Finite Group Databasebeta

← #13106170

Subgroups — #13106170

Order 1600.

920 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×C4×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×#87886C211yesmaximal, characteristic
3C5×#4075C212yesmaximal, complemented
4#11717092C211yesmaximal, characteristic, complemented
5#11717080C212yesmaximal, complemented
6#11717039C212yesmaximal, complemented
7#11717001C212yesmaximal, complemented
8#11716971C214yesmaximal, complemented
9C23×C5⋊D5C2211yescharacteristic
10C22×D52C2211yescharacteristic
11C22×C4×C52C2211yescharacteristic, complemented
12C22×#1039C2211yescharacteristic, complemented
13C2×C4×C5⋊D5—24no—
14C2×C4×C5⋊D5C2212yes—
15C2×C4×C5⋊D5C2212yescomplemented
16C2×C5×C22⋊D5C2212yes—
17#87886—24no—
18C2×#6038C2212yes—
19C2×#6036C2211yescharacteristic
20C2×#6036C2212yes—
21C5×C22⋊Dic5C2212yescomplemented
22C5×#685C2214yescomplemented
23C5×#683C2214yescomplemented
24#87780C2214yescomplemented
25#87779C2212yescomplemented
26#87778C2214yescomplemented
27D5×#123—52nomaximal
28C23×C52C2311yescharacteristic
29C22×C5⋊D5—24no—
30C22×C5⋊D5D414yes—
31C22×C5⋊D5C2311yescharacteristic
32C22×C5⋊D5C2311yescharacteristic
33C22×C5×D5C2312yes—
34C2×D52—22no—
35C2×D52—24no—
36C2×C4×C52—22no—
37C2×C4×C52C2312yes—
38C2×#1039—22no—
39C2×#1039C2312yes—
40C4×C5⋊D5—24no—
41C4×C5⋊D5—24no—
42C5×C22⋊D5—24no—
43C2×C5×Dic5C2312yes—
44C2×C5×Dic5C2314yes—
45#6038—24no—
46#6036—22no—
47#6036—24no—
48C2×D4×D5—52no—
49C22×C4×D5—52no—
50C22×C4×D5—102no—
51C5×#123—52no—
52#4081—52no—
53#4075D512yescomplemented
54#4040—54no—
55D5×C4⋊C4—54no—
56#4032—52no—
57#4029—54no—
58D5×C22⋊C4—52no—
59D5×C22⋊C4—54no—
60C22×C52—22no—
61C22×C52C2×D412yes—
62C22×C52C2411yescharacteristic
63C2×C5⋊D5—22no—
64C2×C5⋊D5—28no—
65C2×C5⋊D5C2×D412yes—
66C2×C5⋊D5C2×D412yes—
67C2×C5⋊D5#4114yes—
68C2×C5×D5—22no—
69C2×C5×D5—24no—
70D52—44no—
71C4×C52—22no—
72#1039—22no—
73C5×Dic5—22no—
74C5×Dic5—24no—
75C23×D5—52no—
76C23×D5—52no—
77C23×D5—102no—
78C2×C5×D4—52no—
79C22×C4×C5—22no—
80C22×C4×C5C2×D512yescomplemented
81C2×C22⋊D5—52no—
82C2×C22⋊D5C2×D512yes—
83C22×Dic5—52no—
84C22×Dic5—102no—
85D4×D5—108no—
86C2×D20—52no—
87C2×C4×D5—54no—
88C2×C4×D5—54no—
89C2×C4×D5—52no—
90C2×C4×D5—54no—
91C2×C4×D5—104no—
92C2×C4×D5—104no—
93C2×C4×D5—108no—
94C2×C4×D5—1016no—
95C5×C4⋊C4—54no—
96C5×C22⋊C4—52no—
97C5×C22⋊C4—54no—
98C22⋊Dic5—54no—
99C22⋊Dic5C2×D512yescomplemented
100#685—54no—
101#685—54no—
102#685C2×D514yescomplemented
103C4⋊Dic5—54no—
104#683—54no—
105#683C2×D514yescomplemented
106C2×#123—251noSylow
107C2×C52—22no—
108C2×C52C22×D411yescharacteristic
109C2×C52C2×#4112yes—
110C5⋊D5—22no—
111C5⋊D5#12314yescomplemented
112C5×D5—42no—
113C23×C5—22no—
114C23×C5—52no—
115C23×C5C22×D512yes—
116C22×D5—52no—
117C22×D5—58no—
118C22×D5—52no—
119C22×D5—108no—
120C22×D5—102no—
121C22×D5—102no—
122C22×D5—104no—
123C22×D5—108no—
124C22×D5—108no—
125C22×D5—1016no—
126C22×D5C22×D512yes—
127C5×D4—104no—
128C2×C4×C5—28no—
129C2×C4×C5—24no—
130C2×C4×C5—24no—
131C2×C4×C5—52no—
132C2×C4×C5—54no—
133C2×C4×C5C22×D514yes—
134C22⋊D5—24no—
135C22⋊D5—104no—
136C2×Dic5—54no—
137C2×Dic5—104no—
138C2×Dic5—108no—
139C2×Dic5—104no—
140C2×Dic5C22×D514yes—
141C2×Dic5C22×D512yes—
142D20—104no—
143C4×D5—108no—
144C4×D5—104no—
145C4×D5—108no—
146C4×D5—108no—
147C4×D5—1016no—
148C4×D5—1016no—
149C22×D4—251no—
150C23×C4—251no—
151#123—52no—
152#123—254no—
153#123—252no—
154C2×C4⋊C4—252no—
155C2×C22⋊C4—251no—
156C2×C22⋊C4—252no—
157C52C2×#12311yescharacteristic, complemented, Sylow
158C22×C5—28no—
159C22×C5—24no—
160C22×C5—24no—
161C22×C5—22no—
162C22×C5—102no—
163C22×C5—104no—
164C22×C5D4×D514yes—
165C22×C5C23×D512yes—
166C2×D5—22no—
167C2×D5—24no—
168C2×D5—54no—
169C2×D5—58no—
170C2×D5—54no—
171C2×D5—104no—
172C2×D5—104no—
173C2×D5—108no—
174C2×D5—1016no—
175C2×D5—1032no—
176C2×D5—108no—
177C2×D5—104no—
178C2×D5—208no—
179C4×C5—28no—
180C4×C5—24no—
181C4×C5—102no—
182C4×C5—104no—
183Dic5—22no—
184Dic5—24no—
185Dic5—108no—
186Dic5—104no—
187C24—251no—
188C24—251no—
189C2×D4—52no—
190C2×D4—252no—
191C2×D4—504no—
192C22×C4—252no—
193C22×C4—251no—
194C22×C4—251no—
195C22×C4—252no—
196C22×C4—252no—
197C22×C4—504no—
198C22×C4D5211yescharacteristic, complemented
199C4⋊C4—54no—
200C4⋊C4—254no—
201C22⋊C4—52no—
202C22⋊C4—54no—
203C22⋊C4—252no—
204C22⋊C4—254no—
205C2×C5—28no—
206C2×C5—24no—
207C2×C5—24no—
208C2×C5—22no—
209C2×C5—202no—
210C2×C5C2×D4×D512yes—
211C2×C5D5×#4114yes—
212D5—42no—
213D5—58no—
214D5—108no—
215D5—108no—
216D5—104no—
217C23—52no—
218C23—251no—
219C23—254no—
220C23—251no—
221C23—504no—
222C23—502no—
223C23—504no—
224C23C2×D5211yescharacteristic
225D4—104no—
226D4—504no—
227C2×C4—22no—
228C2×C4—54no—
229C2×C4—52no—
230C2×C4—252no—
231C2×C4—504no—
232C2×C4—504no—
233C2×C4—502no—
234C2×C4—504no—
235C2×C4—502no—
236C2×C4C2×D5212yes—
237C5—22no—
238C5D5×#12312yescomplemented
239C22—22no—
240C22—104no—
241C22—102no—
242C22—254no—
243C22—252no—
244C22—252no—
245C22—508no—
246C22—502no—
247C22—1004no—
248C22#8788612yes—
249C22C22×D5211yescharacteristic
250C4—22no—
251C4—104no—
252C4—102no—
253C4—502no—
254C2—22no—
255C2—202no—
256C2—254no—
257C2—502no—
258C2#1171747112yes—
259C2C2×#8788611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radicalC2×#87886C2
hypercenterC22×C4D52
hypocenterC52C2×#123
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C5⋊D5C2semidirect
3. C5×#4075C2semidirect
4. #11717092C2semidirect
5. #11717080C2semidirect
6. #11717039C2semidirect
7. #11717001C2semidirect
8. #11716971C2semidirect
11. C22×C4×C52C22semidirect
12. C22×#1039C22semidirect
15. C2×C4×C5⋊D5C22semidirect
21. C5×C22⋊Dic5C22semidirect
22. C5×#685C22semidirect
23. C5×#683C22semidirect
24. #87780C22semidirect
25. #87779C22semidirect
26. #87778C22semidirect
53. #4075D5semidirect
80. C22×C4×C5C2×D5semidirect
99. C22⋊Dic5C2×D5semidirect
102. #685C2×D5semidirect
105. #683C2×D5semidirect
111. C5⋊D5#123semidirect
157. C52C2×#123semidirect
198. C22×C4D52semidirect
238. C5D5×#123semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5⋊D52
C2×#878862
C5×#40752
#117170922
#117170802
#117170392
#117170012
#117169712
D5×#1235