Finite Group Databasebeta

← C22⋊Q8×C5⋊D5 #13107158

Subgroups — C22⋊Q8×C5⋊D5 #13107158

Order 1600.

1288 conjugacy classes of subgroups; only the 265 normal subgroups are shown.

normal subgroups only

This group has 1288 conjugacy classes of subgroups — too many to draw. Only its 265 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×Q8×C5⋊D5C211yesmaximal, characteristic, complemented
2C22×C4×C5⋊D5C211yesmaximal, characteristic
3C52×C22⋊Q8C211yesmaximal, characteristic, complemented
4#11717266C211yesmaximal, characteristic, complemented
5#11717244C211yesmaximal, characteristic, complemented
6#11717217C211yesmaximal, characteristic, complemented
7#11717216C212yesmaximal, complemented
8C4⋊C4×C5⋊D5C212yesmaximal, complemented
9C4⋊C4×C5⋊D5C211yesmaximal, characteristic, complemented
10C22⋊C4×C5⋊D5C212yesmaximal
11#11717198C212yesmaximal, complemented
12C23×C5⋊D5C2211yescharacteristic
13C2×C52×Q8C2211yescharacteristic, complemented
14C22×C4×C52C2211yescharacteristic
15C22×#1039C2211yescharacteristic
16Q8×C5⋊D5—24no—
17C2×C4×C5⋊D5—24no—
18C2×C4×C5⋊D5C2212yes—
19C2×C4×C5⋊D5C2212yes—
20C2×C4×C5⋊D5C2211yescharacteristic
21C2×C4×C5⋊D5C2211yescharacteristic
22C2×C4×C5⋊D5C2211yescharacteristic
23C2×C4×C5⋊D5C2211yescharacteristic
24C2×#6045C2212yescomplemented
25C2×#6045C2211yescharacteristic, complemented
26C52×C4⋊C4C2212yescomplemented
27C52×C4⋊C4C2211yescharacteristic, complemented
28C52×C22⋊C4C2212yes—
29C22⋊#1039C2212yes—
30#87808C2212yes—
31#87808C2212yes—
32C4⋊#1039C2212yescomplemented
33C4⋊#1039C2211yescharacteristic, complemented
34#87806C2212yescomplemented
35#87806C2212yescomplemented
36#87806C2212yescomplemented
37D5×C22⋊Q8—56nomaximal
38C23×C52C2311yescharacteristic
39C22×C5⋊D5—22no—
40C22×C5⋊D5—22no—
41C22×C5⋊D5Q814yescomplemented
42C22×C5⋊D5C2311yescharacteristic
43C22×C5⋊D5C2311yescharacteristic
44C52×Q8—22no—
45C2×C4×C52—22no—
46C2×C4×C52C2312yes—
47C2×C4×C52C2312yes—
48C2×C4×C52C2311yescharacteristic
49C2×C4×C52C2311yescharacteristic
50C2×#1039—22no—
51C2×#1039C2312yes—
52C2×#1039C2312yes—
53C2×#1039C2311yescharacteristic
54C2×#1039C2311yescharacteristic
55C4×C5⋊D5—22no—
56C4×C5⋊D5—24no—
57C4×C5⋊D5—24no—
58C4×C5⋊D5—22no—
59C4×C5⋊D5—22no—
60C4×C5⋊D5D414yes—
61#6045—22no—
62#6045—24no—
63C2×Q8×D5—56no—
64C22×C4×D5—56no—
65C5×C22⋊Q8D516yescomplemented
66#4092—56no—
67#4070—56no—
68#4043—56no—
69#4042—512no—
70D5×C4⋊C4—56no—
71D5×C4⋊C4—512no—
72D5×C22⋊C4—512no—
73#4024—512no—
74C22×C52—21no—
75C22×C52—21no—
76C22×C52C2×Q812yescomplemented
77C22×C52C2411yescharacteristic
78C2×C5⋊D5—21no—
79C2×C5⋊D5—21no—
80C2×C5⋊D5—28no—
81C2×C5⋊D5C2×D412yes—
82C2×C5⋊D5C2×Q812yes—
83C2×C5⋊D5C2×Q812yescomplemented
84C2×C5⋊D5#4112yes—
85C4×C52—22no—
86C4×C52—21no—
87C4×C52—22no—
88C4×C52C2×D412yes—
89#1039—22no—
90#1039—21no—
91#1039—22no—
92#1039C2×D412yes—
93C23×D5—56no—
94C2×C5×Q8C2×D516yescomplemented
95C22×C4×C5C2×D516yes—
96C22×Dic5—56no—
97Q8×D5—1024no—
98C2×C4×D5—56no—
99C2×C4×D5—56no—
100C2×C4×D5—56no—
101C2×C4×D5—56no—
102C2×C4×D5—512no—
103C2×C4×D5—512no—
104C2×C4×D5—1024no—
105C2×Dic10—512no—
106C2×Dic10—56no—
107C5×C4⋊C4C2×D5112yescomplemented
108C5×C4⋊C4C2×D516yescomplemented
109C5×C22⋊C4C2×D5112yes—
110C22⋊Dic5—512no—
111#685—512no—
112#685—512no—
113C4⋊Dic5—512no—
114C4⋊Dic5—56no—
115#683—512no—
116#683—512no—
117#683—512no—
118C2×C22⋊Q8—251noSylow
119C2×C52—22no—
120C2×C52C22×D411yescharacteristic
121C2×C52C22×Q811yescharacteristic
122C2×C52C2×#4111yescharacteristic
123C5⋊D5—22no—
124C5⋊D5C22⋊Q814yesdirect product factor
125C23×C5C22×D516yes—
126C22×D5—56no—
127C22×D5—524no—
128C22×D5—56no—
129C22×D5—1012no—
130C22×D5—1012no—
131C5×Q8—212no—
132C2×C4×C5—212no—
133C2×C4×C5C22×D516yes—
134C2×C4×C5C22×D5112yes—
135C2×C4×C5C22×D5112yes—
136C2×C4×C5C22×D516yes—
137C2×Dic5—56no—
138C2×Dic5—512no—
139C2×Dic5—56no—
140C2×Dic5—512no—
141C2×Dic5—1012no—
142C4×D5—524no—
143C4×D5—1012no—
144C4×D5—1024no—
145C4×D5—1012no—
146C4×D5—1024no—
147C4×D5—1012no—
148Dic10—1024no—
149Dic10—1012no—
150C22×Q8—251no—
151C23×C4—251no—
152C22⋊Q8—251no—
153C22⋊Q8—251no—
154C22⋊Q8—252no—
155C22⋊Q8—251no—
156C22⋊Q8—252no—
157C22⋊Q8C5⋊D511yescharacteristic, direct product factor
158C2×C4⋊C4—251no—
159C2×C4⋊C4—252no—
160C2×C22⋊C4—252no—
161C52C2×C22⋊Q811yescharacteristic, complemented, Sylow
162C22×C5—26no—
163C22×C5—26no—
164C22×C5Q8×D5112yescomplemented
165C22×C5C23×D516yes—
166C2×D5—512no—
167C2×D5—512no—
168C2×D5—512no—
169C2×D5—512no—
170C2×D5—1048no—
171C2×D5—106no—
172C2×D5—106no—
173C4×C5—212no—
174C4×C5—26no—
175C4×C5—212no—
176C4×C5D4×D5112yes—
177Dic5—512no—
178Dic5—1012no—
179Dic5—106no—
180Dic5—1012no—
181C24—251no—
182C2×Q8—252no—
183C2×Q8—251no—
184C2×Q8—504no—
185C2×Q8C2×C5⋊D511yescharacteristic, complemented
186C22×C4—251no—
187C22×C4—251no—
188C22×C4—252no—
189C22×C4—252no—
190C22×C4—251no—
191C22×C4—251no—
192C22×C4—251no—
193C22×C4—504no—
194C22×C4C2×C5⋊D511yescharacteristic
195C4⋊C4—252no—
196C4⋊C4—252no—
197C4⋊C4—252no—
198C4⋊C4—251no—
199C4⋊C4—252no—
200C4⋊C4C2×C5⋊D511yescharacteristic, complemented
201C4⋊C4C2×C5⋊D512yescomplemented
202C22⋊C4—252no—
203C22⋊C4—252no—
204C22⋊C4—252no—
205C22⋊C4C2×C5⋊D512yes—
206C2×C5—212no—
207C2×C5C2×D4×D516yes—
208C2×C5C2×Q8×D516yes—
209C2×C5D5×#4116yes—
210D5—524no—
211D5—1012no—
212C23—251no—
213C23—254no—
214C23—251no—
215C23—502no—
216C23—502no—
217C23C22×C5⋊D511yescharacteristic
218Q8—22no—
219Q8—504no—
220Q8—502no—
221C2×C4—22no—
222C2×C4—251no—
223C2×C4—254no—
224C2×C4—252no—
225C2×C4—252no—
226C2×C4—251no—
227C2×C4—502no—
228C2×C4—502no—
229C2×C4—504no—
230C2×C4—504no—
231C2×C4—502no—
232C2×C4—502no—
233C2×C4C22×C5⋊D512yes—
234C2×C4C22×C5⋊D512yes—
235C2×C4C22×C5⋊D511yescharacteristic
236C2×C4C22×C5⋊D511yescharacteristic
237C5D5×C22⋊Q816yescomplemented
238C22—21no—
239C22—21no—
240C22—252no—
241C22—252no—
242C22—252no—
243C22—252no—
244C22—508no—
245C22—501no—
246C22—501no—
247C22Q8×C5⋊D512yescomplemented
248C22C23×C5⋊D511yescharacteristic
249C4—21no—
250C4—22no—
251C4—22no—
252C4—252no—
253C4—501no—
254C4—502no—
255C4—502no—
256C4D4×C5⋊D512yes—
257C2—22no—
258C2—254no—
259C2—502no—
260C2C2×D4×C5⋊D511yescharacteristic
261C2C2×Q8×C5⋊D511yescharacteristic
262C2#41×C5⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊Q8C2
prime radicalC23×C5⋊D5C22
hypercenterC22⋊Q8C5⋊D5
hypocenterC52C2×C22⋊Q8
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C5⋊D5C2semidirect
3. C52×C22⋊Q8C2semidirect
4. #11717266C2semidirect
5. #11717244C2semidirect
6. #11717217C2semidirect
7. #11717216C2semidirect
8. C4⋊C4×C5⋊D5C2semidirect
9. C4⋊C4×C5⋊D5C2semidirect
11. #11717198C2semidirect
13. C2×C52×Q8C22semidirect
24. C2×#6045C22semidirect
25. C2×#6045C22semidirect
26. C52×C4⋊C4C22semidirect
27. C52×C4⋊C4C22semidirect
32. C4⋊#1039C22semidirect
33. C4⋊#1039C22semidirect
34. #87806C22semidirect
35. #87806C22semidirect
36. #87806C22semidirect
41. C22×C5⋊D5Q8semidirect
65. C5×C22⋊Q8D5semidirect
76. C22×C52C2×Q8semidirect
83. C2×C5⋊D5C2×Q8semidirect
94. C2×C5×Q8C2×D5semidirect
107. C5×C4⋊C4C2×D5semidirect
108. C5×C4⋊C4C2×D5semidirect
124. C5⋊D5C22⋊Q8direct (N × H)
157. C22⋊Q8C5⋊D5direct (N × H)
161. C52C2×C22⋊Q8semidirect
164. C22×C5Q8×D5semidirect
185. C2×Q8C2×C5⋊D5semidirect
200. C4⋊C4C2×C5⋊D5semidirect
201. C4⋊C4C2×C5⋊D5semidirect
237. C5D5×C22⋊Q8semidirect
247. C22Q8×C5⋊D5semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C5⋊D52
C22×C4×C5⋊D52
C52×C22⋊Q82
#117172662
#117172442
#117172172
#117172162
C4⋊C4×C5⋊D52
C22⋊C4×C5⋊D52
#117171982
D5×C22⋊Q85