Finite Group Databasebeta

← #13107145

Subgroups — #13107145

Order 1600.

1000 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 227 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×#121C211yesmaximal, characteristic, complemented
2#11717259C212yesmaximal, complemented
3#11717257C212yesmaximal, complemented
4D4×#1039C212yesmaximal, complemented
5C2×#87808C211yesmaximal, characteristic, complemented
6#11717244C211yesmaximal, characteristic, complemented
7#11717218C212yesmaximal, complemented
8#11717206C212yesmaximal, complemented
9#11717205C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C2×C52×D4C2212yes—
12C2×C52×D4C2211yescharacteristic, complemented
13C22×C4×C52C2211yescharacteristic
14C2×C22⋊(C5⋊D5)C2212yes—
15C2×C22⋊(C5⋊D5)C2212yes—
16C22×#1039C2211yescharacteristic
17C22×#1039C2212yescomplemented
18C2×#6045C2211yescharacteristic, complemented
19C52×C4⋊C4C2211yescharacteristic, complemented
20C52×C22⋊C4C2212yescomplemented
21C22⋊#1039C2212yes—
22C22⋊#1039C2212yescomplemented
23#87808C2212yes—
24#87808C2212yes—
25#87808C2212yes—
26#87808C2212yes—
27C4⋊#1039C2211yescharacteristic, complemented
28C4⋊#1039C2212yescomplemented
29#87806C2212yescomplemented
30C4×#1039C2212yescomplemented
31#65951—56nomaximal
32C23×C52C2312yes—
33C23×C52C2311yescharacteristic
34C22×C5⋊D5—22no—
35C22×C5⋊D5—41no—
36C22×C5⋊D5—41no—
37C22×C5⋊D5C2312yes—
38C52×D4—24no—
39C52×D4—41no—
40C2×C4×C52—41no—
41C2×C4×C52C2311yescharacteristic
42C2×C4×C52C2312yes—
43C2×C4×C52C2311yescharacteristic
44C22⋊(C5⋊D5)—42no—
45C22⋊(C5⋊D5)—42no—
46C2×#1039—41no—
47C2×#1039—42no—
48C2×#1039C2312yes—
49C2×#1039C2312yes—
50C2×#1039C2312yes—
51#6045—41no—
52C5×#121D516yescomplemented
53#4085—512no—
54#4083—512no—
55D4×Dic5—512no—
56C2×#685—56no—
57#4070—56no—
58#4044—512no—
59#4032—512no—
60#4031—512no—
61C22×C52—21no—
62C22×C52—21no—
63C22×C52—22no—
64C22×C52—22no—
65C22×C52—22no—
66C22×C52#4112yes—
67C22×C52C2411yescharacteristic
68C2×C5⋊D5—22no—
69C2×C5⋊D5—22no—
70C2×C5⋊D5—22no—
71C2×C5⋊D5—44no—
72C4×C52—22no—
73C4×C52—21no—
74C4×C52—21no—
75#1039—22no—
76#1039—22no—
77#1039—22no—
78C23×D5—56no—
79C2×C5×D4C2×D5112yes—
80C2×C5×D4C2×D516yescomplemented
81C22×C4×C5C2×D516yes—
82C2×C22⋊D5—512no—
83C2×C22⋊D5—512no—
84C22×Dic5—56no—
85C22×Dic5—512no—
86C2×Dic10—56no—
87C5×C4⋊C4C2×D516yescomplemented
88C5×C22⋊C4C2×D5112yescomplemented
89C22⋊Dic5—512no—
90C22⋊Dic5—512no—
91#685—512no—
92#685—512no—
93#685—512no—
94#685—512no—
95C4⋊Dic5—56no—
96C4⋊Dic5—512no—
97#683—512no—
98C4×Dic5—512no—
99#538—251noSylow
100C2×C52—22no—
101C2×C52—42no—
102C2×C52C2×#4111yescharacteristic
103C2×C52H(5,𝔽2)11yescharacteristic
104C2×C52H(5,𝔽2)11yescharacteristic
105C5⋊D5—42no—
106C23×C5C22×D5112yes—
107C23×C5C22×D516yes—
108C22×D5—512no—
109C22×D5—1012no—
110C22×D5—206no—
111C22×D5—206no—
112C5×D4—224no—
113C5×D4—46no—
114C2×C4×C5—46no—
115C2×C4×C5C22×D5112yes—
116C2×C4×C5C22×D516yes—
117C2×C4×C5C22×D516yes—
118C22⋊D5—2012no—
119C22⋊D5—2012no—
120C2×Dic5—512no—
121C2×Dic5—512no—
122C2×Dic5—512no—
123C2×Dic5—206no—
124C2×Dic5—2012no—
125Dic10—206no—
126#126—252no—
127#124—252no—
128#123—252no—
129C22⋊Q8—251no—
130#121—252no—
131#121C5⋊D511yescharacteristic, complemented
132#120—252no—
133C4×D4—252no—
134C2×C22⋊C4—251no—
135C52#53811yescharacteristic, complemented, Sylow
136C22×C5—26no—
137C22×C5—26no—
138C22×C5—212no—
139C22×C5—212no—
140C22×C5—212no—
141C22×C5#711112yes—
142C22×C5C23×D516yes—
143C2×D5—1012no—
144C2×D5—1012no—
145C2×D5—1012no—
146C2×D5—2024no—
147C4×C5—212no—
148C4×C5—26no—
149C4×C5—26no—
150Dic5—1012no—
151Dic5—1012no—
152Dic5—1012no—
153C24—251no—
154C2×Q8—251no—
155C2×D4—252no—
156C2×D4—252no—
157C2×D4C2×C5⋊D512yes—
158C2×D4C2×C5⋊D511yescharacteristic, complemented
159C22×C4—251no—
160C22×C4—252no—
161C22×C4C2×C5⋊D511yescharacteristic
162C4⋊C4—252no—
163C4⋊C4—252no—
164C4⋊C4—251no—
165C4⋊C4C2×C5⋊D511yescharacteristic, complemented
166C22⋊C4—252no—
167C22⋊C4—252no—
168C22⋊C4—252no—
169C22⋊C4—252no—
170C22⋊C4—252no—
171C22⋊C4—252no—
172C22⋊C4C2×C5⋊D512yescomplemented
173C42—252no—
174C2×C5—212no—
175C2×C5—412no—
176C2×C5C2×#71116yes—
177C2×C5#414416yes—
178C2×C5#414916yes—
179D5—2012no—
180C23—252no—
181C23—502no—
182C23—1001no—
183C23—1001no—
184C23C22×C5⋊D512yes—
185C23C22×C5⋊D511yescharacteristic
186Q8—1001no—
187D4—24no—
188D4—41no—
189D4—1002no—
190D4—1002no—
191C2×C4—41no—
192C2×C4—252no—
193C2×C4—252no—
194C2×C4—252no—
195C2×C4—1001no—
196C2×C4—1002no—
197C2×C4C22×C5⋊D512yes—
198C2×C4C22×C5⋊D511yescharacteristic
199C2×C4C22×C5⋊D511yescharacteristic
200C5#6595116yescomplemented
201C22—21no—
202C22—21no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—502no—
207C22—502no—
208C22—502no—
209C22—1004no—
210C22#8790212yes—
211C22C23×C5⋊D511yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—502no—
216C4—502no—
217C4—502no—
218C2—22no—
219C2—42no—
220C2—1002no—
221C2C2×#8790211yescharacteristic
222C2#1171751711yescharacteristic
223C2#1171752211yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#121C2
hypercenter#121C5⋊D5
hypocenterC52#538
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C52×#121C2semidirect
2. #11717259C2semidirect
3. #11717257C2semidirect
4. D4×#1039C2semidirect
5. C2×#87808C2semidirect
6. #11717244C2semidirect
7. #11717218C2semidirect
8. #11717206C2semidirect
9. #11717205C2semidirect
12. C2×C52×D4C22semidirect
17. C22×#1039C22semidirect
18. C2×#6045C22semidirect
19. C52×C4⋊C4C22semidirect
20. C52×C22⋊C4C22semidirect
22. C22⋊#1039C22semidirect
27. C4⋊#1039C22semidirect
28. C4⋊#1039C22semidirect
29. #87806C22semidirect
30. C4×#1039C22semidirect
52. C5×#121D5semidirect
80. C2×C5×D4C2×D5semidirect
87. C5×C4⋊C4C2×D5semidirect
88. C5×C22⋊C4C2×D5semidirect
131. #121C5⋊D5semidirect
135. C52#538semidirect
158. C2×D4C2×C5⋊D5semidirect
165. C4⋊C4C2×C5⋊D5semidirect
172. C22⋊C4C2×C5⋊D5semidirect
200. C5#65951semidirect

Maximal subgroups

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

subgroupindex
C52×#1212
#117172592
#117172572
D4×#10392
C2×#878082
#117172442
#117172182
#117172062
#117172052
#659515