94C5×Q8—122no—
95C5×D4—41no—
96C5×D4—122no—
97C5×D4—122no—
98C5×D4—124no—
99C2×C4×C5—34no—
100C2×C4×C5C22×D312yes—
101C2×C4×C5C22×D311yescharacteristic
102C2×C4×C5C22×D312yes—
103#124—32no—
104#124—34no—
105#124—32no—
106#124C5×D311yescharacteristic, complemented
107#120—32no—
108#120—34no—
109C2×C3×C5—42no—
110C2×C3×C5H(5,𝔽2)11yescharacteristic
111C2×C3×C5H(5,𝔽2)12yes—
112C5×D3—44no—
113C23×C3C23×C512yes—
114C22×D3—42no—
115C22×D3—44no—
116C22×D3C23×C514yes—
117C3×Q8—41no—
118C3×D4—41no—
119C2×C3×C4C23×C512yes—
120C2×C3×C4C23×C511yescharacteristic
121C2×C3×C4C23×C512yes—
122#67—44no—
123C2×Dic3C23×C514yes—
124D12—42no—
125D12—42no—
126Dic6—42no—
127C22×C5—22no—
128C22×C5—24no—
129C22×C5—64no—
130C22×C5—64no—
131C22×C5—64no—
132C22×C5—128no—
133C22×C5C23×D311yescharacteristic
134C4×C5—21no—
135C4×C5—22no—
136C4×C5—22no—
137C4×C5—64no—
138C24—32no—
139C2×Q8—32no—
140C2×Q8C2×C5×D311yescharacteristic, complemented
141C2×D4—34no—
142C2×D4—32no—
143C2×D4—32no—
144C2×D4C2×C5×D311yescharacteristic
145C22⋊C4—32no—
146C22⋊C4—34no—
147C22⋊C4—34no—
148C22⋊C4—34no—
149C22⋊C4C2×C5×D314yescomplemented
150C42—32no—
151C42C2×C5×D311yescharacteristic, complemented
152C3×C5H(3,𝔽4)11yescharacteristic, complemented, Hall
153C22×C3—24no—
154C22×C3—22no—
155C22×C3C24×C511yescharacteristic
156C2×D3—24no—
157C2×D3—24no—
158C2×D3—24no—
159C2×D3—48no—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—21no—
163Dic3—24no—
164C2×C5—42no—
165C2×C5—124no—
166C2×C5#100411yescharacteristic
167C2×C5#100912yes—
168C23—34no—
169C23—122no—
170C23—124no—
171C23C22×C5×D312yes—
172Q8—41no—
173Q8—122no—
174D4—41no—
175D4—122no—
176D4—122no—
177D4—124no—
178C2×C4—34no—
179C2×C4C22×C5×D312yes—
180C2×C4C22×C5×D311yescharacteristic
181C2×C4C22×C5×D312yes—
182C2×C3—42no—
183C2×C3C5×H(5,𝔽2)11yescharacteristic
184C2×C3C5×H(5,𝔽2)12yes—
185D3—44no—
186C5#568311yescharacteristic, direct product factor, Sylow
187C22—22no—
188C22—24no—
189C22—64no—
190C22—64no—
191C22—64no—
192C22—128no—
193C22C23×C5×D311yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—64no—
198C3C5×H(3,𝔽4)11yescharacteristic, complemented, Sylow
199C2—42no—
200C2—124no—
201C2C5×#100411yescharacteristic
202C2C5×#100912yes—
subgroupG/H
centerC22×C5C23×D3
commutator, cosocleC22×C3C24×C5
FrattiniC22C23×C5×D3
FittingC3×C5×#124C2
socleC22×C3×C5C24
hypercenterC5×#124D3
hypocenterC3C5×H(3,𝔽4)
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C5×#124

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C5×#124C2semidirect
2. C5×#947C2semidirect
3. C5×#937C2semidirect
4. C5×#885C2semidirect
5. C5×#882C2semidirect
6. C5×#875C2semidirect
8. C23×C5×D3C22semidirect
9. C2×C3×C5×Q8C22semidirect
14. C2×C5×Dic6C22semidirect
15. C3×C5×C22⋊C4C22semidirect
16. C3×C42×C5C22semidirect
17. C5×#217C22semidirect
18. C5×(C2×D3)⋊C4C22semidirect
19. C5×(C2×D3)⋊C4C22semidirect
20. C5×(C2×D3)⋊C4C22semidirect
21. C4×C5×Dic3C22semidirect
22. #5683C5direct (N × H)
26. C5×#124D3semidirect
43. C3×#124C2×C5semidirect
44. #947C2×C5semidirect
45. #937C2×C5semidirect
46. #885C2×C5semidirect
47. #882C2×C5semidirect
48. #875C2×C5semidirect
51. C2×C5×Q8C2×D3semidirect
60. C5×C22⋊C4C2×D3semidirect
62. C42×C5C2×D3semidirect
75. C23×D3C22×C5semidirect
76. C2×C3×Q8C22×C5semidirect
81. C2×Dic6C22×C5semidirect
82. C3×C22⋊C4C22×C5semidirect
83. C3×C42C22×C5semidirect
84. #217C22×C5semidirect
85. (C2×D3)⋊C4C22×C5semidirect
86. (C2×D3)⋊C4C22×C5semidirect
87. (C2×D3)⋊C4C22×C5semidirect
88. C4×Dic3C22×C5semidirect
106. #124C5×D3semidirect
140. C2×Q8C2×C5×D3semidirect
149. C22⋊C4C2×C5×D3semidirect
151. C42C2×C5×D3semidirect
152. C3×C5H(3,𝔽4)semidirect
186. C5#5683direct (N × H)
198. C3C5×H(3,𝔽4)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C5×#1242
C5×#9472
C5×#9372
C5×#8852
C5×#8822
C5×#8752
C5×H(3,𝔽4)3
#56835