Finite Group Databasebeta

← #123×C5⋊D5 #13107184

Subgroups — #123×C5⋊D5 #13107184

Order 1600.

1368 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1368 conjugacy classes of subgroups — too many to draw. Only its 243 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
1C2×D4×C5⋊D5C211yesmaximal, characteristic
2C22×C4×C5⋊D5C211yesmaximal, characteristic, complemented
3C52×#123C211yesmaximal, characteristic, complemented
4#11717255C211yesmaximal, characteristic, complemented
5#11717249C211yesmaximal, characteristic, complemented
6#11717214C212yesmaximal, complemented
7C4⋊C4×C5⋊D5C212yesmaximal, complemented
8#11717206C211yesmaximal, characteristic, complemented
9#11717203C212yesmaximal, complemented
10C22⋊C4×C5⋊D5C211yesmaximal, characteristic, complemented
11C22⋊C4×C5⋊D5C212yesmaximal, complemented
12C23×C5⋊D5C2211yescharacteristic
13C23×C5⋊D5C2211yescharacteristic
14C2×C52×D4C2211yescharacteristic
15C22×C4×C52C2211yescharacteristic, complemented
16C2×C22⋊(C5⋊D5)C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C22×#1039C2211yescharacteristic, complemented
19D4×C5⋊D5—24no—
20C2×C4⋊(C5⋊D5)C2211yescharacteristic
21C2×C4×C5⋊D5—24no—
22C2×C4×C5⋊D5C2212yes—
23C2×C4×C5⋊D5C2212yes—
24C2×C4×C5⋊D5C2211yescharacteristic
25C2×C4×C5⋊D5C2212yescomplemented
26C52×C4⋊C4C2212yescomplemented
27C52×C22⋊C4C2212yescomplemented
28C52×C22⋊C4C2211yescharacteristic, complemented
29C22⋊#1039C2212yescomplemented
30C22⋊#1039C2211yescharacteristic, complemented
31#87808C2212yescomplemented
32#87808C2212yescomplemented
33#87808C2212yescomplemented
34C4⋊#1039C2212yescomplemented
35#87806C2212yescomplemented
36#87806C2212yescomplemented
37D5×#123—56nomaximal
38C23×C52C2311yescharacteristic
39C23×C52C2311yescharacteristic
40C22×C5⋊D5—24no—
41C22×C5⋊D5—22no—
42C22×C5⋊D5—24no—
43C22×C5⋊D5D414yes—
44C22×C5⋊D5C2311yescharacteristic
45C22×C5⋊D5C2311yescharacteristic
46C22×C5⋊D5C2311yescharacteristic
47C52×D4—22no—
48C2×C4×C52—22no—
49C2×C4×C52C2312yes—
50C2×C4×C52C2312yes—
51C2×C4×C52C2311yescharacteristic
52C22⋊(C5⋊D5)—22no—
53C22⋊(C5⋊D5)—22no—
54C2×#1039—22no—
55C2×#1039C2312yes—
56C2×#1039C2312yes—
57C2×#1039C2311yescharacteristic
58C4⋊(C5⋊D5)—22no—
59C4×C5⋊D5—24no—
60C4×C5⋊D5—24no—
61C4×C5⋊D5—22no—
62C4×C5⋊D5—24no—
63C2×D4×D5—56no—
64C22×C4×D5—56no—
65C5×#123D516yescomplemented
66#4081—56no—
67#4075—56no—
68#4040—512no—
69D5×C4⋊C4—512no—
70#4032—56no—
71#4029—512no—
72D5×C22⋊C4—56no—
73D5×C22⋊C4—512no—
74C22×C52—22no—
75C22×C52—21no—
76C22×C52—22no—
77C22×C52C2×D412yes—
78C22×C52C2411yescharacteristic
79C2×C5⋊D5—22no—
80C2×C5⋊D5—28no—
81C2×C5⋊D5—21no—
82C2×C5⋊D5—22no—
83C2×C5⋊D5—44no—
84C2×C5⋊D5C2×D412yes—
85C2×C5⋊D5C2×D412yes—
86C2×C5⋊D5#4114yes—
87C4×C52—22no—
88C4×C52—21no—
89C4×C52—22no—
90#1039—22no—
91#1039—21no—
92#1039—22no—
93C23×D5—56no—
94C23×D5—56no—
95C2×C5×D4C2×D516yes—
96C22×C4×C5C2×D516yescomplemented
97C2×C22⋊D5—56no—
98C2×C22⋊D5—56no—
99C22×Dic5—56no—
100D4×D5—1024no—
101C2×D20—56no—
102C2×C4×D5—512no—
103C2×C4×D5—512no—
104C2×C4×D5—56no—
105C2×C4×D5—512no—
106C2×C4×D5—1024no—
107C5×C4⋊C4C2×D5112yescomplemented
108C5×C22⋊C4C2×D5112yescomplemented
109C5×C22⋊C4C2×D516yescomplemented
110C22⋊Dic5—512no—
111C22⋊Dic5—56no—
112#685—512no—
113#685—512no—
114#685—512no—
115C4⋊Dic5—512no—
116#683—512no—
117#683—512no—
118C2×#123—251noSylow
119C2×C52—22no—
120C2×C52—41no—
121C2×C52C22×D411yescharacteristic
122C2×C52C2×#4112yes—
123C5⋊D5—22no—
124C5⋊D5—41no—
125C5⋊D5#12314yesdirect product factor
126C23×C5C22×D516yes—
127C23×C5C22×D516yes—
128C22×D5—56no—
129C22×D5—524no—
130C22×D5—56no—
131C22×D5—56no—
132C22×D5—1024no—
133C22×D5—1012no—
134C22×D5—1024no—
135C5×D4—212no—
136C2×C4×C5—212no—
137C2×C4×C5C22×D516yes—
138C2×C4×C5C22×D5112yes—
139C2×C4×C5C22×D5112yes—
140C22⋊D5—1012no—
141C22⋊D5—1012no—
142C2×Dic5—56no—
143C2×Dic5—512no—
144C2×Dic5—512no—
145C2×Dic5—1012no—
146D20—1012no—
147C4×D5—1024no—
148C4×D5—1024no—
149C4×D5—1024no—
150C4×D5—1012no—
151C22×D4—251no—
152C23×C4—251no—
153#123—251no—
154#123—251no—
155#123—252no—
156#123—252no—
157#123—251no—
158#123C5⋊D511yescharacteristic, direct product factor
159C2×C4⋊C4—252no—
160C2×C22⋊C4—251no—
161C2×C22⋊C4—252no—
162C52C2×#12311yescharacteristic, complemented, Sylow
163C22×C5—212no—
164C22×C5—26no—
165C22×C5—212no—
166C22×C5D4×D5112yes—
167C22×C5C23×D516yes—
168C2×D5—512no—
169C2×D5—524no—
170C2×D5—512no—
171C2×D5—1048no—
172C2×D5—1012no—
173C2×D5—106no—
174C2×D5—1012no—
175C2×D5—2024no—
176C4×C5—212no—
177C4×C5—26no—
178C4×C5—212no—
179Dic5—1012no—
180Dic5—106no—
181Dic5—1012no—
182C24—251no—
183C24—251no—
184C2×D4—251no—
185C2×D4—251no—
186C2×D4—251no—
187C2×D4—504no—
188C2×D4C2×C5⋊D511yescharacteristic
189C22×C4—251no—
190C22×C4—252no—
191C22×C4—252no—
192C22×C4—251no—
193C22×C4—252no—
194C22×C4—504no—
195C22×C4C2×C5⋊D511yescharacteristic, complemented
196C4⋊C4—252no—
197C4⋊C4—252no—
198C4⋊C4—252no—
199C4⋊C4C2×C5⋊D512yescomplemented
200C22⋊C4—251no—
201C22⋊C4—252no—
202C22⋊C4—252no—
203C22⋊C4—252no—
204C22⋊C4—252no—
205C22⋊C4C2×C5⋊D511yescharacteristic, complemented
206C22⋊C4C2×C5⋊D512yescomplemented
207C2×C5—212no—
208C2×C5—46no—
209C2×C5C2×D4×D516yes—
210C2×C5D5×#41112yes—
211D5—524no—
212D5—1012no—
213D5—206no—
214C23—251no—
215C23—254no—
216C23—251no—
217C23—251no—
218C23—504no—
219C23—502no—
220C23—504no—
221C23C22×C5⋊D511yescharacteristic
222C23C22×C5⋊D511yescharacteristic
223D4—22no—
224D4—502no—
225D4—502no—
226D4—502no—
227C2×C4—22no—
228C2×C4—252no—
229C2×C4—252no—
230C2×C4—251no—
231C2×C4—504no—
232C2×C4—504no—
233C2×C4—502no—
234C2×C4—504no—
235C2×C4—502no—
236C2×C4C22×C5⋊D512yes—
237C2×C4C22×C5⋊D512yes—
238C2×C4C22×C5⋊D511yescharacteristic
239C5D5×#12316yescomplemented
240C22—22no—
241C22—21no—
242C22—22no—
243C22—252no—
244C22—254no—
245C22—252no—
246C22—508no—
247C22—502no—
248C22—501no—
249C22—502no—
250C22—1004no—
251C22D4×C5⋊D512yes—
252C22C23×C5⋊D511yescharacteristic
253C4—22no—
254C4—21no—
255C4—22no—
256C4—502no—
257C4—501no—
258C4—502no—
259C2—22no—
260C2—41no—
261C2—254no—
262C2—502no—
263C2—1001no—
264C2C2×D4×C5⋊D511yescharacteristic
265C2#41×C5⋊D512yes—
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#123C2
prime radicalC2×D4×C5⋊D5C2
hypercenter#123C5⋊D5
hypocenterC52C2×#123
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×C5⋊D5C2semidirect
3. C52×#123C2semidirect
4. #11717255C2semidirect
5. #11717249C2semidirect
6. #11717214C2semidirect
7. C4⋊C4×C5⋊D5C2semidirect
8. #11717206C2semidirect
9. #11717203C2semidirect
10. C22⋊C4×C5⋊D5C2semidirect
11. C22⋊C4×C5⋊D5C2semidirect
15. C22×C4×C52C22semidirect
18. C22×#1039C22semidirect
25. C2×C4×C5⋊D5C22semidirect
26. C52×C4⋊C4C22semidirect
27. C52×C22⋊C4C22semidirect
28. C52×C22⋊C4C22semidirect
29. C22⋊#1039C22semidirect
30. C22⋊#1039C22semidirect
31. #87808C22semidirect
32. #87808C22semidirect
33. #87808C22semidirect
34. C4⋊#1039C22semidirect
35. #87806C22semidirect
36. #87806C22semidirect
65. C5×#123D5semidirect
96. C22×C4×C5C2×D5semidirect
107. C5×C4⋊C4C2×D5semidirect
108. C5×C22⋊C4C2×D5semidirect
109. C5×C22⋊C4C2×D5semidirect
125. C5⋊D5#123direct (N × H)
158. #123C5⋊D5direct (N × H)
162. C52C2×#123semidirect
195. C22×C4C2×C5⋊D5semidirect
199. C4⋊C4C2×C5⋊D5semidirect
205. C22⋊C4C2×C5⋊D5semidirect
206. C22⋊C4C2×C5⋊D5semidirect
239. C5D5×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C5⋊D52
C22×C4×C5⋊D52
C52×#1232
#117172552
#117172492
#117172142
C4⋊C4×C5⋊D52
#117172062
#117172032
C22⋊C4×C5⋊D52
D5×#1235