Finite Group Databasebeta

← #13107176

Subgroups — #13107176

Order 1600.

1320 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 239 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×#87902C211yesmaximal, characteristic, complemented
2C22×C4×C5⋊D5C211yesmaximal, characteristic, complemented
3C52×#123C211yesmaximal, characteristic, complemented
4#11717257C211yesmaximal, characteristic, complemented
5#11717249C211yesmaximal, characteristic, complemented
6#11717216C212yesmaximal, complemented
7#11717213C212yesmaximal, complemented
8#11717204C212yesmaximal, complemented
9#11717202C211yesmaximal, characteristic, complemented
10#11717201C212yesmaximal, complemented
11#11717197C211yesmaximal, characteristic, complemented
12C23×C5⋊D5C2211yescharacteristic
13C2×C52×D4C2211yescharacteristic
14C22×C4×C52C2211yescharacteristic
15C2×C22⋊(C5⋊D5)C2212yes—
16C2×C22⋊(C5⋊D5)C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
18C22×#1039C2211yescharacteristic
19C22×#1039C2211yescharacteristic, complemented
20#87902—24no—
21C2×C4⋊(C5⋊D5)C2212yes—
22C2×C4×C5⋊D5—24no—
23C2×C4×C5⋊D5C2212yes—
24C2×C4×C5⋊D5C2212yes—
25C2×C4×C5⋊D5C2211yescharacteristic, complemented
26C2×#6045C2211yescharacteristic, complemented
27C52×C4⋊C4C2212yescomplemented
28C52×C22⋊C4C2211yescharacteristic
29C52×C22⋊C4C2212yescomplemented
30C22⋊#1039C2211yescharacteristic
31#87808C2212yes—
32#87808C2212yes—
33#87808C2212yescomplemented
34#87806C2212yescomplemented
35#87806C2212yescomplemented
36C4×#1039C2212yescomplemented
37#65990—56nomaximal
38C23×C52C2311yescharacteristic
39C23×C52C2311yescharacteristic
40C22×C5⋊D5—22no—
41C22×C5⋊D5—24no—
42C22×C5⋊D5C2311yescharacteristic
43C22×C5⋊D5C2312yes—
44C52×D4—22no—
45C2×C4×C52—22no—
46C2×C4×C52C2311yescharacteristic
47C2×C4×C52C2312yes—
48C2×C4×C52C2312yes—
49C22⋊(C5⋊D5)—22no—
50C22⋊(C5⋊D5)—24no—
51C22⋊(C5⋊D5)—22no—
52C2×#1039—22no—
53C2×#1039D414yescomplemented
54C2×#1039C2311yescharacteristic
55C2×#1039C2312yes—
56C2×#1039C2311yescharacteristic
57C2×#1039C2311yescharacteristic
58C4⋊(C5⋊D5)—24no—
59C4×C5⋊D5—24no—
60C4×C5⋊D5—24no—
61C4×C5⋊D5—22no—
62#6045—22no—
63C2×#711—56no—
64C22×C4×D5—56no—
65C5×#123D516yescomplemented
66#4083—56no—
67#4075—56no—
68#4042—512no—
69#4039—512no—
70#4030—512no—
71#4028—56no—
72#4027—512no—
73#4023—56no—
74C22×C52—22no—
75C22×C52—21no—
76C22×C52—22no—
77C22×C52C2×D412yes—
78C22×C52C2411yescharacteristic
79C2×C5⋊D5—28no—
80C2×C5⋊D5—22no—
81C2×C5⋊D5—22no—
82C2×C5⋊D5—21no—
83C2×C5⋊D5—22no—
84C2×C5⋊D5#4114yescomplemented
85C4×C52—21no—
86C4×C52—22no—
87C4×C52—22no—
88#1039—21no—
89#1039—22no—
90#1039C2×D412yes—
91#1039C2×D412yes—
92C23×D5—56no—
93C2×C5×D4C2×D516yes—
94C22×C4×C5C2×D516yes—
95C2×C22⋊D5—56no—
96C2×C22⋊D5—56no—
97C2×C22⋊D5—512no—
98C22×Dic5—56no—
99C22×Dic5—56no—
100#711—1024no—
101C2×D20—512no—
102C2×C4×D5—512no—
103C2×C4×D5—512no—
104C2×C4×D5—56no—
105C2×C4×D5—1024no—
106C2×Dic10—56no—
107C5×C4⋊C4C2×D5112yescomplemented
108C5×C22⋊C4C2×D516yes—
109C5×C22⋊C4C2×D5112yescomplemented
110C22⋊Dic5—56no—
111#685—512no—
112#685—512no—
113#685—512no—
114#683—512no—
115#683—512no—
116C4×Dic5—512no—
117#525—251noSylow
118C2×C52—22no—
119C2×C52—41no—
120C2×C52C22×D411yescharacteristic
121C2×C52C2×#4112yes—
122C5⋊D5—24no—
123C5⋊D5—41no—
124C23×C5C22×D516yes—
125C23×C5C22×D516yes—
126C22×D5—512no—
127C22×D5—56no—
128C22×D5—1024no—
129C22×D5—1012no—
130C5×D4—212no—
131C2×C4×C5—212no—
132C2×C4×C5C22×D5112yes—
133C2×C4×C5C22×D5112yes—
134C2×C4×C5C22×D516yes—
135C22⋊D5—1012no—
136C22⋊D5—1024no—
137C22⋊D5—1012no—
138C2×Dic5—524no—
139C2×Dic5—56no—
140C2×Dic5—56no—
141C2×Dic5—512no—
142C2×Dic5—56no—
143C2×Dic5—1012no—
144D20—1024no—
145C4×D5—1024no—
146C4×D5—1024no—
147C4×D5—1012no—
148Dic10—1012no—
149C2×#41—251no—
150C23×C4—251no—
151#123—251no—
152#123C5⋊D511yescharacteristic, complemented
153C22⋊Q8—252no—
154#121—252no—
155#120—251no—
156#120—251no—
157C4×D4—252no—
158C4×D4—252no—
159#117—251no—
160C52#52511yescharacteristic, complemented, Sylow
161C22×C5—212no—
162C22×C5—26no—
163C22×C5—212no—
164C22×C5D4×D5112yes—
165C22×C5C23×D516yes—
166C2×D5—524no—
167C2×D5—1048no—
168C2×D5—1012no—
169C2×D5—1012no—
170C2×D5—106no—
171C2×D5—1012no—
172C4×C5—26no—
173C4×C5—212no—
174C4×C5—212no—
175Dic5—512no—
176Dic5—512no—
177Dic5—106no—
178Dic5—1012no—
179C24—251no—
180#41—504no—
181C2×Q8—251no—
182C2×D4—252no—
183C2×D4—251no—
184C2×D4—251no—
185C2×D4—252no—
186C2×D4C2×C5⋊D511yescharacteristic
187C22×C4—251no—
188C22×C4—251no—
189C22×C4—252no—
190C22×C4—251no—
191C22×C4—252no—
192C22×C4—504no—
193C22×C4C2×C5⋊D511yescharacteristic
194C4⋊C4—252no—
195C4⋊C4—252no—
196C4⋊C4C2×C5⋊D512yescomplemented
197C22⋊C4—251no—
198C22⋊C4—252no—
199C22⋊C4—252no—
200C22⋊C4—252no—
201C22⋊C4C2×C5⋊D511yescharacteristic
202C22⋊C4C2×C5⋊D512yescomplemented
203C42—252no—
204C2×C5—212no—
205C2×C5—46no—
206C2×C5C2×D4×D516yes—
207C2×C5D5×#41112yes—
208D5—1024no—
209D5—206no—
210C23—252no—
211C23—251no—
212C23—504no—
213C23—502no—
214C23C22×C5⋊D511yescharacteristic
215C23C22×C5⋊D511yescharacteristic
216Q8—502no—
217D4—22no—
218D4—504no—
219D4—502no—
220D4—504no—
221D4—502no—
222C2×C4—22no—
223C2×C4—254no—
224C2×C4—251no—
225C2×C4—251no—
226C2×C4—252no—
227C2×C4—251no—
228C2×C4—504no—
229C2×C4—504no—
230C2×C4—502no—
231C2×C4—502no—
232C2×C4C22×C5⋊D512yes—
233C2×C4C22×C5⋊D511yescharacteristic
234C2×C4C22×C5⋊D512yes—
235C5#6599016yescomplemented
236C22—22no—
237C22—21no—
238C22—22no—
239C22—254no—
240C22—508no—
241C22—502no—
242C22—502no—
243C22—501no—
244C22—502no—
245C22D4×C5⋊D512yes—
246C22C23×C5⋊D511yescharacteristic
247C4—22no—
248C4—21no—
249C4—22no—
250C4—252no—
251C4—252no—
252C4—501no—
253C4—502no—
254C2—22no—
255C2—41no—
256C2—504no—
257C2—1001no—
258C2C2×D4×C5⋊D511yescharacteristic
259C2#41×C5⋊D512yes—
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#123C2
hypercenter#123C5⋊D5
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#87902C2semidirect
2. C22×C4×C5⋊D5C2semidirect
3. C52×#123C2semidirect
4. #11717257C2semidirect
5. #11717249C2semidirect
6. #11717216C2semidirect
7. #11717213C2semidirect
8. #11717204C2semidirect
9. #11717202C2semidirect
10. #11717201C2semidirect
11. #11717197C2semidirect
17. C2×C22⋊(C5⋊D5)C22semidirect
19. C22×#1039C22semidirect
25. C2×C4×C5⋊D5C22semidirect
26. C2×#6045C22semidirect
27. C52×C4⋊C4C22semidirect
29. C52×C22⋊C4C22semidirect
33. #87808C22semidirect
34. #87806C22semidirect
35. #87806C22semidirect
36. C4×#1039C22semidirect
53. C2×#1039D4semidirect
65. C5×#123D5semidirect
84. C2×C5⋊D5#41semidirect
107. C5×C4⋊C4C2×D5semidirect
109. C5×C22⋊C4C2×D5semidirect
152. #123C5⋊D5semidirect
160. C52#525semidirect
196. C4⋊C4C2×C5⋊D5semidirect
202. C22⋊C4C2×C5⋊D5semidirect
235. C5#65990semidirect

Maximal subgroups

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

subgroupindex
C2×#879022
C22×C4×C5⋊D52
C52×#1232
#117172572
#117172492
#117172162
#117172132
#117172042
#117172022
#117172012
#117171972
#659905