Finite Group Databasebeta

← #13107115

Subgroups — #13107115

Order 1600.

1336 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C22×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C2×#87902C211yesmaximal, characteristic, complemented
3C52×#120C211yesmaximal, characteristic, complemented
4#11717273C211yesmaximal, characteristic, complemented
5#11717260C212yesmaximal, complemented
6#11717259C212yesmaximal, complemented
7#11717257C211yesmaximal, characteristic, complemented
8#11717205C212yesmaximal, complemented
9#11717204C212yesmaximal, complemented
10#11717202C211yesmaximal, characteristic, complemented
11#11717197C211yesmaximal, characteristic, complemented
12C24×C52C2211yescharacteristic
13C23×C5⋊D5C2211yescharacteristic
14C2×C52×D4C2211yescharacteristic
15C2×C52×D4C2212yescomplemented
16C2×C22⋊(C5⋊D5)—24no—
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2212yes—
19C2×C22⋊(C5⋊D5)C2212yes—
20C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
21C2×C22⋊(C5⋊D5)C2212yescomplemented
22C2×C22⋊(C5⋊D5)C2212yescomplemented
23C22×#1039C2211yescharacteristic
24C22×#1039C2211yescharacteristic, complemented
25#87902—24no—
26C2×C4⋊(C5⋊D5)C2212yescomplemented
27C2×C4×C5⋊D5C2211yescharacteristic, complemented
28C2×#6045C2211yescharacteristic, complemented
29C52×C22⋊C4C2211yescharacteristic
30C52×C22⋊C4C2212yescomplemented
31C22⋊#1039C2211yescharacteristic
32C22⋊#1039C2212yescomplemented
33#87808C2212yescomplemented
34#87808C2212yescomplemented
35#87806C2212yescomplemented
36C4×#1039C2212yescomplemented
37#65926—56nomaximal
38C23×C52—22no—
39C23×C52—42no—
40C23×C52C2312yes—
41C23×C52C2311yescharacteristic
42C23×C52C2311yescharacteristic
43C22×C5⋊D5—22no—
44C22×C5⋊D5—42no—
45C22×C5⋊D5C2312yes—
46C22×C5⋊D5C2311yescharacteristic
47C52×D4—22no—
48C52×D4—42no—
49C2×C4×C52C2311yescharacteristic
50C2×C4×C52C2312yes—
51C22⋊(C5⋊D5)—22no—
52C22⋊(C5⋊D5)—22no—
53C22⋊(C5⋊D5)—24no—
54C22⋊(C5⋊D5)—24no—
55C22⋊(C5⋊D5)—42no—
56C22⋊(C5⋊D5)—42no—
57C2×#1039—22no—
58C2×#1039D414yescomplemented
59C2×#1039C2311yescharacteristic
60C2×#1039C2312yes—
61C2×#1039C2311yescharacteristic
62C2×#1039C2311yescharacteristic
63C4⋊(C5⋊D5)—42no—
64C4×C5⋊D5—22no—
65#6045—22no—
66C22×C22⋊D5—56no—
67C2×#711—56no—
68C5×#120D516yescomplemented
69#4099—56no—
70#4086—512no—
71#4085—512no—
72#4083—56no—
73#4031—512no—
74#4030—512no—
75#4028—56no—
76#4023—56no—
77C22×C52—22no—
78C22×C52—21no—
79C22×C52—22no—
80C22×C52—22no—
81C22×C52—22no—
82C22×C52—22no—
83C22×C52—44no—
84C22×C52C2×D412yes—
85C22×C52C2411yescharacteristic
86C2×C5⋊D5—21no—
87C2×C5⋊D5—22no—
88C2×C5⋊D5—22no—
89C2×C5⋊D5—22no—
90C2×C5⋊D5—22no—
91C2×C5⋊D5—44no—
92C4×C52—21no—
93C4×C52—22no—
94#1039—21no—
95#1039—22no—
96#1039C2×D412yes—
97#1039C2×D412yes—
98C24×C5C2×D516yes—
99C23×D5—56no—
100C2×C5×D4C2×D516yes—
101C2×C5×D4C2×D5112yescomplemented
102C2×C22⋊D5—56no—
103C2×C22⋊D5—56no—
104C2×C22⋊D5—512no—
105C2×C22⋊D5—512no—
106C2×C22⋊D5—512no—
107C2×C22⋊D5—512no—
108C2×C22⋊D5—1024no—
109C22×Dic5—56no—
110C22×Dic5—56no—
111#711—1024no—
112C2×D20—512no—
113C2×C4×D5—56no—
114C2×Dic10—56no—
115C5×C22⋊C4C2×D516yes—
116C5×C22⋊C4C2×D5112yescomplemented
117C22⋊Dic5—56no—
118C22⋊Dic5—512no—
119#685—512no—
120#685—512no—
121#683—512no—
122C4×Dic5—512no—
123#535—251noSylow
124C2×C52—22no—
125C2×C52—41no—
126C2×C52—42no—
127C2×C52C22×D411yescharacteristic
128C2×C52H(5,𝔽2)12yes—
129C5⋊D5—41no—
130C5⋊D5—42no—
131C23×C5—212no—
132C23×C5—412no—
133C23×C5C22×D5112yes—
134C23×C5C22×D516yes—
135C23×C5C22×D516yes—
136C22×D5—512no—
137C22×D5—56no—
138C22×D5—1012no—
139C22×D5—2012no—
140C5×D4—212no—
141C5×D4—412no—
142C2×C4×C5C22×D5112yes—
143C2×C4×C5C22×D516yes—
144C22⋊D5—1012no—
145C22⋊D5—1024no—
146C22⋊D5—1024no—
147C22⋊D5—1012no—
148C22⋊D5—2012no—
149C22⋊D5—2012no—
150C2×Dic5—524no—
151C2×Dic5—56no—
152C2×Dic5—56no—
153C2×Dic5—512no—
154C2×Dic5—56no—
155C2×Dic5—1012no—
156D20—2012no—
157C4×D5—1012no—
158Dic10—1012no—
159C2×#41—251no—
160C22×D4—251no—
161#127—252no—
162#124—252no—
163#121—252no—
164#121—252no—
165#120—251no—
166#120—251no—
167#120—251no—
168#120C5⋊D511yescharacteristic, complemented
169#117—251no—
170C52#53511yescharacteristic, complemented, Sylow
171C22×C5—212no—
172C22×C5—26no—
173C22×C5—212no—
174C22×C5—212no—
175C22×C5—212no—
176C22×C5—212no—
177C22×C5—424no—
178C22×C5D4×D5112yes—
179C22×C5C23×D516yes—
180C2×D5—106no—
181C2×D5—1012no—
182C2×D5—1012no—
183C2×D5—1012no—
184C2×D5—1012no—
185C2×D5—2024no—
186C4×C5—26no—
187C4×C5—212no—
188Dic5—512no—
189Dic5—512no—
190Dic5—106no—
191Dic5—1012no—
192C24—251no—
193C24C2×C5⋊D511yescharacteristic
194#41—504no—
195C2×Q8—251no—
196C2×D4—251no—
197C2×D4—252no—
198C2×D4—252no—
199C2×D4—251no—
200C2×D4—252no—
201C2×D4—252no—
202C2×D4—252no—
203C2×D4—504no—
204C2×D4C2×C5⋊D511yescharacteristic
205C2×D4C2×C5⋊D512yescomplemented
206C22×C4—251no—
207C22×C4—251no—
208C22×C4—251no—
209C4⋊C4—252no—
210C22⋊C4—251no—
211C22⋊C4—252no—
212C22⋊C4—252no—
213C22⋊C4—252no—
214C22⋊C4C2×C5⋊D511yescharacteristic
215C22⋊C4C2×C5⋊D512yescomplemented
216C42—252no—
217C2×C5—212no—
218C2×C5—46no—
219C2×C5—412no—
220C2×C5C2×D4×D516yes—
221C2×C5#4144112yes—
222D5—206no—
223D5—2012no—
224C23—22no—
225C23—42no—
226C23—252no—
227C23—251no—
228C23—502no—
229C23—1002no—
230C23C22×C5⋊D512yes—
231C23C22×C5⋊D511yescharacteristic
232C23C22×C5⋊D511yescharacteristic
233Q8—502no—
234D4—22no—
235D4—42no—
236D4—504no—
237D4—504no—
238D4—502no—
239D4—502no—
240D4—1002no—
241D4—1002no—
242D4—1002no—
243C2×C4—254no—
244C2×C4—251no—
245C2×C4—251no—
246C2×C4—252no—
247C2×C4—251no—
248C2×C4—502no—
249C2×C4—502no—
250C2×C4C22×C5⋊D512yes—
251C2×C4C22×C5⋊D511yescharacteristic
252C5#6592616yescomplemented
253C22—22no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—44no—
260C22—501no—
261C22—502no—
262C22—502no—
263C22—502no—
264C22—502no—
265C22—1004no—
266C22D4×C5⋊D512yes—
267C22C23×C5⋊D511yescharacteristic
268C4—21no—
269C4—22no—
270C4—252no—
271C4—252no—
272C4—501no—
273C4—502no—
274C2—22no—
275C2—42no—
276C2—41no—
277C2—1002no—
278C2—1001no—
279C2C2×D4×C5⋊D511yescharacteristic
280C2#1171751712yes—
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#120C2
hypercenter#120C5⋊D5
hypocenterC52#535
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C5⋊D5)C2semidirect
2. C2×#87902C2semidirect
3. C52×#120C2semidirect
4. #11717273C2semidirect
5. #11717260C2semidirect
6. #11717259C2semidirect
7. #11717257C2semidirect
8. #11717205C2semidirect
9. #11717204C2semidirect
10. #11717202C2semidirect
11. #11717197C2semidirect
15. C2×C52×D4C22semidirect
20. C2×C22⋊(C5⋊D5)C22semidirect
21. C2×C22⋊(C5⋊D5)C22semidirect
22. C2×C22⋊(C5⋊D5)C22semidirect
24. C22×#1039C22semidirect
26. C2×C4⋊(C5⋊D5)C22semidirect
27. C2×C4×C5⋊D5C22semidirect
28. C2×#6045C22semidirect
30. C52×C22⋊C4C22semidirect
32. C22⋊#1039C22semidirect
33. #87808C22semidirect
34. #87808C22semidirect
35. #87806C22semidirect
36. C4×#1039C22semidirect
58. C2×#1039D4semidirect
68. C5×#120D5semidirect
101. C2×C5×D4C2×D5semidirect
116. C5×C22⋊C4C2×D5semidirect
168. #120C5⋊D5semidirect
170. C52#535semidirect
205. C2×D4C2×C5⋊D5semidirect
215. C22⋊C4C2×C5⋊D5semidirect
252. C5#65926semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C5⋊D5)2
C2×#879022
C52×#1202
#117172732
#117172602
#117172592
#117172572
#117172052
#117172042
#117172022
#117171972
#659265