Finite Group Databasebeta

← #13107159

Subgroups — #13107159

Order 1600.

1320 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 253 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×#87904C211yesmaximal, characteristic, complemented
2C22×C4×C5⋊D5C211yesmaximal, characteristic, complemented
3C52×C22⋊Q8C211yesmaximal, characteristic, complemented
4#11717266C211yesmaximal, characteristic, complemented
5#11717250C211yesmaximal, characteristic, complemented
6#11717215C211yesmaximal, characteristic, complemented
7#11717214C212yesmaximal, complemented
8#11717213C212yesmaximal, complemented
9#11717212C211yesmaximal, characteristic, complemented
10#11717202C212yesmaximal, complemented
11#11717201C212yesmaximal, complemented
12C23×C5⋊D5C2211yescharacteristic
13C2×C52×Q8C2211yescharacteristic, complemented
14C22×C4×C52C2211yescharacteristic
15C2×C22⋊(C5⋊D5)C2212yes—
16C22×#1039C2211yescharacteristic
17#87904—24no—
18C2×C4⋊(C5⋊D5)C2212yes—
19C2×C4⋊(C5⋊D5)C2212yes—
20C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
21C2×C4×C5⋊D5—24no—
22C2×C4×C5⋊D5C2211yescharacteristic
23C2×C4×C5⋊D5C2211yescharacteristic
24C2×C4×C5⋊D5C2211yescharacteristic
25C2×C4×C5⋊D5C2211yescharacteristic
26C2×C4×C5⋊D5C2212yescomplemented
27C52×C4⋊C4C2211yescharacteristic, complemented
28C52×C4⋊C4C2212yescomplemented
29C52×C22⋊C4C2212yes—
30#87808C2212yes—
31#87808C2212yes—
32#87808C2212yescomplemented
33#87808C2212yescomplemented
34C4⋊#1039C2211yescharacteristic, complemented
35#87806C2212yescomplemented
36C4×#1039C2212yescomplemented
37#65961—56nomaximal
38C23×C52C2311yescharacteristic
39C22×C5⋊D5—22no—
40C22×C5⋊D5—22no—
41C22×C5⋊D5—22no—
42C22×C5⋊D5C2312yes—
43C22×C5⋊D5C2311yescharacteristic
44C22×C5⋊D5C2311yescharacteristic
45C52×Q8—22no—
46C2×C4×C52—22no—
47C2×C4×C52C2312yes—
48C2×C4×C52C2312yes—
49C2×C4×C52C2311yescharacteristic
50C2×C4×C52C2311yescharacteristic
51C22⋊(C5⋊D5)—24no—
52C2×#1039—22no—
53C2×#1039C2312yes—
54C2×#1039C2311yescharacteristic
55C2×#1039C2311yescharacteristic
56C4⋊(C5⋊D5)—24no—
57C4⋊(C5⋊D5)—24no—
58C4⋊(C5⋊D5)—22no—
59C4×C5⋊D5—22no—
60C4×C5⋊D5—22no—
61C4×C5⋊D5—22no—
62C4×C5⋊D5—24no—
63C4×C5⋊D5D414yescomplemented
64C2×#713—56no—
65C22×C4×D5—56no—
66C5×C22⋊Q8D516yescomplemented
67#4092—56no—
68#4076—56no—
69#4041—56no—
70#4040—512no—
71#4039—512no—
72#4038—56no—
73#4028—512no—
74#4027—512no—
75C22×C52—21no—
76C22×C52—21no—
77C22×C52#4112yescomplemented
78C22×C52C2411yescharacteristic
79C2×C5⋊D5—28no—
80C2×C5⋊D5—21no—
81C2×C5⋊D5—21no—
82C2×C5⋊D5—21no—
83C2×C5⋊D5—21no—
84C2×C5⋊D5—22no—
85C2×C5⋊D5—22no—
86C2×C5⋊D5—22no—
87C2×C5⋊D5C2×D412yes—
88C2×C5⋊D5#4112yescomplemented
89C4×C52—21no—
90C4×C52—22no—
91C4×C52—22no—
92C4×C52C2×D412yes—
93#1039—21no—
94#1039—22no—
95#1039C2×D412yes—
96C23×D5—56no—
97C2×C5×Q8C2×D516yescomplemented
98C22×C4×C5C2×D516yes—
99C2×C22⋊D5—512no—
100C22×Dic5—56no—
101#713—1024no—
102C2×D20—512no—
103C2×D20—56no—
104C2×D20—512no—
105C2×C4×D5—56no—
106C2×C4×D5—56no—
107C2×C4×D5—56no—
108C2×C4×D5—56no—
109C2×C4×D5—512no—
110C2×C4×D5—1024no—
111C5×C4⋊C4C2×D516yescomplemented
112C5×C4⋊C4C2×D5112yescomplemented
113C5×C22⋊C4C2×D5112yes—
114#685—512no—
115#685—512no—
116#685—512no—
117#685—512no—
118C4⋊Dic5—56no—
119#683—512no—
120C4×Dic5—512no—
121#525—251noSylow
122C2×C52—22no—
123C2×C52C22×D411yescharacteristic
124C2×C52C2×#4111yescharacteristic
125C2×C52C2×#4111yescharacteristic
126C5⋊D5—22no—
127C5⋊D5—22no—
128C5⋊D5—42no—
129C23×C5C22×D516yes—
130C22×D5—56no—
131C22×D5—56no—
132C22×D5—512no—
133C22×D5—1012no—
134C22×D5—1012no—
135C22×D5—1012no—
136C5×Q8—212no—
137C2×C4×C5—212no—
138C2×C4×C5C22×D5112yes—
139C2×C4×C5C22×D5112yes—
140C2×C4×C5C22×D516yes—
141C2×C4×C5C22×D516yes—
142C22⋊D5—1024no—
143C2×Dic5—56no—
144C2×Dic5—512no—
145C2×Dic5—56no—
146C2×Dic5—1012no—
147D20—1024no—
148D20—1024no—
149D20—1012no—
150C4×D5—524no—
151C4×D5—1012no—
152C4×D5—1024no—
153C4×D5—1012no—
154C4×D5—1012no—
155C2×#41—251no—
156C23×C4—251no—
157#123—252no—
158C22⋊Q8—251no—
159C22⋊Q8C5⋊D511yescharacteristic, complemented
160#121—251no—
161#121—251no—
162#120—252no—
163C4×D4—252no—
164C4×D4—252no—
165#117—251no—
166C52#52511yescharacteristic, complemented, Sylow
167C22×C5—26no—
168C22×C5—26no—
169C22×C5#713112yescomplemented
170C22×C5C23×D516yes—
171C2×D5—512no—
172C2×D5—512no—
173C2×D5—1048no—
174C2×D5—106no—
175C2×D5—106no—
176C2×D5—106no—
177C2×D5—106no—
178C2×D5—1012no—
179C2×D5—1012no—
180C2×D5—1012no—
181C4×C5—26no—
182C4×C5—212no—
183C4×C5—212no—
184C4×C5D4×D5112yes—
185Dic5—512no—
186Dic5—106no—
187Dic5—1012no—
188C24—251no—
189#41—504no—
190C2×Q8C2×C5⋊D511yescharacteristic, complemented
191C2×D4—252no—
192C2×D4—252no—
193C2×D4—252no—
194C2×D4—251no—
195C22×C4—251no—
196C22×C4—251no—
197C22×C4—251no—
198C22×C4—252no—
199C22×C4—251no—
200C22×C4—251no—
201C22×C4—504no—
202C22×C4C2×C5⋊D511yescharacteristic
203C4⋊C4—251no—
204C4⋊C4—252no—
205C4⋊C4C2×C5⋊D511yescharacteristic, complemented
206C4⋊C4C2×C5⋊D512yescomplemented
207C22⋊C4—252no—
208C22⋊C4—252no—
209C22⋊C4—252no—
210C22⋊C4—252no—
211C22⋊C4C2×C5⋊D512yes—
212C42—252no—
213C2×C5—212no—
214C2×C5C2×D4×D516yes—
215C2×C5C2×#71316yes—
216C2×C5D5×#4116yes—
217D5—1012no—
218D5—1012no—
219D5—2012no—
220C23—251no—
221C23—251no—
222C23—252no—
223C23—502no—
224C23—502no—
225C23—502no—
226C23C22×C5⋊D511yescharacteristic
227Q8—22no—
228D4—504no—
229D4—504no—
230D4—504no—
231D4—502no—
232C2×C4—22no—
233C2×C4—251no—
234C2×C4—254no—
235C2×C4—252no—
236C2×C4—251no—
237C2×C4—502no—
238C2×C4—504no—
239C2×C4—502no—
240C2×C4—502no—
241C2×C4—502no—
242C2×C4C22×C5⋊D512yes—
243C2×C4C22×C5⋊D511yescharacteristic
244C2×C4C22×C5⋊D512yes—
245C2×C4C22×C5⋊D511yescharacteristic
246C5#6596116yescomplemented
247C22—21no—
248C22—21no—
249C22—252no—
250C22—252no—
251C22—508no—
252C22—501no—
253C22—501no—
254C22—501no—
255C22—501no—
256C22—502no—
257C22—502no—
258C22—502no—
259C22#8790412yescomplemented
260C22C23×C5⋊D511yescharacteristic
261C4—21no—
262C4—22no—
263C4—22no—
264C4—252no—
265C4—501no—
266C4—502no—
267C4D4×C5⋊D512yes—
268C2—22no—
269C2—502no—
270C2—502no—
271C2—1002no—
272C2C2×D4×C5⋊D511yescharacteristic
273C2C2×#8790411yescharacteristic
274C2#41×C5⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊Q8C2
hypercenterC22⋊Q8C5⋊D5
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#87904C2semidirect
2. C22×C4×C5⋊D5C2semidirect
3. C52×C22⋊Q8C2semidirect
4. #11717266C2semidirect
5. #11717250C2semidirect
6. #11717215C2semidirect
7. #11717214C2semidirect
8. #11717213C2semidirect
9. #11717212C2semidirect
10. #11717202C2semidirect
11. #11717201C2semidirect
13. C2×C52×Q8C22semidirect
20. C2×C4⋊(C5⋊D5)C22semidirect
26. C2×C4×C5⋊D5C22semidirect
27. C52×C4⋊C4C22semidirect
28. C52×C4⋊C4C22semidirect
32. #87808C22semidirect
33. #87808C22semidirect
34. C4⋊#1039C22semidirect
35. #87806C22semidirect
36. C4×#1039C22semidirect
63. C4×C5⋊D5D4semidirect
66. C5×C22⋊Q8D5semidirect
77. C22×C52#41semidirect
88. C2×C5⋊D5#41semidirect
97. C2×C5×Q8C2×D5semidirect
111. C5×C4⋊C4C2×D5semidirect
112. C5×C4⋊C4C2×D5semidirect
159. C22⋊Q8C5⋊D5semidirect
166. C52#525semidirect
169. C22×C5#713semidirect
190. C2×Q8C2×C5⋊D5semidirect
205. C4⋊C4C2×C5⋊D5semidirect
206. C4⋊C4C2×C5⋊D5semidirect
246. C5#65961semidirect
259. C22#87904semidirect

Maximal subgroups

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

subgroupindex
C2×#879042
C22×C4×C5⋊D52
C52×C22⋊Q82
#117172662
#117172502
#117172152
#117172142
#117172132
#117172122
#117172022
#117172012
#659615