Finite Group Databasebeta

← #13106619

Subgroups — #13106619

Order 1600.

928 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 136 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×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C2×#87886C211yesmaximal, characteristic, complemented
3C5×#4099C212yesmaximal, complemented
4#11717101C212yesmaximal, complemented
5#11717093C212yesmaximal, complemented
6#11717092C211yesmaximal, characteristic, complemented
7#11717080C212yesmaximal, complemented
8#11717076C214yesmaximal, complemented
9C24×C52C2211yescharacteristic, complemented
10C23×C5⋊D5C2211yescharacteristic
11C22×D52C2211yescharacteristic, complemented
12C2×C22⋊(C5⋊D5)—24no—
13C2×C22⋊(C5⋊D5)C2212yes—
14C2×C22⋊(C5⋊D5)C2212yescomplemented
15C22×#1039C2211yescharacteristic, complemented
16C2×C5×C22⋊D5C2212yes—
17C2×C5×C22⋊D5C2214yescomplemented
18#87886—24no—
19C2×#6038C2212yescomplemented
20C2×#6036C2211yescharacteristic, complemented
21C2×#6036C2212yescomplemented
22C5×C22⋊Dic5C2214yescomplemented
23C5×C22⋊Dic5C2212yescomplemented
24#87780C2214yescomplemented
25#87779C2212yescomplemented
26#87778C2214yescomplemented
27#65923—52nomaximal
28C23×C52—22no—
29C23×C52—42no—
30C23×C52C2312yes—
31C23×C52C2311yescharacteristic
32C22×C5⋊D5—42no—
33C22×C5⋊D5D414yescomplemented
34C22×C5⋊D5C2311yescharacteristic
35C22×C5⋊D5C2311yescharacteristic
36C22×C5×D5C2312yes—
37C2×D52—22no—
38C2×D52—42no—
39C22⋊(C5⋊D5)—24no—
40C22⋊(C5⋊D5)—24no—
41C2×#1039—22no—
42C2×#1039C2312yes—
43C5×C22⋊D5—24no—
44C5×C22⋊D5—44no—
45C2×C5×Dic5C2312yes—
46C2×C5×Dic5C2314yes—
47#6038—24no—
48#6036—22no—
49#6036—42no—
50C22×C22⋊D5—52no—
51C22×C22⋊D5—102no—
52C2×D4×D5—52no—
53C5×#120—52no—
54#4099D512yescomplemented
55#4084—54no—
56#4083—54no—
57#4081—52no—
58#4032—52no—
59#4030—54no—
60#4029—54no—
61D5×C22⋊C4—52no—
62C22×C52—22no—
63C22×C52—22no—
64C22×C52—24no—
65C22×C52—44no—
66C22×C52C2×D412yes—
67C22×C52C2411yescharacteristic
68C2×C5⋊D5—22no—
69C2×C5⋊D5—22no—
70C2×C5⋊D5—44no—
71C2×C5⋊D5C2×D412yes—
72C2×C5⋊D5C2×D412yes—
73C2×C5×D5—22no—
74C2×C5×D5—24no—
75D52—44no—
76#1039—22no—
77C5×Dic5—22no—
78C5×Dic5—24no—
79C24×C5—22no—
80C24×C5C2×D512yescomplemented
81C23×D5—52no—
82C23×D5—52no—
83C23×D5—102no—
84C2×C5×D4—52no—
85C2×C5×D4—54no—
86C2×C22⋊D5—54no—
87C2×C22⋊D5—54no—
88C2×C22⋊D5—52no—
89C2×C22⋊D5—104no—
90C2×C22⋊D5—104no—
91C2×C22⋊D5—108no—
92C2×C22⋊D5—1016no—
93C2×C22⋊D5C2×D512yes—
94C2×C22⋊D5C2×D514yescomplemented
95C22×Dic5—52no—
96C22×Dic5—102no—
97D4×D5—108no—
98C2×D20—52no—
99C2×C4×D5—52no—
100C2×C4×D5—54no—
101C5×C22⋊C4—52no—
102C5×C22⋊C4—54no—
103C22⋊Dic5—54no—
104C22⋊Dic5C2×D512yescomplemented
105C22⋊Dic5C2×D514yescomplemented
106#685—54no—
107#685—54no—
108C4⋊Dic5—54no—
109#683—54no—
110#534—251noSylow
111C2×C52—22no—
112C2×C52—42no—
113C2×C52C22×D411yescharacteristic
114C2×C52H(5,𝔽2)12yes—
115C5⋊D5—22no—
116C5⋊D5—22no—
117C5×D5—42no—
118C23×C5—28no—
119C23×C5—24no—
120C23×C5—22no—
121C23×C5—24no—
122C23×C5—44no—
123C23×C5—48no—
124C23×C5—52no—
125C23×C5C22×D514yes—
126C23×C5C22×D512yes—
127C22×D5—52no—
128C22×D5—52no—
129C22×D5—58no—
130C22×D5—108no—
131C22×D5—102no—
132C22×D5—102no—
133C22×D5—104no—
134C22×D5—204no—
135C22×D5—204no—
136C22×D5—208no—
137C22×D5C22×D512yes—
138C5×D4—104no—
139C5×D4—204no—
140C2×C4×C5—52no—
141C2×C4×C5—54no—
142C22⋊D5—24no—
143C22⋊D5—44no—
144C22⋊D5—104no—
145C22⋊D5—108no—
146C22⋊D5—108no—
147C22⋊D5—1016no—
148C22⋊D5—1016no—
149C2×Dic5—54no—
150C2×Dic5—104no—
151C2×Dic5—108no—
152C2×Dic5—104no—
153C2×Dic5C22×D514yes—
154C2×Dic5C22×D512yes—
155D20—104no—
156C4×D5—104no—
157C4×D5—204no—
158C22×D4—251no—
159C22×D4—251no—
160#123—252no—
161#123—252no—
162#121—254no—
163#120—52no—
164#120—252no—
165C2×C22⋊C4—251no—
166C52#53411yescharacteristic, complemented, Sylow
167C22×C5—28no—
168C22×C5—24no—
169C22×C5—28no—
170C22×C5—24no—
171C22×C5—24no—
172C22×C5—24no—
173C22×C5—24no—
174C22×C5—22no—
175C22×C5—48no—
176C22×C5—416no—
177C22×C5—48no—
178C22×C5—102no—
179C22×C5—104no—
180C22×C5D4×D514yes—
181C22×C5C23×D512yes—
182C2×D5—22no—
183C2×D5—24no—
184C2×D5—54no—
185C2×D5—54no—
186C2×D5—104no—
187C2×D5—104no—
188C2×D5—104no—
189C2×D5—104no—
190C2×D5—204no—
191C2×D5—204no—
192C2×D5—208no—
193C2×D5—208no—
194C2×D5—2016no—
195C4×C5—102no—
196C4×C5—104no—
197Dic5—22no—
198Dic5—24no—
199Dic5—108no—
200Dic5—104no—
201C24—251no—
202C24—251no—
203C24D5211yescharacteristic, complemented
204C2×D4—52no—
205C2×D4—54no—
206C2×D4—252no—
207C2×D4—252no—
208C2×D4—252no—
209C2×D4—504no—
210C2×D4—504no—
211C22×C4—251no—
212C22×C4—252no—
213C22×C4—251no—
214C4⋊C4—254no—
215C22⋊C4—52no—
216C22⋊C4—54no—
217C22⋊C4—252no—
218C22⋊C4—254no—
219C2×C5—28no—
220C2×C5—24no—
221C2×C5—22no—
222C2×C5—24no—
223C2×C5—48no—
224C2×C5—44no—
225C2×C5—202no—
226C2×C5C2×D4×D512yes—
227C2×C5#414414yes—
228D5—42no—
229D5—104no—
230D5—104no—
231D5—204no—
232D5—204no—
233C23—22no—
234C23—42no—
235C23—52no—
236C23—251no—
237C23—251no—
238C23—254no—
239C23—502no—
240C23—1002no—
241C23—1002no—
242C23C2×D5212yes—
243C23C2×D5211yescharacteristic
244D4—104no—
245D4—204no—
246D4—504no—
247D4—504no—
248D4—504no—
249C2×C4—54no—
250C2×C4—52no—
251C2×C4—252no—
252C2×C4—502no—
253C2×C4—502no—
254C2×C4—1002no—
255C5—22no—
256C5#6592312yescomplemented
257C22—22no—
258C22—22no—
259C22—24no—
260C22—44no—
261C22—102no—
262C22—104no—
263C22—252no—
264C22—252no—
265C22—502no—
266C22—502no—
267C22—1004no—
268C22—1004no—
269C22#8788612yes—
270C22C22×D5211yescharacteristic
271C4—102no—
272C4—104no—
273C4—502no—
274C2—22no—
275C2—42no—
276C2—202no—
277C2—502no—
278C2—502no—
279C2C2×#8788611yescharacteristic
280C2#1171749512yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC24×C52C22
hypercenterC24D52
hypocenterC52#534
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C5⋊D5)C2semidirect
2. C2×#87886C2semidirect
3. C5×#4099C2semidirect
4. #11717101C2semidirect
5. #11717093C2semidirect
6. #11717092C2semidirect
7. #11717080C2semidirect
8. #11717076C2semidirect
9. C24×C52C22semidirect
11. C22×D52C22semidirect
14. C2×C22⋊(C5⋊D5)C22semidirect
15. C22×#1039C22semidirect
17. C2×C5×C22⋊D5C22semidirect
19. C2×#6038C22semidirect
20. C2×#6036C22semidirect
21. C2×#6036C22semidirect
22. C5×C22⋊Dic5C22semidirect
23. C5×C22⋊Dic5C22semidirect
24. #87780C22semidirect
25. #87779C22semidirect
26. #87778C22semidirect
33. C22×C5⋊D5D4semidirect
54. #4099D5semidirect
80. C24×C5C2×D5semidirect
94. C2×C22⋊D5C2×D5semidirect
104. C22⋊Dic5C2×D5semidirect
105. C22⋊Dic5C2×D5semidirect
166. C52#534semidirect
203. C24D52semidirect
256. C5#65923semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C5⋊D5)2
C2×#878862
C5×#40992
#117171012
#117170932
#117170922
#117170802
#117170762
#659235