Finite Group Databasebeta

← #10612599

Subgroups — #10612599

Order 640.

514 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 113 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
1D5×#121C212yesmaximal, complemented
2C5×#469C211yesmaximal, characteristic, complemented
3#65445C212yesmaximal, complemented
4#65424C211yesmaximal, characteristic, complemented
5#65415C211yesmaximal, characteristic, complemented
6#65328C212yesmaximal, complemented
7#65152C212yesmaximal, complemented
8#65066C212yesmaximal, complemented
9#65062C212yesmaximal, complemented
10C2×D4×D5—22no—
11C2×D4×D5C2212yescomplemented
12C22×C4×D5C2211yescharacteristic
13C2×C5×D8C2211yescharacteristic, complemented
14C2×C5×M16C2211yescharacteristic, complemented
15C5×#121C2212yescomplemented
16#4085—22no—
17#4084C2212yes—
18C2×#686C2212yescomplemented
19#4076C2212yescomplemented
20C2×#681C2211yescharacteristic, complemented
21C2×D40C2211yescharacteristic, complemented
22#4041C2212yes—
23D5×C4⋊C4C2212yescomplemented
24#4030—22no—
25D5×C22⋊C4—22no—
26C5×#106C2211yescharacteristic, complemented
27C5×#102C2212yescomplemented
28#3964C2212yescomplemented
29#3953C2212yescomplemented
30#3952C2212yescomplemented
31#3952C2212yescomplemented
32#3949C2211yescharacteristic, complemented
33#3941C2212yescomplemented
34#3940C2212yescomplemented
35#3192—51nomaximal, Sylow
36C23×D5—22no—
37C23×D5D411yescharacteristic
38C2×C5×D4—22no—
39C2×C5×D4C2312yes—
40C22×C4×C5C2311yescharacteristic
41C2×C22⋊D5—22no—
42C2×C22⋊D5—22no—
43C2×C22⋊D5—22no—
44C22×Dic5D411yescharacteristic
45D4×D5—42no—
46D4×D5—42no—
47D4×D5—48no—
48C2×D20—22no—
49C2×D20C2312yes—
50C2×C4×D5—22no—
51C2×C4×D5—41no—
52C2×C4×D5—41no—
53C2×C4×D5D412yes—
54C2×C4×D5C2312yes—
55C5×D8—41no—
56C5×M16—41no—
57C2×C5×C8C2311yescharacteristic
58C2×C5×C8C2311yescharacteristic
59C5×C4⋊C4C2312yes—
60C5×C22⋊C4—22no—
61C22⋊Dic5—22no—
62#686—42no—
63#685—22no—
64#685—22no—
65C4⋊Dic5C2312yes—
66#683—22no—
67#681—41no—
68C2×#168C2312yes—
69D40—41no—
70C2×#121—52no—
71#481—52no—
72#481—52no—
73#469—51no—
74#469—52no—
75#469D511yescharacteristic, complemented
76#447—52no—
77#447—52no—
78#407—51no—
79C23×C5—22no—
80C23×C5C2×D411yescharacteristic
81C22×D5—24no—
82C22×D5—22no—
83C22×D5—41no—
84C22×D5—44no—
85C22×D5—44no—
86C22×D5—44no—
87C22×D5C2×D412yes—
88C5×D4—22no—
89C5×D4—22no—
90C5×D4—44no—
91C2×C4×C5—21no—
92C2×C4×C5—21no—
93C2×C4×C5—22no—
94C2×C4×C5C2×D411yescharacteristic
95C2×C4×C5C2411yescharacteristic
96C22⋊D5—44no—
97C22⋊D5—44no—
98C22⋊D5—44no—
99C2×Dic5—22no—
100C2×Dic5—41no—
101C2×Dic5C2×D412yes—
102D20—22no—
103D20—22no—
104D20—44no—
105C4×D5—22no—
106C4×D5—22no—
107C4×D5—42no—
108C4×D5—44no—
109C5×C8—21no—
110C5×C8—21no—
111#168—22no—
112C22×D4—52no—
113C22×D4—102no—
114C23×C4—51no—
115C2×D8—52no—
116C2×D8—51no—
117C2×D8C2×D511yescharacteristic, complemented
118C2×M16—51no—
119C2×M16C2×D511yescharacteristic, complemented
120#121—52no—
121#121—52no—
122#121—52no—
123#121—102no—
124#121—102no—
125#121C2×D512yescomplemented
126C2×C4⋊C4—52no—
127C2×C22⋊C4—102no—
128#106—51no—
129#106—52no—
130#106C2×D511yescharacteristic, complemented
131#102—52no—
132#102—52no—
133#102—52no—
134#102C2×D512yescomplemented
135C22⋊C8—52no—
136C22⋊C8—52no—
137C22×C5—21no—
138C22×C5—22no—
139C22×C5—42no—
140C22×C5—42no—
141C22×C5—42no—
142C22×C5C22×D411yescharacteristic
143C2×D5—22no—
144C2×D5—22no—
145C2×D5—44no—
146C2×D5—42no—
147C2×D5—42no—
148C2×D5—42no—
149C2×D5—88no—
150C2×D5AGL(1,ℤ/8)14yes—
151C4×C5—21no—
152C4×C5—42no—
153C4×C5H(5,𝔽2)11yescharacteristic
154C4×C5H(5,𝔽2)11yescharacteristic
155Dic5—22no—
156Dic5—42no—
157C24—51no—
158C24—102no—
159C2×D4—22no—
160C2×D4—52no—
161C2×D4—102no—
162C2×D4—102no—
163C2×D4—102no—
164C2×D4—102no—
165C2×D4—202no—
166C2×D4—202no—
167C2×D4—208no—
168C2×D4C22×D512yes—
169C22×C4—51no—
170C22×C4—52no—
171C22×C4—52no—
172C22×C4—102no—
173C22×C4—201no—
174C22×C4—201no—
175C22×C4C22×D511yescharacteristic
176D8—41no—
177D8—201no—
178D8—202no—
179M16—41no—
180M16—201no—
181C2×C8—52no—
182C2×C8C22×D511yescharacteristic
183C2×C8C22×D511yescharacteristic
184C4⋊C4—52no—
185C4⋊C4—102no—
186C4⋊C4C22×D512yes—
187C22⋊C4—22no—
188C22⋊C4—102no—
189C22⋊C4—102no—
190C22⋊C4—102no—
191C2×C5—41no—
192C2×C5—82no—
193C2×C5#53411yescharacteristic
194C2×C5C2×AGL(1,ℤ/8)12yes—
195D5—24no—
196D5—82no—
197C23—22no—
198C23—52no—
199C23—104no—
200C23—102no—
201C23—201no—
202C23—204no—
203C23—204no—
204C23—204no—
205C23D4×D511yescharacteristic
206D4—22no—
207D4—22no—
208D4—44no—
209D4—102no—
210D4—102no—
211D4—204no—
212D4—204no—
213D4—204no—
214D4—204no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—52no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—202no—
223C2×C4—204no—
224C2×C4—201no—
225C2×C4D4×D511yescharacteristic
226C2×C4C23×D511yescharacteristic
227C8—21no—
228C8—21no—
229C8—102no—
230C5#319211yescharacteristic, complemented, Sylow
231C22—22no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—54no—
237C22—102no—
238C22—102no—
239C22—204no—
240C22—202no—
241C22—202no—
242C22—202no—
243C22—408no—
244C22C2×D4×D511yescharacteristic
245C4—21no—
246C4—42no—
247C4—102no—
248C4—202no—
249C4#414411yescharacteristic
250C4#414911yescharacteristic
251C2—41no—
252C2—82no—
253C2—104no—
254C2—402no—
255C2#6593911yescharacteristic
256C2D5×AGL(1,ℤ/8)12yes—
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#469C2
socleC22×C5C22×D4
hypercenter#469D5
hypocenterC5#3192
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. D5×#121C2semidirect
2. C5×#469C2semidirect
3. #65445C2semidirect
4. #65424C2semidirect
5. #65415C2semidirect
6. #65328C2semidirect
7. #65152C2semidirect
8. #65066C2semidirect
9. #65062C2semidirect
11. C2×D4×D5C22semidirect
13. C2×C5×D8C22semidirect
14. C2×C5×M16C22semidirect
15. C5×#121C22semidirect
18. C2×#686C22semidirect
19. #4076C22semidirect
20. C2×#681C22semidirect
21. C2×D40C22semidirect
23. D5×C4⋊C4C22semidirect
26. C5×#106C22semidirect
27. C5×#102C22semidirect
28. #3964C22semidirect
29. #3953C22semidirect
30. #3952C22semidirect
31. #3952C22semidirect
32. #3949C22semidirect
33. #3941C22semidirect
34. #3940C22semidirect
75. #469D5semidirect
117. C2×D8C2×D5semidirect
119. C2×M16C2×D5semidirect
125. #121C2×D5semidirect
130. #106C2×D5semidirect
134. #102C2×D5semidirect
230. C5#3192semidirect

Maximal subgroups

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

subgroupindex
D5×#1212
C5×#4692
#654452
#654242
#654152
#653282
#651522
#650662
#650622
#31925