Finite Group Databasebeta

← #10617747

Subgroups — #10617747

Order 640.

1748 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 1748 conjugacy classes of subgroups — too many to draw. Only its 557 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×#4149C212yesmaximal, complemented
2C4×C5×#41C211yesmaximal, characteristic, complemented
3#65911C214yesmaximal, complemented
4D4×D20C2112yesmaximal, complemented
5#65859C216yesmaximal, complemented
6#65811C213yesmaximal, complemented
7C2×#4020C213yesmaximal, complemented
8C2×C5×#41C2211yescharacteristic
9#4149—216no—
10C2×#713C2214yescomplemented
11C2×D4×D5C22112yescomplemented
12C2×#709C2216yescomplemented
13C22×D20C2216yescomplemented
14C22×D20C22112yescomplemented
15C4×C5×Q8C2212yescomplemented
16C4×C5×D4C2216yescomplemented
17C5×#117C2213yescomplemented
18C2×C42×C5C2213yescomplemented
19#4076C22112yescomplemented
20#4041C22124yescomplemented
21#4028C22124yescomplemented
22#4021C2216yescomplemented
23#4020C2213yescomplemented
24#4020C2216yescomplemented
25#4020C22112yescomplemented
26C4×D20C22112yescomplemented
27#4015C2211yescharacteristic, complemented
28#3498—51nomaximal, Sylow
29C23×D5C23112yes—
30C5×#41D418yescomplemented
31C2×C5×Q8C2311yescharacteristic
32C2×C5×D4C2313yes—
33C22×C4×C5C2316yes—
34C22×C4×C5C2313yes—
35C2×C22⋊D5C23112yes—
36#713—216no—
37D4×D5—248no—
38#709—224no—
39C2×D20—224no—
40C2×D20—424no—
41C2×D20C2316yes—
42C2×D20C23112yes—
43C2×D20C23112yes—
44C2×D20C23124yes—
45C2×C4×D5C23112yes—
46C2×Dic10C2312yes—
47C5×C4⋊C4C2316yes—
48C5×C22⋊C4C2316yes—
49C42×C5C2311yescharacteristic
50C42×C5C2313yes—
51C42×C5C2316yes—
52#685C23124yes—
53C4⋊Dic5C2314yes—
54C2×H(5,𝔽2)—52no—
55#550—54no—
56D42—512no—
57#535—56no—
58#532—53no—
59C2×#127—53no—
60C4×#41D511yescharacteristic, complemented
61C23×C5C2413yes—
62C22×D5—224no—
63C22×D5—424no—
64C22×D5C24112yes—
65C5×Q8C2×D414yescomplemented
66C5×D4C2×D4112yescomplemented
67C2×C4×C5—212no—
68C2×C4×C5C2×D4112yescomplemented
69C2×C4×C5C2416yes—
70C2×C4×C5C2411yescharacteristic
71C2×C4×C5C2413yes—
72C2×C4×C5C2412yes—
73C22⋊D5—224no—
74C2×Dic5C2414yes—
75D20—212no—
76D20—224no—
77D20—224no—
78D20—424no—
79C4×D5—224no—
80Dic10—24no—
81H(5,𝔽2)—1016no—
82C2×#41—54no—
83C2×#41—56no—
84C2×#41C2×D511yescharacteristic
85C22×D4—512no—
86C22×D4—56no—
87C22×D4—512no—
88C4⋊Q8—51no—
89#127—53no—
90#127—56no—
91#127—512no—
92#124—56no—
93#121—512no—
94#121—524no—
95#120—524no—
96C4×Q8C2×D512yescomplemented
97C4×D4—512no—
98C4×D4C2×D516yescomplemented
99#117C2×D513yescomplemented
100C2×C42C2×D513yescomplemented
101C22×C5—26no—
102C22×C5C22×D416yes—
103C22×C5C2511yescharacteristic
104C2×D5—212no—
105C2×D5—224no—
106C2×D5—824no—
107C4×C5—26no—
108C4×C5C22×D412yes—
109C4×C5C22×D416yes—
110C4×C5H(5,𝔽2)14yescomplemented
111Dic5—24no—
112C24—512no—
113#41—1016no—
114#41—1024no—
115#41D2018yescomplemented
116C2×Q8—52no—
117C2×Q8C22×D511yescharacteristic
118C2×D4—512no—
119C2×D4—512no—
120C2×D4—512no—
121C2×D4—56no—
122C2×D4—524no—
123C2×D4—1048no—
124C2×D4—1024no—
125C2×D4—2024no—
126C2×D4C22×D513yes—
127C22×C4—512no—
128C22×C4C22×D516yes—
129C22×C4C22×D513yes—
130C4⋊C4—54no—
131C4⋊C4C22×D516yes—
132C22⋊C4—524no—
133C22⋊C4C22×D516yes—
134C42C22×D511yescharacteristic
135C42C22×D513yes—
136C42C22×D516yes—
137C2×C5—26no—
138C2×C5C23×D411yescharacteristic
139C2×C5C2×H(5,𝔽2)12yes—
140D5—412no—
141C23—512no—
142C23—1024no—
143C23—2024no—
144C23C23×D513yes—
145Q8—104no—
146Q8C2×D2014yescomplemented
147D4—1024no—
148D4—1024no—
149D4—1012no—
150D4—1024no—
151D4—2024no—
152D4C2×D20112yescomplemented
153C2×C4—212no—
154C2×C4—54no—
155C2×C4—1024no—
156C2×C4C2×D20112yescomplemented
157C2×C4C23×D516yes—
158C2×C4C23×D511yescharacteristic
159C2×C4C23×D513yes—
160C2×C4C23×D512yes—
161C5#349811yescharacteristic, complemented, Sylow
162C22—26no—
163C22—1012no—
164C22—1024no—
165C22—4024no—
166C22C22×D2016yes—
167C22C24×D511yescharacteristic
168C4—26no—
169C4—104no—
170C4C22×D2016yes—
171C4C22×D2012yes—
172C4#414914yescomplemented
173C2—26no—
174C2—2012no—
175C2C23×D2011yescharacteristic
176C2C2×#414912yes—
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC4×C5×#41C2
hypercenterC4×#41D5
hypocenterC5#3498
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#4149C2semidirect
2. C4×C5×#41C2semidirect
3. #65911C2semidirect
4. D4×D20C2semidirect
5. #65859C2semidirect
6. #65811C2semidirect
7. C2×#4020C2semidirect
10. C2×#713C22semidirect
11. C2×D4×D5C22semidirect
12. C2×#709C22semidirect
13. C22×D20C22semidirect
14. C22×D20C22semidirect
15. C4×C5×Q8C22semidirect
16. C4×C5×D4C22semidirect
17. C5×#117C22semidirect
18. C2×C42×C5C22semidirect
19. #4076C22semidirect
20. #4041C22semidirect
21. #4028C22semidirect
22. #4021C22semidirect
23. #4020C22semidirect
24. #4020C22semidirect
25. #4020C22semidirect
26. C4×D20C22semidirect
27. #4015C22semidirect
30. C5×#41D4semidirect
60. C4×#41D5semidirect
65. C5×Q8C2×D4semidirect
66. C5×D4C2×D4semidirect
68. C2×C4×C5C2×D4semidirect
96. C4×Q8C2×D5semidirect
98. C4×D4C2×D5semidirect
99. #117C2×D5semidirect
100. C2×C42C2×D5semidirect
110. C4×C5H(5,𝔽2)semidirect
115. #41D20semidirect
146. Q8C2×D20semidirect
152. D4C2×D20semidirect
156. C2×C4C2×D20semidirect
161. C5#3498semidirect
172. C4#4149semidirect

Maximal subgroups

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

subgroupindex
C2×#41492
C4×C5×#412
#659112
D4×D202
#658592
#658112
C2×#40202
#34985