Finite Group Databasebeta

← C5×#5835 #11756220

Subgroups — C5×#5835 #11756220

Order 960.

536 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 214 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×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C5×#1009C211yesmaximal, characteristic, complemented
3C5×#950C213yesmaximal, complemented
4C5×#949C213yesmaximal, complemented
5C5×#948C211yesmaximal, characteristic, complemented
6C5×#932C213yesmaximal, complemented
7C2×C5×#213C213yesmaximal, complemented
8C5×#576—31nomaximal, Hall
9C3×C5×#41—21no—
10C3×C5×#41C2211yescharacteristic
11C3×C5×#41C2213yescomplemented
12C2×C3×C5×D4—23no—
13C2×C3×C5×D4C2213yescomplemented
14C5×#239—21no—
15C5×D3×D4—23no—
16C5×#235C2213yescomplemented
17C2×C5×D12C2213yescomplemented
18C5×#216C2211yescharacteristic, complemented
19C5×#215C2213yescomplemented
20C5×#214C2213yescomplemented
21C5×#213C2213yescomplemented
22C5×#213C2216yescomplemented
23C5×#208C2213yescomplemented
24C2×C5×C3⋊C8C2213yescomplemented
25#5835C511yesmaximal, characteristic, direct product factor, Hall
26C5×H(5,𝔽2)—31no—
27C5×H(5,𝔽2)D311yescharacteristic, complemented
28C5×AGL(1,ℤ/8)—33no—
29C5×AGL(1,ℤ/8)—33no—
30C5×#135—33no—
31C2×C5×D8—33no—
32C5×#131—31no—
33C23×C3×C5—23no—
34C22×C5×D3—23no—
35C3×C5×Q8D411yescharacteristic
36C3×C5×Q8C2311yescharacteristic
37C3×C5×D4—23no—
38C3×C5×D4—23no—
39C3×C5×D4D413yes—
40C3×C5×D4C2313yes—
41C2×C3×C4×C5—23no—
42C2×C3×C4×C5C2313yes—
43C5×#67—23no—
44C5×D12—23no—
45C5×D12C2313yes—
46C4×C5×D3—23no—
47C5×Dic6C2311yescharacteristic
48C5×C3⋊C8C2311yescharacteristic
49C5×C3⋊C8C2313yes—
50C3×H(5,𝔽2)C2×C511yescharacteristic, complemented
51#1009C2×C511yescharacteristic, complemented
52#950C2×C513yescomplemented
53#949C2×C513yescomplemented
54#948C2×C511yescharacteristic, complemented
55#932C2×C513yescomplemented
56C2×#213C2×C513yescomplemented
57C5×#41—21no—
58C5×#41—33no—
59C5×#41—61no—
60C5×#41C2×D311yescharacteristic
61C5×#41C2×D313yescomplemented
62C2×C5×D4—23no—
63C2×C5×D4—33no—
64C2×C5×D4—63no—
65C2×C5×D4C2×D313yescomplemented
66C5×Q16—31no—
67C5×SD16—33no—
68C5×SD16—33no—
69C5×D8—33no—
70C5×D8—36no—
71C5×M16—33no—
72C2×C5×C8—33no—
73#576—31noSylow
74C22×C3×C5—23no—
75C22×C3×C5—83no—
76C22×C3×C5C2×D413yes—
77C2×C5×D3—23no—
78C2×C5×D3—83no—
79C3×C4×C5—21no—
80C3×C4×C5C2×D413yes—
81C3×C4×C5C2411yescharacteristic
82C5×Dic3—21no—
83C3×#41—21no—
84C3×#41C22×C511yescharacteristic
85C3×#41C22×C513yescomplemented
86C2×C3×D4—23no—
87C2×C3×D4C22×C513yescomplemented
88#239—21no—
89D3×D4—23no—
90#235C22×C513yescomplemented
91C2×D12C22×C513yescomplemented
92#216C22×C511yescharacteristic, complemented
93#215C22×C513yescomplemented
94#214C22×C513yescomplemented
95#213C22×C513yescomplemented
96#213C22×C516yescomplemented
97#208C22×C513yescomplemented
98C2×C3⋊C8C22×C513yescomplemented
99C23×C5—23no—
100C23×C5—63no—
101C5×Q8—31no—
102C5×Q8#6711yescharacteristic
103C5×Q8C22×D311yescharacteristic
104C5×D4—23no—
105C5×D4—23no—
106C5×D4—33no—
107C5×D4—63no—
108C5×D4—63no—
109C5×D4#6713yes—
110C5×D4C22×D313yes—
111C2×C4×C5—23no—
112C2×C4×C5—63no—
113C2×C4×C5C22×D313yes—
114C5×C8—31no—
115C5×C8—33no—
116H(5,𝔽2)—31no—
117H(5,𝔽2)C5×D311yescharacteristic, complemented
118AGL(1,ℤ/8)—33no—
119AGL(1,ℤ/8)—33no—
120#135—33no—
121C2×D8—33no—
122#131—31no—
123C2×C3×C5—23no—
124C2×C3×C5—43no—
125C2×C3×C5C22×D411yescharacteristic
126C5×D3—43no—
127C23×C3—23no—
128C22×D3—23no—
129C3×Q8C5×D411yescharacteristic
130C3×Q8C23×C511yescharacteristic
131C3×D4—23no—
132C3×D4—23no—
133C3×D4C5×D413yes—
134C3×D4C23×C513yes—
135C2×C3×C4—23no—
136C2×C3×C4C23×C513yes—
137#67—23no—
138D12—23no—
139D12C23×C513yes—
140C4×D3—23no—
141Dic6C23×C511yescharacteristic
142C3⋊C8C23×C511yescharacteristic
143C3⋊C8C23×C513yes—
144C22×C5—23no—
145C22×C5—63no—
146C22×C5—83no—
147C22×C5—243no—
148C22×C5C2×#6713yes—
149C4×C5—21no—
150C4×C5—61no—
151C4×C5C2×#6713yes—
152C4×C5C23×D311yescharacteristic
153#41—21no—
154#41—33no—
155#41—61no—
156#41C2×C5×D311yescharacteristic
157#41C2×C5×D313yescomplemented
158C2×D4—23no—
159C2×D4—33no—
160C2×D4—63no—
161C2×D4C2×C5×D313yescomplemented
162Q16—31no—
163SD16—33no—
164SD16—33no—
165D8—33no—
166D8—36no—
167M16—33no—
168C2×C8—33no—
169C3×C5#57611yescharacteristic, complemented, Hall
170C22×C3—23no—
171C22×C3—83no—
172C22×C3C2×C5×D413yes—
173C2×D3—23no—
174C2×D3—83no—
175C3×C4—21no—
176C3×C4C2×C5×D413yes—
177C3×C4C24×C511yescharacteristic
178Dic3—21no—
179C2×C5—23no—
180C2×C5—43no—
181C2×C5—123no—
182C2×C5C22×#6711yescharacteristic
183C23—23no—
184C23—63no—
185Q8—31no—
186Q8C5×#6711yescharacteristic
187Q8C22×C5×D311yescharacteristic
188D4—23no—
189D4—23no—
190D4—33no—
191D4—63no—
192D4—63no—
193D4C5×#6713yes—
194D4C22×C5×D313yes—
195C2×C4—23no—
196C2×C4—63no—
197C2×C4C22×C5×D313yes—
198C8—31no—
199C8—33no—
200C2×C3—23no—
201C2×C3—43no—
202C2×C3C22×C5×D411yescharacteristic
203D3—43no—
204C5#583511yescharacteristic, direct product factor, Sylow
205C22—23no—
206C22—63no—
207C22—83no—
208C22—243no—
209C22C2×C5×#6713yes—
210C4—21no—
211C4—61no—
212C4C2×C5×#6713yes—
213C4C23×C5×D311yescharacteristic
214C3C5×#57611yescharacteristic, complemented, Sylow
215C2—23no—
216C2—43no—
217C2—123no—
218C2C22×C5×#6711yescharacteristic
subgroupG/H
centerC2×C5C22×#67
commutator, cosocleC3×C4C24×C5
FrattiniC4C23×C5×D3
FittingC3×C5×H(5,𝔽2)C2
socleC2×C3×C5C22×D4
hypercenterC5×H(5,𝔽2)D3
hypocenterC3C5×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C5×#41 ◁ C5×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×C5×H(5,𝔽2)C2semidirect
2. C5×#1009C2semidirect
3. C5×#950C2semidirect
4. C5×#949C2semidirect
5. C5×#948C2semidirect
6. C5×#932C2semidirect
7. C2×C5×#213C2semidirect
11. C3×C5×#41C22semidirect
13. C2×C3×C5×D4C22semidirect
16. C5×#235C22semidirect
17. C2×C5×D12C22semidirect
18. C5×#216C22semidirect
19. C5×#215C22semidirect
20. C5×#214C22semidirect
21. C5×#213C22semidirect
22. C5×#213C22semidirect
23. C5×#208C22semidirect
24. C2×C5×C3⋊C8C22semidirect
25. #5835C5direct (N × H)
27. C5×H(5,𝔽2)D3semidirect
50. C3×H(5,𝔽2)C2×C5semidirect
51. #1009C2×C5semidirect
52. #950C2×C5semidirect
53. #949C2×C5semidirect
54. #948C2×C5semidirect
55. #932C2×C5semidirect
56. C2×#213C2×C5semidirect
61. C5×#41C2×D3semidirect
65. C2×C5×D4C2×D3semidirect
85. C3×#41C22×C5semidirect
87. C2×C3×D4C22×C5semidirect
90. #235C22×C5semidirect
91. C2×D12C22×C5semidirect
92. #216C22×C5semidirect
93. #215C22×C5semidirect
94. #214C22×C5semidirect
95. #213C22×C5semidirect
96. #213C22×C5semidirect
97. #208C22×C5semidirect
98. C2×C3⋊C8C22×C5semidirect
117. H(5,𝔽2)C5×D3semidirect
157. #41C2×C5×D3semidirect
161. C2×D4C2×C5×D3semidirect
169. C3×C5#576semidirect
204. C5#5835direct (N × H)
214. C3C5×#576semidirect

Maximal subgroups

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

subgroupindex
C3×C5×H(5,𝔽2)2
C5×#10092
C5×#9502
C5×#9492
C5×#9482
C5×#9322
C2×C5×#2132
C5×#5763
#58355