Finite Group Databasebeta

← #13107208

Subgroups — #13107208

Order 1600.

992 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 215 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
1C52×#124C211yesmaximal, characteristic, complemented
2#11717266C212yesmaximal, complemented
3#11717257C211yesmaximal, characteristic
4#11717255C211yesmaximal, characteristic, complemented
5#11717203C212yesmaximal, complemented
6#11717201C212yesmaximal, complemented
7C22⋊C4×C5⋊D5C212yesmaximal, complemented
8#11717199C212yesmaximal, complemented
9#11717192C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C2×C52×Q8C2211yescharacteristic, complemented
12C2×C52×D4C2211yescharacteristic
13C2×C22⋊(C5⋊D5)C2212yes—
14C22×#1039C2211yescharacteristic, complemented
15C2×C4×C5⋊D5C2212yes—
16C2×C4×C5⋊D5C2212yes—
17C52×C22⋊C4C2212yes—
18C52×C22⋊C4C2212yescomplemented
19C42×C52C2211yescharacteristic, complemented
20C22⋊#1039C2211yescharacteristic
21C22⋊#1039C2212yes—
22C22⋊#1039C2211yescharacteristic, complemented
23#87808C2212yes—
24#87808C2212yes—
25#87808C2212yes—
26C4⋊#1039C2212yescomplemented
27#87806C2212yescomplemented
28#87806C2212yescomplemented
29#87806C2212yescomplemented
30C4×#1039C2212yescomplemented
31#66013—56nomaximal
32C23×C52C2311yescharacteristic
33C23×C52C2311yescharacteristic
34C22×C5⋊D5—24no—
35C22×C5⋊D5—41no—
36C22×C5⋊D5C2312yes—
37C52×Q8—41no—
38C52×D4—41no—
39C2×C4×C52C2311yescharacteristic
40C2×C4×C52C2312yes—
41C2×C4×C52C2312yes—
42C22⋊(C5⋊D5)—24no—
43C2×#1039—41no—
44C2×#1039C2312yes—
45C2×#1039C2312yes—
46C2×#1039C2312yes—
47C4×C5⋊D5—24no—
48C4×C5⋊D5—42no—
49C5×#124D516yescomplemented
50#4092—512no—
51#4083—56no—
52#4081—56no—
53#4029—512no—
54#4027—512no—
55D5×C22⋊C4—512no—
56#4025—512no—
57#4018—512no—
58C22×C52—22no—
59C22×C52—22no—
60C22×C52—21no—
61C22×C52—21no—
62C22×C52C2411yescharacteristic
63C2×C5⋊D5—22no—
64C2×C5⋊D5—22no—
65C2×C5⋊D5—44no—
66C2×C5⋊D5#4114yes—
67C4×C52—22no—
68C4×C52—22no—
69C4×C52—21no—
70#1039—22no—
71#1039—22no—
72#1039—22no—
73C23×D5—56no—
74C2×C5×Q8C2×D516yescomplemented
75C2×C5×D4C2×D516yes—
76C2×C22⋊D5—512no—
77C22×Dic5—56no—
78C2×C4×D5—512no—
79C2×C4×D5—512no—
80C5×C22⋊C4C2×D5112yes—
81C5×C22⋊C4C2×D5112yescomplemented
82C42×C5C2×D516yescomplemented
83C22⋊Dic5—56no—
84C22⋊Dic5—512no—
85C22⋊Dic5—56no—
86#685—512no—
87#685—512no—
88#685—512no—
89C4⋊Dic5—512no—
90#683—512no—
91#683—512no—
92#683—512no—
93C4×Dic5—512no—
94#551—251noSylow
95C2×C52—41no—
96C2×C52—41no—
97C2×C52C2×#4112yes—
98C2×C52H(5,𝔽2)11yescharacteristic
99C5⋊D5—24no—
100C23×C5C22×D516yes—
101C23×C5C22×D516yes—
102C22×D5—512no—
103C22×D5—1024no—
104C22×D5—206no—
105C5×Q8—46no—
106C5×D4—46no—
107C2×C4×C5C22×D5112yes—
108C2×C4×C5C22×D516yes—
109C2×C4×C5C22×D5112yes—
110C22⋊D5—1024no—
111C2×Dic5—512no—
112C2×Dic5—512no—
113C2×Dic5—512no—
114C2×Dic5—206no—
115C4×D5—1024no—
116C4×D5—2012no—
117#126—252no—
118#124C5⋊D511yescharacteristic, complemented
119#123—251no—
120#123—252no—
121C22⋊Q8—252no—
122#120—251no—
123C4×D4—252no—
124#117—252no—
125C2×C22⋊C4—252no—
126C52#55111yescharacteristic, complemented, Sylow
127C22×C5—26no—
128C22×C5—26no—
129C22×C5—212no—
130C22×C5—212no—
131C22×C5C23×D516yes—
132C2×D5—524no—
133C2×D5—1012no—
134C2×D5—1012no—
135C2×D5—2024no—
136C4×C5—212no—
137C4×C5—26no—
138C4×C5—212no—
139Dic5—1012no—
140Dic5—1012no—
141Dic5—1012no—
142C24—251no—
143C2×Q8C2×C5⋊D511yescharacteristic, complemented
144C2×D4—252no—
145C2×D4C2×C5⋊D511yescharacteristic
146C22×C4—251no—
147C22×C4—252no—
148C22×C4—252no—
149C4⋊C4—252no—
150C4⋊C4—252no—
151C4⋊C4—252no—
152C4⋊C4—252no—
153C22⋊C4—252no—
154C22⋊C4—251no—
155C22⋊C4—251no—
156C22⋊C4—252no—
157C22⋊C4—252no—
158C22⋊C4—252no—
159C22⋊C4C2×C5⋊D512yes—
160C22⋊C4C2×C5⋊D512yescomplemented
161C42—252no—
162C42C2×C5⋊D511yescharacteristic, complemented
163C2×C5—46no—
164C2×C5—46no—
165C2×C5#414416yes—
166C2×C5D5×#41112yes—
167D5—1024no—
168C23—252no—
169C23—504no—
170C23—1001no—
171C23C22×C5⋊D511yescharacteristic
172C23C22×C5⋊D511yescharacteristic
173Q8—41no—
174D4—41no—
175D4—504no—
176C2×C4—252no—
177C2×C4—252no—
178C2×C4—252no—
179C2×C4—504no—
180C2×C4—1002no—
181C2×C4—1001no—
182C2×C4C22×C5⋊D512yes—
183C2×C4C22×C5⋊D511yescharacteristic
184C2×C4C22×C5⋊D512yes—
185C5#6601316yescomplemented
186C22—21no—
187C22—21no—
188C22—22no—
189C22—22no—
190C22—254no—
191C22—502no—
192C22—502no—
193C22—1004no—
194C22C23×C5⋊D511yescharacteristic
195C4—22no—
196C4—21no—
197C4—22no—
198C4—502no—
199C4—502no—
200C4—502no—
201C2—41no—
202C2—41no—
203C2—504no—
204C2#1171751711yescharacteristic
205C2#41×C5⋊D512yes—
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#124C2
prime radical#11717257C2
hypercenter#124C5⋊D5
hypocenterC52#551
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C52×#124C2semidirect
2. #11717266C2semidirect
4. #11717255C2semidirect
5. #11717203C2semidirect
6. #11717201C2semidirect
7. C22⋊C4×C5⋊D5C2semidirect
8. #11717199C2semidirect
9. #11717192C2semidirect
11. C2×C52×Q8C22semidirect
14. C22×#1039C22semidirect
18. C52×C22⋊C4C22semidirect
19. C42×C52C22semidirect
22. C22⋊#1039C22semidirect
26. C4⋊#1039C22semidirect
27. #87806C22semidirect
28. #87806C22semidirect
29. #87806C22semidirect
30. C4×#1039C22semidirect
49. C5×#124D5semidirect
74. C2×C5×Q8C2×D5semidirect
81. C5×C22⋊C4C2×D5semidirect
82. C42×C5C2×D5semidirect
118. #124C5⋊D5semidirect
126. C52#551semidirect
143. C2×Q8C2×C5⋊D5semidirect
160. C22⋊C4C2×C5⋊D5semidirect
162. C42C2×C5⋊D5semidirect
185. C5#66013semidirect

Maximal subgroups

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

subgroupindex
C52×#1242
#117172662
#117172572
#117172552
#117172032
#117172012
C22⋊C4×C5⋊D52
#117171992
#117171922
#660135