Finite Group Databasebeta

← #13105387

Subgroups — #13105387

Order 1600.

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

normal subgroups only

This group has 816 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
1C2×#87886C211yesmaximal, characteristic
2C22×#6036C211yesmaximal, characteristic, complemented
3C22⋊C4×C5⋊D5C211yesmaximal, characteristic, complemented
4C5×#4032C212yesmaximal, complemented
5#11717079C212yesmaximal, complemented
6#11717039C212yesmaximal, complemented
7#11717018C212yesmaximal, complemented
8#11716972C214yesmaximal, complemented
9C23×C5⋊D5C2211yescharacteristic
10C22×D52C2211yescharacteristic
11C2×C4×C5⋊D5C2212yes—
12C2×C5×C22⋊D5C2212yes—
13C22×C5×Dic5C2212yescomplemented
14#87886—24no—
15C2×#6038C2212yes—
16C2×#6036—24no—
17C2×#6036C2212yes—
18C2×#6036C2211yescharacteristic
19C2×#6036C2212yescomplemented
20C52×C22⋊C4C2211yescharacteristic, complemented
21C22⋊#1039C2211yescharacteristic, complemented
22#87808C2212yescomplemented
23C5×#685C2214yescomplemented
24C5×C4⋊Dic5C2214yescomplemented
25#87781C2214yescomplemented
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×C52C2312yes—
37C2×#1039C2312yes—
38C4×C5⋊D5—24no—
39C5×C22⋊D5—24no—
40C2×C5×Dic5—24no—
41C2×C5×Dic5C2314yes—
42C2×C5×Dic5C2312yes—
43#6038—24no—
44#6036—24no—
45#6036—24no—
46#6036—22no—
47C2×D4×D5—52no—
48C22×C4×D5—52no—
49C5×#123—52no—
50#4081—52no—
51#4075—52no—
52#4040—54no—
53D5×C4⋊C4—54no—
54#4032D512yescomplemented
55#4029—54no—
56D5×C22⋊C4—52no—
57D5×C22⋊C4—54no—
58D5×C22⋊C4—102no—
59C22×C52—22no—
60C22×C52C2×D412yes—
61C22×C52C2411yescharacteristic
62C2×C5⋊D5—22no—
63C2×C5⋊D5—28no—
64C2×C5⋊D5C2×D412yes—
65C2×C5⋊D5C2×D412yes—
66C2×C5⋊D5#4114yes—
67C2×C5×D5—22no—
68C2×C5×D5—24no—
69D52—44no—
70C4×C52—22no—
71#1039—22no—
72C5×Dic5—24no—
73C5×Dic5—22no—
74C23×D5—52no—
75C23×D5—52no—
76C23×D5—102no—
77C2×C5×D4—52no—
78C22×C4×C5—52no—
79C2×C22⋊D5—52no—
80C2×C22⋊D5C2×D512yes—
81C22×Dic5C2×D512yescomplemented
82D4×D5—108no—
83C2×D20—52no—
84C2×C4×D5—54no—
85C2×C4×D5—54no—
86C2×C4×D5—54no—
87C2×C4×D5—52no—
88C2×C4×D5—104no—
89C2×C4×D5—108no—
90C5×C4⋊C4—54no—
91C5×C22⋊C4—22no—
92C5×C22⋊C4—54no—
93C5×C22⋊C4C2×D512yescomplemented
94C22⋊Dic5—52no—
95C22⋊Dic5—54no—
96C22⋊Dic5—102no—
97#685—54no—
98#685—54no—
99#685—104no—
100#685C2×D514yescomplemented
101C4⋊Dic5C2×D514yescomplemented
102#683—54no—
103#683—54no—
104C2×#123—251noSylow
105C2×C52—22no—
106C2×C52C22×D411yescharacteristic
107C2×C52C2×#4112yes—
108C5⋊D5—22no—
109C5⋊D5#12314yescomplemented
110C5×D5—42no—
111C23×C5—22no—
112C23×C5—52no—
113C23×C5C22×D512yes—
114C22×D5—52no—
115C22×D5—58no—
116C22×D5—52no—
117C22×D5—102no—
118C22×D5—102no—
119C22×D5—108no—
120C22×D5—104no—
121C22×D5—108no—
122C22×D5—108no—
123C22×D5—208no—
124C22×D5C22×D512yes—
125C5×D4—104no—
126C2×C4×C5—24no—
127C2×C4×C5—52no—
128C2×C4×C5—54no—
129C2×C4×C5—104no—
130C2×C4×C5C22×D514yes—
131C22⋊D5—24no—
132C22⋊D5—104no—
133C2×Dic5—24no—
134C2×Dic5—54no—
135C2×Dic5—104no—
136C2×Dic5C22×D512yes—
137C2×Dic5C22×D514yes—
138D20—104no—
139C4×D5—108no—
140C4×D5—108no—
141C4×D5—104no—
142C4×D5—108no—
143C4×D5—208no—
144C22×D4—251no—
145C23×C4—251no—
146#123—52no—
147#123—254no—
148#123—252no—
149C2×C4⋊C4—252no—
150C2×C22⋊C4—252no—
151C2×C22⋊C4—251no—
152C52C2×#12311yescharacteristic, complemented, Sylow
153C22×C5—24no—
154C22×C5—24no—
155C22×C5—22no—
156C22×C5—44no—
157C22×C5—102no—
158C22×C5—104no—
159C22×C5D4×D514yes—
160C22×C5C23×D512yes—
161C2×D5—22no—
162C2×D5—24no—
163C2×D5—54no—
164C2×D5—58no—
165C2×D5—54no—
166C2×D5—104no—
167C2×D5—108no—
168C2×D5—104no—
169C2×D5—1016no—
170C2×D5—104no—
171C2×D5—2016no—
172C2×D5—204no—
173C2×D5—208no—
174C4×C5—24no—
175C4×C5—44no—
176C4×C5—104no—
177C4×C5—102no—
178Dic5—24no—
179Dic5—22no—
180Dic5—104no—
181Dic5—204no—
182C24—251no—
183C24—251no—
184C2×D4—52no—
185C2×D4—252no—
186C2×D4—504no—
187C22×C4—52no—
188C22×C4—252no—
189C22×C4—251no—
190C22×C4—252no—
191C22×C4—252no—
192C22×C4—504no—
193C4⋊C4—54no—
194C4⋊C4—254no—
195C22⋊C4—54no—
196C22⋊C4—254no—
197C22⋊C4—251no—
198C22⋊C4—252no—
199C22⋊C4D5211yescharacteristic, complemented
200C2×C5—24no—
201C2×C5—22no—
202C2×C5—24no—
203C2×C5—44no—
204C2×C5—202no—
205C2×C5C2×D4×D512yes—
206C2×C5D5×#4114yes—
207D5—42no—
208D5—58no—
209D5—108no—
210D5—104no—
211D5—204no—
212C23—52no—
213C23—251no—
214C23—254no—
215C23—251no—
216C23—504no—
217C23—502no—
218C23—504no—
219C23C2×D5211yescharacteristic
220D4—104no—
221D4—504no—
222C2×C4—54no—
223C2×C4—52no—
224C2×C4—104no—
225C2×C4—252no—
226C2×C4—504no—
227C2×C4—504no—
228C2×C4—504no—
229C2×C4—502no—
230C2×C4C2×D5212yes—
231C5—22no—
232C5D5×#12312yescomplemented
233C22—22no—
234C22—102no—
235C22—104no—
236C22—252no—
237C22—254no—
238C22—252no—
239C22—508no—
240C22—502no—
241C22—1004no—
242C22#8787712yes—
243C22C22×D5211yescharacteristic
244C4—22no—
245C4—104no—
246C4—102no—
247C4—502no—
248C2—22no—
249C2—202no—
250C2—254no—
251C2—502no—
252C2C2×#8787711yescharacteristic
253C2#1171747712yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
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
2. C22×#6036C2semidirect
3. C22⋊C4×C5⋊D5C2semidirect
4. C5×#4032C2semidirect
5. #11717079C2semidirect
6. #11717039C2semidirect
7. #11717018C2semidirect
8. #11716972C2semidirect
13. C22×C5×Dic5C22semidirect
19. C2×#6036C22semidirect
20. C52×C22⋊C4C22semidirect
21. C22⋊#1039C22semidirect
22. #87808C22semidirect
23. C5×#685C22semidirect
24. C5×C4⋊Dic5C22semidirect
25. #87781C22semidirect
26. #87778C22semidirect
54. #4032D5semidirect
81. C22×Dic5C2×D5semidirect
93. C5×C22⋊C4C2×D5semidirect
100. #685C2×D5semidirect
101. C4⋊Dic5C2×D5semidirect
109. C5⋊D5#123semidirect
152. C52C2×#123semidirect
199. C22⋊C4D52semidirect
232. C5D5×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#878862
C22×#60362
C22⋊C4×C5⋊D52
C5×#40322
#117170792
#117170392
#117170182
#117169722
D5×#1235