Finite Group Databasebeta

← #13107296

Subgroups — #13107296

Order 1600.

1048 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 219 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
1#41×C5⋊D5C212yesmaximal, complemented
2C52×#135C211yesmaximal, characteristic, complemented
3#11717270C212yesmaximal, complemented
4#11717238C212yesmaximal, complemented
5#11717235C212yesmaximal, complemented
6#11717234C212yesmaximal, complemented
7#11717231C212yesmaximal, complemented
8#11717224C211yesmaximal, characteristic, complemented
9C2×#87799C211yesmaximal, characteristic, complemented
10C52×#41C2212yescomplemented
11#87904C2212yes—
12Q8×C5⋊D5C2212yescomplemented
13#87902—22no—
14#87902C2212yes—
15D4×C5⋊D5—22no—
16D4×C5⋊D5C2212yescomplemented
17#87900—22no—
18#87900C2212yescomplemented
19C2×C4×C5⋊D5—22no—
20C2×C4×C5⋊D5C2211yescharacteristic
21C52×Q16C2211yescharacteristic, complemented
22C52×SD16C2212yescomplemented
23C52×D8C2211yescharacteristic, complemented
24C2×C52×C8C2211yescharacteristic, complemented
25#87812C2212yescomplemented
26#87811C2212yescomplemented
27#87810C2212yescomplemented
28#87809C2212yescomplemented
29C2×#6029C2211yescharacteristic, complemented
30#87802C2211yescharacteristic, complemented
31#87801C2211yescharacteristic, complemented
32#87800C2212yescomplemented
33#87799C2212yescomplemented
34#87799C2212yescomplemented
35#66102—56nomaximal
36C22×C5⋊D5—22no—
37C22×C5⋊D5D411yescharacteristic
38C52×Q8C2312yes—
39C52×D4—22no—
40C52×D4C2312yes—
41C2×C4×C52—22no—
42C2×C4×C52C2311yescharacteristic
43C22⋊(C5⋊D5)—22no—
44C22⋊(C5⋊D5)—22no—
45C22⋊(C5⋊D5)—22no—
46C2×#1039—22no—
47C2×#1039D411yescharacteristic
48C4⋊(C5⋊D5)—22no—
49C4⋊(C5⋊D5)C2312yes—
50C4×C5⋊D5—22no—
51C4×C5⋊D5—22no—
52C4×C5⋊D5—22no—
53C4×C5⋊D5D412yes—
54C4×C5⋊D5C2312yes—
55#6045—22no—
56#6045C2312yes—
57C52×C8C2311yescharacteristic
58C52×C8C2311yescharacteristic
59#6029C2312yes—
60D5×#41—512no—
61C5×#135D516yescomplemented
62#4096—512no—
63#4064—512no—
64#4061—512no—
65#4060—512no—
66#4057—512no—
67#4050—56no—
68C2×#676—56no—
69C22×C52—22no—
70C22×C52C2×D411yescharacteristic
71C2×C5⋊D5—22no—
72C2×C5⋊D5—41no—
73C2×C5⋊D5—44no—
74C2×C5⋊D5C2×D412yes—
75C4×C52—22no—
76C4×C52C2×D411yescharacteristic
77C4×C52C2411yescharacteristic
78#1039—22no—
79#1039C2×D412yes—
80C5×#41C2×D5112yescomplemented
81#713—512no—
82Q8×D5—512no—
83#711—512no—
84#711—1012no—
85D4×D5—512no—
86D4×D5—1012no—
87#709—512no—
88#709—1012no—
89C2×C4×D5—56no—
90C2×C4×D5—1012no—
91C5×Q16C2×D516yescomplemented
92C5×SD16C2×D5112yescomplemented
93C5×D8C2×D516yescomplemented
94C2×C5×C8C2×D516yescomplemented
95#689—512no—
96#688—512no—
97#687—512no—
98#686—512no—
99C2×#168—56no—
100Dic20—56no—
101D40—56no—
102#677—512no—
103#676—512no—
104#676—512no—
105#575—251noSylow
106C2×C52—21no—
107C2×C52—42no—
108C2×C52C22×D411yescharacteristic
109C5⋊D5—22no—
110C5⋊D5—42no—
111C22×D5—56no—
112C22×D5—1012no—
113C5×Q8C22×D5112yes—
114C5×D4—212no—
115C5×D4C22×D5112yes—
116C2×C4×C5—212no—
117C2×C4×C5C22×D516yes—
118C22⋊D5—1012no—
119C22⋊D5—1012no—
120C22⋊D5—1012no—
121C2×Dic5—56no—
122C2×Dic5—1012no—
123D20—512no—
124D20—1012no—
125C4×D5—512no—
126C4×D5—512no—
127C4×D5—1012no—
128C4×D5—1012no—
129C4×D5—1012no—
130Dic10—512no—
131Dic10—1012no—
132C5×C8C22×D516yes—
133C5×C8C22×D516yes—
134#168—512no—
135C2×#41—252no—
136#137—252no—
137#137—252no—
138AGL(1,ℤ/8)—252no—
139AGL(1,ℤ/8)—252no—
140#135—252no—
141#135—251no—
142#135C5⋊D511yescharacteristic, complemented
143C2×M16—251no—
144C52#57511yescharacteristic, complemented, Sylow
145C22×C5—212no—
146C22×C5D4×D516yes—
147C2×D5—512no—
148C2×D5—1012no—
149C2×D5—206no—
150C2×D5—2024no—
151C4×C5—212no—
152C4×C5D4×D516yes—
153C4×C5C23×D516yes—
154Dic5—512no—
155Dic5—1012no—
156#41—252no—
157#41—252no—
158#41—252no—
159#41—502no—
160#41—502no—
161#41C2×C5⋊D512yescomplemented
162C2×Q8—252no—
163C2×D4—252no—
164C2×D4—502no—
165C22×C4—251no—
166C22×C4—502no—
167Q16—252no—
168Q16—251no—
169Q16C2×C5⋊D511yescharacteristic, complemented
170SD16—252no—
171SD16—252no—
172SD16—252no—
173SD16C2×C5⋊D512yescomplemented
174D8—252no—
175D8—251no—
176D8C2×C5⋊D511yescharacteristic, complemented
177M16—252no—
178M16—252no—
179C2×C8—251no—
180C2×C8C2×C5⋊D511yescharacteristic, complemented
181C2×C5—26no—
182C2×C5—412no—
183C2×C5C2×D4×D516yes—
184D5—1012no—
185D5—2012no—
186C23—251no—
187C23—502no—
188Q8—252no—
189Q8—502no—
190Q8C22×C5⋊D512yes—
191D4—22no—
192D4—252no—
193D4—502no—
194D4—502no—
195D4—502no—
196D4—502no—
197D4C22×C5⋊D512yes—
198C2×C4—22no—
199C2×C4—251no—
200C2×C4—252no—
201C2×C4—252no—
202C2×C4—502no—
203C2×C4—502no—
204C2×C4—502no—
205C2×C4—502no—
206C2×C4C22×C5⋊D511yescharacteristic
207C8—252no—
208C8C22×C5⋊D511yescharacteristic
209C8C22×C5⋊D511yescharacteristic
210C5#6610216yescomplemented
211C22—22no—
212C22—252no—
213C22—502no—
214C22—1001no—
215C22—1004no—
216C22D4×C5⋊D511yescharacteristic
217C4—22no—
218C4—252no—
219C4—502no—
220C4D4×C5⋊D511yescharacteristic
221C4C23×C5⋊D511yescharacteristic
222C2—21no—
223C2—42no—
224C2—502no—
225C2—1002no—
226C2C2×D4×C5⋊D511yescharacteristic
subgroupG/H
centerC4D4×C5⋊D5
commutator, cosocleC4×C52C24
FrattiniC4C23×C5⋊D5
FittingC52×#135C2
socleC2×C52C22×D4
hypercenter#135C5⋊D5
hypocenterC52#575
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×C5⋊D5C2semidirect
2. C52×#135C2semidirect
3. #11717270C2semidirect
4. #11717238C2semidirect
5. #11717235C2semidirect
6. #11717234C2semidirect
7. #11717231C2semidirect
8. #11717224C2semidirect
9. C2×#87799C2semidirect
10. C52×#41C22semidirect
12. Q8×C5⋊D5C22semidirect
16. D4×C5⋊D5C22semidirect
18. #87900C22semidirect
21. C52×Q16C22semidirect
22. C52×SD16C22semidirect
23. C52×D8C22semidirect
24. C2×C52×C8C22semidirect
25. #87812C22semidirect
26. #87811C22semidirect
27. #87810C22semidirect
28. #87809C22semidirect
29. C2×#6029C22semidirect
30. #87802C22semidirect
31. #87801C22semidirect
32. #87800C22semidirect
33. #87799C22semidirect
34. #87799C22semidirect
61. C5×#135D5semidirect
80. C5×#41C2×D5semidirect
91. C5×Q16C2×D5semidirect
92. C5×SD16C2×D5semidirect
93. C5×D8C2×D5semidirect
94. C2×C5×C8C2×D5semidirect
142. #135C5⋊D5semidirect
144. C52#575semidirect
161. #41C2×C5⋊D5semidirect
169. Q16C2×C5⋊D5semidirect
173. SD16C2×C5⋊D5semidirect
176. D8C2×C5⋊D5semidirect
180. C2×C8C2×C5⋊D5semidirect
210. C5#66102semidirect

Maximal subgroups

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

subgroupindex
#41×C5⋊D52
C52×#1352
#117172702
#117172382
#117172352
#117172342
#117172312
#117172242
C2×#877992
#661025