Finite Group Databasebeta

← #124×C5⋊D5 #13107202

Subgroups — #124×C5⋊D5 #13107202

Order 1600.

1320 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 243 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×Q8×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×D4×C5⋊D5C211yesmaximal, characteristic
3C52×#124C211yesmaximal, characteristic, complemented
4#11717267C211yesmaximal, characteristic, complemented
5#11717256C211yesmaximal, characteristic, complemented
6#11717205C214yesmaximal, complemented
7C22⋊C4×C5⋊D5C214yesmaximal, complemented
8#11717195C211yesmaximal, characteristic, complemented
9C42×C5⋊D5C211yesmaximal, characteristic, complemented
10C23×C5⋊D5C2212yes—
11C2×C52×Q8C2211yescharacteristic, complemented
12C2×C52×D4C2211yescharacteristic
13C2×C22⋊(C5⋊D5)C2212yes—
14Q8×C5⋊D5—24no—
15D4×C5⋊D5—24no—
16C2×C4⋊(C5⋊D5)C2211yescharacteristic
17C2×C4×C5⋊D5C2212yes—
18C2×C4×C5⋊D5C2212yes—
19C2×C4×C5⋊D5C2211yescharacteristic
20C2×#6045C2211yescharacteristic, complemented
21C2×#6045C2212yescomplemented
22C52×C22⋊C4C2214yescomplemented
23C42×C52C2211yescharacteristic, complemented
24C22⋊#1039C2214yescomplemented
25#87808C2214yescomplemented
26#87808C2214yescomplemented
27C4×#1039C2212yescomplemented
28C4×#1039C2211yescharacteristic, complemented
29D5×#124—56nomaximal
30C23×C52C2312yes—
31C22×C5⋊D5—24no—
32C22×C5⋊D5—28no—
33C22×C5⋊D5C2312yes—
34C22×C5⋊D5C2311yescharacteristic
35C52×Q8—22no—
36C52×D4—22no—
37C2×C4×C52C2312yes—
38C2×C4×C52C2312yes—
39C2×C4×C52C2311yescharacteristic
40C22⋊(C5⋊D5)—24no—
41C2×#1039C2312yes—
42C2×#1039C2312yes—
43C2×#1039C2311yescharacteristic
44C4⋊(C5⋊D5)—22no—
45C4×C5⋊D5—24no—
46C4×C5⋊D5—24no—
47C4×C5⋊D5D414yes—
48#6045—22no—
49#6045—24no—
50C2×Q8×D5—56no—
51C2×D4×D5—56no—
52C5×#124D516yescomplemented
53#4093—56no—
54#4082—56no—
55#4031—524no—
56D5×C22⋊C4—524no—
57#4021—56no—
58C42×D5—56no—
59C22×C52—22no—
60C22×C52—24no—
61C22×C52C2411yescharacteristic
62C2×C5⋊D5—22no—
63C2×C5⋊D5—24no—
64C2×C5⋊D5—48no—
65C2×C5⋊D5C2×D412yes—
66C2×C5⋊D5#4114yes—
67C4×C52—22no—
68C4×C52—22no—
69C4×C52C2×D412yes—
70#1039—22no—
71#1039—22no—
72#1039C2×D412yes—
73C23×D5—512no—
74C2×C5×Q8C2×D516yescomplemented
75C2×C5×D4C2×D516yes—
76C2×C22⋊D5—512no—
77Q8×D5—1024no—
78D4×D5—1024no—
79C2×D20—56no—
80C2×C4×D5—56no—
81C2×C4×D5—512no—
82C2×C4×D5—512no—
83C2×Dic10—56no—
84C2×Dic10—512no—
85C5×C22⋊C4C2×D5124yescomplemented
86C42×C5C2×D516yescomplemented
87C22⋊Dic5—524no—
88#685—524no—
89#685—524no—
90C4×Dic5—512no—
91C4×Dic5—56no—
92C2×#124—251noSylow
93C2×C52—42no—
94C2×C52C22×D411yescharacteristic
95C2×C52C2×#4112yes—
96C5⋊D5—42no—
97C5⋊D5#12414yesdirect product factor
98C23×C5C22×D5112yes—
99C22×D5—56no—
100C22×D5—512no—
101C22×D5—1024no—
102C22×D5—1048no—
103C5×Q8—212no—
104C5×D4—212no—
105C2×C4×C5C22×D5112yes—
106C2×C4×C5C22×D5112yes—
107C2×C4×C5C22×D516yes—
108C22⋊D5—1024no—
109C2×Dic5—56no—
110C2×Dic5—512no—
111C2×Dic5—512no—
112D20—1012no—
113C4×D5—524no—
114C4×D5—1024no—
115C4×D5—1024no—
116Dic10—1024no—
117Dic10—1012no—
118C22×Q8—251no—
119C22×D4—251no—
120#124—251no—
121#124—251no—
122#124—254no—
123#124—251no—
124#124C5⋊D511yescharacteristic, direct product factor
125C2×C22⋊C4—254no—
126C2×C42—251no—
127C52C2×#12411yescharacteristic, complemented, Sylow
128C22×C5—212no—
129C22×C5—224no—
130C22×C5C23×D516yes—
131C2×D5—512no—
132C2×D5—524no—
133C2×D5—1012no—
134C2×D5—1024no—
135C2×D5—2048no—
136C4×C5—212no—
137C4×C5—212no—
138C4×C5D4×D5112yes—
139Dic5—512no—
140Dic5—1012no—
141Dic5—1012no—
142C24—252no—
143C2×Q8—252no—
144C2×Q8—251no—
145C2×Q8—504no—
146C2×Q8C2×C5⋊D511yescharacteristic, complemented
147C2×D4—252no—
148C2×D4—251no—
149C2×D4—504no—
150C2×D4C2×C5⋊D511yescharacteristic
151C22×C4—251no—
152C22×C4—252no—
153C22×C4—252no—
154C22⋊C4—254no—
155C22⋊C4—254no—
156C22⋊C4—254no—
157C22⋊C4C2×C5⋊D514yescomplemented
158C42—252no—
159C42—251no—
160C42C2×C5⋊D511yescharacteristic, complemented
161C2×C5—412no—
162C2×C5C2×D4×D516yes—
163C2×C5D5×#41112yes—
164D5—524no—
165D5—2012no—
166C23—251no—
167C23—252no—
168C23—504no—
169C23—508no—
170C23C22×C5⋊D512yes—
171Q8—22no—
172Q8—504no—
173Q8—502no—
174D4—22no—
175D4—504no—
176D4—502no—
177C2×C4—251no—
178C2×C4—254no—
179C2×C4—252no—
180C2×C4—252no—
181C2×C4—504no—
182C2×C4—504no—
183C2×C4C22×C5⋊D512yes—
184C2×C4C22×C5⋊D512yes—
185C2×C4C22×C5⋊D511yescharacteristic
186C5D5×#12416yescomplemented
187C22—22no—
188C22—24no—
189C22—252no—
190C22—254no—
191C22—502no—
192C22—504no—
193C22—1008no—
194C22C23×C5⋊D511yescharacteristic
195C4—22no—
196C4—22no—
197C4—252no—
198C4—502no—
199C4—502no—
200C4D4×C5⋊D512yes—
201C2—42no—
202C2—254no—
203C2—1002no—
204C2C2×D4×C5⋊D511yescharacteristic
205C2#41×C5⋊D512yes—
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#124C2
prime radicalC2×D4×C5⋊D5C2
hypercenter#124C5⋊D5
hypocenterC52C2×#124
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C5⋊D5C2semidirect
3. C52×#124C2semidirect
4. #11717267C2semidirect
5. #11717256C2semidirect
6. #11717205C2semidirect
7. C22⋊C4×C5⋊D5C2semidirect
8. #11717195C2semidirect
9. C42×C5⋊D5C2semidirect
11. C2×C52×Q8C22semidirect
20. C2×#6045C22semidirect
21. C2×#6045C22semidirect
22. C52×C22⋊C4C22semidirect
23. C42×C52C22semidirect
24. C22⋊#1039C22semidirect
25. #87808C22semidirect
26. #87808C22semidirect
27. C4×#1039C22semidirect
28. C4×#1039C22semidirect
52. C5×#124D5semidirect
74. C2×C5×Q8C2×D5semidirect
85. C5×C22⋊C4C2×D5semidirect
86. C42×C5C2×D5semidirect
97. C5⋊D5#124direct (N × H)
124. #124C5⋊D5direct (N × H)
127. C52C2×#124semidirect
146. C2×Q8C2×C5⋊D5semidirect
157. C22⋊C4C2×C5⋊D5semidirect
160. C42C2×C5⋊D5semidirect
186. C5D5×#124semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C5⋊D52
C2×D4×C5⋊D52
C52×#1242
#117172672
#117172562
#117172052
C22⋊C4×C5⋊D52
#117171952
C42×C5⋊D52
D5×#1245