Finite Group Databasebeta

← C5×#5836 #11756221

Subgroups — C5×#5836 #11756221

Order 960.

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

normal subgroups only

This group has 516 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×#1010C211yesmaximal, characteristic, complemented
3C5×#951C213yesmaximal, complemented
4C5×#950C213yesmaximal, complemented
5C5×#948C211yesmaximal, characteristic, complemented
6C2×C5×#214C213yesmaximal, complemented
7C5×#932C213yesmaximal, complemented
8C5×#577—31nomaximal, Hall
9C3×C5×#41—21no—
10C3×C5×#41C2213yes—
11C3×C5×#41C2211yescharacteristic
12C2×C3×C5×D4—23no—
13C2×C3×C5×D4C2213yes—
14C5×D3×Q8—21no—
15C5×#237—23no—
16C5×#235C2213yescomplemented
17C2×C5×Dic6C2213yescomplemented
18C5×#216C2213yescomplemented
19C5×#215C2211yescharacteristic, complemented
20C5×#214C2216yescomplemented
21C5×#214C2213yescomplemented
22C5×#213C2213yescomplemented
23C5×#208C2213yescomplemented
24C2×C5×C3⋊C8C2213yescomplemented
25#5836C511yesmaximal, characteristic, direct product factor, Hall
26C5×2−1+4—31no—
27C5×H(5,𝔽2)D311yescharacteristic, complemented
28C5×#137—33no—
29C5×AGL(1,ℤ/8)—33no—
30C5×#135—33no—
31C2×C5×SD16—33no—
32C5×#131—31no—
33C23×C3×C5—23no—
34C3×C5×Q8D411yescharacteristic
35C3×C5×Q8C2311yescharacteristic
36C3×C5×D4—23no—
37C3×C5×D4—23no—
38C3×C5×D4D413yes—
39C3×C5×D4C2313yes—
40C2×C3×C4×C5—23no—
41C2×C3×C4×C5C2313yes—
42C5×#67—23no—
43C2×C5×Dic3—23no—
44C5×D12C2311yescharacteristic
45C4×C5×D3—23no—
46C5×Dic6—23no—
47C5×Dic6C2313yes—
48C5×C3⋊C8C2311yescharacteristic
49C5×C3⋊C8C2313yes—
50C3×H(5,𝔽2)C2×C511yescharacteristic, complemented
51#1010C2×C511yescharacteristic, complemented
52#951C2×C513yescomplemented
53#950C2×C513yescomplemented
54#948C2×C511yescharacteristic, complemented
55C2×#214C2×C513yescomplemented
56#932C2×C513yescomplemented
57C5×#41—21no—
58C5×#41—33no—
59C5×#41—63no—
60C5×#41C2×D313yes—
61C5×#41C2×D311yescharacteristic
62C2×C5×Q8—33no—
63C2×C5×Q8—61no—
64C2×C5×D4—23no—
65C2×C5×D4C2×D313yes—
66C5×Q16—33no—
67C5×SD16—31no—
68C5×SD16—33no—
69C5×SD16—36no—
70C5×D8—33no—
71C5×M16—33no—
72C2×C5×C8—33no—
73#577—31noSylow
74C22×C3×C5—23no—
75C22×C3×C5—83no—
76C22×C3×C5C2×D413yes—
77C2×C5×D3—21no—
78C3×C4×C5—21no—
79C3×C4×C5C2×D413yes—
80C3×C4×C5C2411yescharacteristic
81C5×Dic3—23no—
82C3×#41—21no—
83C3×#41C22×C513yes—
84C3×#41C22×C511yescharacteristic
85C2×C3×D4—23no—
86C2×C3×D4C22×C513yes—
87D3×Q8—21no—
88#237—23no—
89#235C22×C513yescomplemented
90C2×Dic6C22×C513yescomplemented
91#216C22×C513yescomplemented
92#215C22×C511yescharacteristic, complemented
93#214C22×C516yescomplemented
94#214C22×C513yescomplemented
95#213C22×C513yescomplemented
96#208C22×C513yescomplemented
97C2×C3⋊C8C22×C513yescomplemented
98C23×C5—23no—
99C5×Q8—33no—
100C5×Q8—63no—
101C5×Q8#6711yescharacteristic
102C5×Q8C22×D311yescharacteristic
103C5×D4—23no—
104C5×D4—23no—
105C5×D4—31no—
106C5×D4—63no—
107C5×D4#6713yes—
108C5×D4C22×D313yes—
109C2×C4×C5—23no—
110C2×C4×C5—63no—
111C2×C4×C5—63no—
112C2×C4×C5C22×D313yes—
113C5×C8—31no—
114C5×C8—33no—
1152−1+4—31no—
116H(5,𝔽2)C5×D311yescharacteristic, complemented
117#137—33no—
118AGL(1,ℤ/8)—33no—
119#135—33no—
120C2×SD16—33no—
121#131—31no—
122C2×C3×C5—23no—
123C2×C3×C5—43no—
124C2×C3×C5C22×D411yescharacteristic
125C5×D3—41no—
126C23×C3—23no—
127C3×Q8C5×D411yescharacteristic
128C3×Q8C23×C511yescharacteristic
129C3×D4—23no—
130C3×D4—23no—
131C3×D4C5×D413yes—
132C3×D4C23×C513yes—
133C2×C3×C4—23no—
134C2×C3×C4C23×C513yes—
135#67—23no—
136C2×Dic3—23no—
137D12C23×C511yescharacteristic
138C4×D3—23no—
139Dic6—23no—
140Dic6C23×C513yes—
141C3⋊C8C23×C511yescharacteristic
142C3⋊C8C23×C513yes—
143C22×C5—23no—
144C22×C5—61no—
145C22×C5—83no—
146C22×C5C2×#6713yes—
147C4×C5—21no—
148C4×C5—63no—
149C4×C5C2×#6713yes—
150C4×C5C23×D311yescharacteristic
151#41—21no—
152#41—33no—
153#41—63no—
154#41C2×C5×D313yes—
155#41C2×C5×D311yescharacteristic
156C2×Q8—33no—
157C2×Q8—61no—
158C2×D4—23no—
159C2×D4C2×C5×D313yes—
160Q16—33no—
161SD16—31no—
162SD16—33no—
163SD16—36no—
164D8—33no—
165M16—33no—
166C2×C8—33no—
167C3×C5#57711yescharacteristic, complemented, Hall
168C22×C3—23no—
169C22×C3—83no—
170C22×C3C2×C5×D413yes—
171C2×D3—21no—
172C3×C4—21no—
173C3×C4C2×C5×D413yes—
174C3×C4C24×C511yescharacteristic
175Dic3—23no—
176C2×C5—23no—
177C2×C5—43no—
178C2×C5—121no—
179C2×C5C22×#6711yescharacteristic
180C23—23no—
181Q8—33no—
182Q8—63no—
183Q8C5×#6711yescharacteristic
184Q8C22×C5×D311yescharacteristic
185D4—23no—
186D4—23no—
187D4—31no—
188D4—63no—
189D4C5×#6713yes—
190D4C22×C5×D313yes—
191C2×C4—23no—
192C2×C4—63no—
193C2×C4—63no—
194C2×C4C22×C5×D313yes—
195C8—31no—
196C8—33no—
197C2×C3—23no—
198C2×C3—43no—
199C2×C3C22×C5×D411yescharacteristic
200D3—41no—
201C5#583611yescharacteristic, direct product factor, Sylow
202C22—23no—
203C22—61no—
204C22—83no—
205C22C2×C5×#6713yes—
206C4—21no—
207C4—63no—
208C4C2×C5×#6713yes—
209C4C23×C5×D311yescharacteristic
210C3C5×#57711yescharacteristic, complemented, Sylow
211C2—23no—
212C2—43no—
213C2—121no—
214C2C22×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×#577
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×#1010C2semidirect
3. C5×#951C2semidirect
4. C5×#950C2semidirect
5. C5×#948C2semidirect
6. C2×C5×#214C2semidirect
7. C5×#932C2semidirect
16. C5×#235C22semidirect
17. C2×C5×Dic6C22semidirect
18. C5×#216C22semidirect
19. C5×#215C22semidirect
20. C5×#214C22semidirect
21. C5×#214C22semidirect
22. C5×#213C22semidirect
23. C5×#208C22semidirect
24. C2×C5×C3⋊C8C22semidirect
25. #5836C5direct (N × H)
27. C5×H(5,𝔽2)D3semidirect
50. C3×H(5,𝔽2)C2×C5semidirect
51. #1010C2×C5semidirect
52. #951C2×C5semidirect
53. #950C2×C5semidirect
54. #948C2×C5semidirect
55. C2×#214C2×C5semidirect
56. #932C2×C5semidirect
89. #235C22×C5semidirect
90. C2×Dic6C22×C5semidirect
91. #216C22×C5semidirect
92. #215C22×C5semidirect
93. #214C22×C5semidirect
94. #214C22×C5semidirect
95. #213C22×C5semidirect
96. #208C22×C5semidirect
97. C2×C3⋊C8C22×C5semidirect
116. H(5,𝔽2)C5×D3semidirect
167. C3×C5#577semidirect
201. C5#5836direct (N × H)
210. C3C5×#577semidirect

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×#10102
C5×#9512
C5×#9502
C5×#9482
C2×C5×#2142
C5×#9322
C5×#5773
#58365