Finite Group Databasebeta

← C2×C5×#4149 #13107866

Subgroups — C2×C5×#4149 #13107866

Order 1600.

1960 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C52×#41C211yesmaximal, characteristic, complemented
2C5×#4149C2116yesmaximal, direct product factor
3C2×C5×#713C212yesmaximal, complemented
4C2×C5×D4×D5C216yesmaximal, complemented
5C2×C5×#709C213yesmaximal, complemented
6C22×C5×D20C213yesmaximal, complemented
7C23×C5×D5C2216yescomplemented
8C52×#41C2218yescomplemented
9C2×C52×Q8C2211yescharacteristic, complemented
10C2×C52×D4C2213yescomplemented
11C22×C4×C52C2213yescomplemented
12C2×C5×C22⋊D5C2216yescomplemented
13C5×#713C22116yescomplemented
14C5×D4×D5C22148yescomplemented
15C5×#709C22124yescomplemented
16C2×C5×D20C2213yescomplemented
17C2×C5×D20C2216yescomplemented
18C2×C5×D20C22124yescomplemented
19C2×C4×C5×D5C2216yescomplemented
20C2×C5×Dic10C2211yescharacteristic, complemented
21C2×C5×H(5,𝔽2)—51nomaximal
22C2×#4149C511yesmaximal, characteristic, direct product factor
23C23×C52C2313yes—
24C22×C5×D5—412no—
25C22×C5×D5C2316yes—
26C22×C5×D5C23124yescomplemented
27C52×Q8C2314yescomplemented
28C52×D4C23112yescomplemented
29C2×C4×C52C2311yescharacteristic
30C2×C4×C52C2313yes—
31C2×C4×C52C23112yescomplemented
32C5×C22⋊D5C23124yescomplemented
33C2×C5×Dic5C2312yes—
34C5×D20C23112yescomplemented
35C5×D20C23124yescomplemented
36C4×C5×D5C23124yescomplemented
37C5×Dic10C2314yescomplemented
38C5×H(5,𝔽2)—516no—
39C2×C5×#41—22no—
40C2×C5×#41—52no—
41C2×C5×#41—53no—
42C2×C5×#41D511yescharacteristic, complemented
43C2×C5×#41C2×C511yescharacteristic, complemented
44C22×C5×D4—56no—
45C22×C5×D4—53no—
46#4149C2×C5116yesdirect product factor
47C2×#713C2×C512yescomplemented
48C2×D4×D5C2×C516yescomplemented
49C2×#709C2×C513yescomplemented
50C22×D20C2×C513yescomplemented
51C22×C52—26no—
52C22×C52C2416yes—
53C22×C52C2411yescharacteristic
54C2×C5×D5—212no—
55C2×C5×D5—424no—
56C2×C5×D5C24112yes—
57C4×C52C2412yes—
58C4×C52C2416yes—
59C5×Dic5C2414yes—
60C24×C5—56no—
61C23×D5C22×C516yescomplemented
62C5×#41—216no—
63C5×#41—516no—
64C5×#41—524no—
65C5×#41C2×D518yescomplemented
66C5×#41C22×C518yescomplemented
67C2×C5×Q8—22no—
68C2×C5×Q8—51no—
69C2×C5×Q8C2×D511yescharacteristic, complemented
70C2×C5×Q8C22×C511yescharacteristic, complemented
71C2×C5×D4—26no—
72C2×C5×D4—56no—
73C2×C5×D4—548no—
74C2×C5×D4—56no—
75C2×C5×D4—53no—
76C2×C5×D4—524no—
77C2×C5×D4C2×D513yescomplemented
78C2×C5×D4C22×C513yescomplemented
79C22×C4×C5—26no—
80C22×C4×C5—56no—
81C22×C4×C5C2×D513yescomplemented
82C22×C4×C5C22×C513yescomplemented
83C2×C22⋊D5C22×C516yescomplemented
84#713C22×C5116yescomplemented
85D4×D5C22×C5148yescomplemented
86#709C22×C5124yescomplemented
87C2×D20C22×C513yescomplemented
88C2×D20C22×C516yescomplemented
89C2×D20C22×C5124yescomplemented
90C2×C4×D5C22×C516yescomplemented
91C2×Dic10C22×C511yescharacteristic, complemented
92C2×H(5,𝔽2)—51noSylow
93C2×C52—26no—
94C2×C52H(5,𝔽2)12yescomplemented
95C2×C52C2511yescharacteristic
96C5×D5—212no—
97C23×C5—26no—
98C23×C5—56no—
99C23×C5—524no—
100C23×C5—2012no—
101C23×C5C22×D513yes—
102C23×C5C23×C513yes—
103C22×D5—412no—
104C22×D5C23×C516yes—
105C22×D5C23×C5124yescomplemented
106C5×Q8—28no—
107C5×Q8—54no—
108C5×Q8C22×D514yescomplemented
109C5×Q8C23×C514yescomplemented
110C5×D4—224no—
111C5×D4—524no—
112C5×D4—524no—
113C5×D4—512no—
114C5×D4C22×D5112yescomplemented
115C5×D4C23×C5112yescomplemented
116C2×C4×C5—22no—
117C2×C4×C5—26no—
118C2×C4×C5—224no—
119C2×C4×C5—52no—
120C2×C4×C5—524no—
121C2×C4×C5C22×D513yes—
122C2×C4×C5C22×D511yescharacteristic
123C2×C4×C5C22×D5112yescomplemented
124C2×C4×C5C23×C511yescharacteristic
125C2×C4×C5C23×C513yes—
126C2×C4×C5C23×C5112yescomplemented
127C22⋊D5C23×C5124yescomplemented
128C2×Dic5C23×C512yes—
129D20C23×C5112yescomplemented
130D20C23×C5124yescomplemented
131C4×D5C23×C5124yescomplemented
132Dic10C23×C514yescomplemented
133H(5,𝔽2)—516no—
134C2×#41—52no—
135C2×#41—53no—
136C2×#41C5×D511yescharacteristic, complemented
137C22×D4—56no—
138C22×D4—53no—
139C52C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
140C22×C5—26no—
141C22×C5—26no—
142C22×C5—22no—
143C22×C5—212no—
144C22×C5—412no—
145C22×C5—512no—
146C22×C5—1012no—
147C22×C5—2024no—
148C22×C5C23×D516yes—
149C22×C5C23×D511yescharacteristic
150C22×C5C24×C516yes—
151C22×C5C24×C511yescharacteristic
152C2×D5—212no—
153C2×D5—424no—
154C2×D5C24×C5112yes—
155C4×C5—24no—
156C4×C5—212no—
157C4×C5—54no—
158C4×C5C23×D516yes—
159C4×C5C23×D512yes—
160C4×C5C24×C512yes—
161C4×C5C24×C516yes—
162Dic5C24×C514yes—
163C24—56no—
164#41—516no—
165#41—524no—
166#41C2×C5×D518yescomplemented
167C2×Q8—51no—
168C2×Q8C2×C5×D511yescharacteristic, complemented
169C2×D4—56no—
170C2×D4—548no—
171C2×D4—56no—
172C2×D4—53no—
173C2×D4—524no—
174C2×D4C2×C5×D513yescomplemented
175C22×C4—56no—
176C22×C4C2×C5×D513yescomplemented
177C2×C5—26no—
178C2×C5—26no—
179C2×C5—24no—
180C2×C5—22no—
181C2×C5—412no—
182C2×C5—1012no—
183C2×C5#414912yesdirect product factor
184C2×C5C5×H(5,𝔽2)12yescomplemented
185C2×C5C24×D511yescharacteristic
186C2×C5C25×C511yescharacteristic
187D5—212no—
188C23—56no—
189C23—524no—
190C23—2012no—
191C23C22×C5×D513yes—
192Q8—54no—
193Q8C22×C5×D514yescomplemented
194D4—524no—
195D4—524no—
196D4—512no—
197D4C22×C5×D5112yescomplemented
198C2×C4—52no—
199C2×C4—524no—
200C2×C4C22×C5×D513yes—
201C2×C4C22×C5×D511yescharacteristic
202C2×C4C22×C5×D5112yescomplemented
203C5—22no—
204C5C2×#414911yescharacteristic, direct product factor
205C5C2×C5×H(5,𝔽2)11yescharacteristic, complemented
206C22—26no—
207C22—512no—
208C22—1012no—
209C22—2024no—
210C22C23×C5×D516yes—
211C22C23×C5×D511yescharacteristic
212C4—54no—
213C4C23×C5×D516yes—
214C4C23×C5×D512yes—
215C2—26no—
216C2—1012no—
217C2C5×#414912yesdirect product factor
218C2C24×C5×D511yescharacteristic
subgroupG/H
centerC22×C5C23×D5
commutator, cosocleC2×C5C25×C5
FrattiniC2C24×C5×D5
FittingC2×C52×#41C2
socleC22×C52C24
hypercenterC2×C5×#41D5
hypocenterC5C2×C5×H(5,𝔽2)
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C5 ◁ C2×C5×#41

Complements

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

NH ≅ G/Ntype
1. C2×C52×#41C2semidirect
2. C5×#4149C2direct (N × H)
3. C2×C5×#713C2semidirect
4. C2×C5×D4×D5C2semidirect
5. C2×C5×#709C2semidirect
6. C22×C5×D20C2semidirect
7. C23×C5×D5C22semidirect
8. C52×#41C22semidirect
9. C2×C52×Q8C22semidirect
10. C2×C52×D4C22semidirect
11. C22×C4×C52C22semidirect
12. C2×C5×C22⋊D5C22semidirect
13. C5×#713C22semidirect
14. C5×D4×D5C22semidirect
15. C5×#709C22semidirect
16. C2×C5×D20C22semidirect
17. C2×C5×D20C22semidirect
18. C2×C5×D20C22semidirect
19. C2×C4×C5×D5C22semidirect
20. C2×C5×Dic10C22semidirect
22. C2×#4149C5direct (N × H)
26. C22×C5×D5C23semidirect
27. C52×Q8C23semidirect
28. C52×D4C23semidirect
31. C2×C4×C52C23semidirect
32. C5×C22⋊D5C23semidirect
34. C5×D20C23semidirect
35. C5×D20C23semidirect
36. C4×C5×D5C23semidirect
37. C5×Dic10C23semidirect
42. C2×C5×#41D5semidirect
43. C2×C5×#41C2×C5semidirect
46. #4149C2×C5direct (N × H)
47. C2×#713C2×C5semidirect
48. C2×D4×D5C2×C5semidirect
49. C2×#709C2×C5semidirect
50. C22×D20C2×C5semidirect
61. C23×D5C22×C5semidirect
65. C5×#41C2×D5semidirect
66. C5×#41C22×C5semidirect
69. C2×C5×Q8C2×D5semidirect
70. C2×C5×Q8C22×C5semidirect
77. C2×C5×D4C2×D5semidirect
78. C2×C5×D4C22×C5semidirect
81. C22×C4×C5C2×D5semidirect
82. C22×C4×C5C22×C5semidirect
83. C2×C22⋊D5C22×C5semidirect
84. #713C22×C5semidirect
85. D4×D5C22×C5semidirect
86. #709C22×C5semidirect
87. C2×D20C22×C5semidirect
88. C2×D20C22×C5semidirect
89. C2×D20C22×C5semidirect
90. C2×C4×D5C22×C5semidirect
91. C2×Dic10C22×C5semidirect
94. C2×C52H(5,𝔽2)semidirect
105. C22×D5C23×C5semidirect
108. C5×Q8C22×D5semidirect
109. C5×Q8C23×C5semidirect
114. C5×D4C22×D5semidirect
115. C5×D4C23×C5semidirect
123. C2×C4×C5C22×D5semidirect
126. C2×C4×C5C23×C5semidirect
127. C22⋊D5C23×C5semidirect
129. D20C23×C5semidirect
130. D20C23×C5semidirect
131. C4×D5C23×C5semidirect
132. Dic10C23×C5semidirect
136. C2×#41C5×D5semidirect
139. C52C2×H(5,𝔽2)semidirect
166. #41C2×C5×D5semidirect
168. C2×Q8C2×C5×D5semidirect
174. C2×D4C2×C5×D5semidirect
176. C22×C4C2×C5×D5semidirect
183. C2×C5#4149direct (N × H)
184. C2×C5C5×H(5,𝔽2)semidirect
193. Q8C22×C5×D5semidirect
197. D4C22×C5×D5semidirect
202. C2×C4C22×C5×D5semidirect
204. C5C2×#4149direct (N × H)
205. C5C2×C5×H(5,𝔽2)semidirect
217. C2C5×#4149direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C52×#412
C5×#41492
C2×C5×#7132
C2×C5×D4×D52
C2×C5×#7092
C22×C5×D202
C2×C5×H(5,𝔽2)5
C2×#41495