Finite Group Databasebeta

← #13105213

Subgroups — #13105213

Order 1600.

626 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 124 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#11717202C211yesmaximal, characteristic
2C5×#4025C212yesmaximal, complemented
3#11717093C211yesmaximal, characteristic, complemented
4#11717092C212yesmaximal, complemented
5#11717071C211yesmaximal, characteristic, complemented
6#11717016C212yesmaximal, complemented
7#11717013C212yesmaximal, complemented
8#11717009C212yesmaximal, complemented
9#11716954C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C2×C22⋊(C5⋊D5)C2211yescharacteristic
12C2×C4⋊(C5⋊D5)C2212yes—
13C2×#6039C2211yescharacteristic, complemented
14C2×#6036C2212yes—
15C2×#6036C2212yes—
16C2×#6036C2211yescharacteristic, complemented
17C52×C22⋊C4C2211yescharacteristic
18#87808C2212yes—
19C5×C22⋊Dic5C2212yes—
20C5×C22⋊Dic5C2212yes—
21C5×C4⋊Dic5C2212yescomplemented
22C5×#683C2212yescomplemented
23C5×#683C2212yescomplemented
24C4×C5×Dic5C2212yescomplemented
25#87780C2212yescomplemented
26#87779C2212yes—
27#87779C2212yes—
28#87779C2212yescomplemented
29#87779C2211yescharacteristic, complemented
30Dic52C2211yescharacteristic, complemented
31#66039—52nomaximal
32C23×C52C2311yescharacteristic
33C22×C5⋊D5—24no—
34C22×C5⋊D5—41no—
35C22×C5⋊D5C2311yescharacteristic
36C22×C5⋊D5C2312yes—
37C2×C4×C52C2312yes—
38C22⋊(C5⋊D5)—41no—
39C2×#1039C2311yescharacteristic
40C4⋊(C5⋊D5)—24no—
41C2×C5×Dic5C2312yes—
42C2×C5×Dic5C2312yes—
43C2×C5×Dic5C2312yes—
44C2×C5×Dic5C2312yes—
45#6039—41no—
46#6036—24no—
47#6036—42no—
48#6036—41no—
49C5×#126—52no—
50#4043—52no—
51#4042—52no—
52#4040—52no—
53#4040—52no—
54#4039—52no—
55#4038—52no—
56#4031—52no—
57#4029—52no—
58#4028—52no—
59#4028—102no—
60D5×C22⋊C4—52no—
61D5×C22⋊C4—52no—
62#4025D512yescomplemented
63C4×D20—52no—
64#4018—52no—
65C22×C52—22no—
66C22×C52—21no—
67C22×C52C2411yescharacteristic
68C2×C5⋊D5—22no—
69C2×C5⋊D5—22no—
70C2×C5⋊D5—22no—
71C2×C5⋊D5—21no—
72C2×C5⋊D5—44no—
73C2×C5⋊D5#4114yes—
74C4×C52—22no—
75#1039—21no—
76C5×Dic5—22no—
77C5×Dic5—22no—
78C5×Dic5—22no—
79C5×Dic5—22no—
80C23×D5—52no—
81C23×D5—102no—
82C2×C22⋊D5—52no—
83C2×C22⋊D5—102no—
84C2×D20—52no—
85C2×D20—52no—
86C2×D20—104no—
87C2×C4×D5—52no—
88C2×C4×D5—52no—
89C2×C4×D5—52no—
90C2×C4×D5—52no—
91C2×C4×D5—52no—
92C2×Dic10—52no—
93C5×C4⋊C4—52no—
94C5×C4⋊C4—52no—
95C5×C4⋊C4—52no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—52no—
98C5×C22⋊C4—52no—
99C5×C22⋊C4C2×D512yes—
100C42×C5—52no—
101C22⋊Dic5C2×D512yes—
102C22⋊Dic5C2×D512yes—
103#685—52no—
104#685—52no—
105#685—52no—
106#685—52no—
107#685—52no—
108#685—52no—
109#685—52no—
110#685—52no—
111#685—52no—
112#685—104no—
113C4⋊Dic5—52no—
114C4⋊Dic5C2×D512yescomplemented
115#683—52no—
116#683C2×D512yescomplemented
117#683C2×D512yescomplemented
118C4×Dic5—52no—
119C4×Dic5C2×D512yescomplemented
120#551—251noSylow
121C2×C52—41no—
122C2×C52C2×#4112yes—
123C2×C52H(5,𝔽2)11yescharacteristic
124C5⋊D5—24no—
125C5⋊D5—41no—
126C23×C5—22no—
127C23×C5C22×D512yes—
128C22×D5—52no—
129C22×D5—52no—
130C22×D5—52no—
131C22×D5—104no—
132C22×D5—102no—
133C22×D5—104no—
134C22×D5—104no—
135C22×D5—208no—
136C22×D5—202no—
137C22×D5—204no—
138C2×C4×C5—24no—
139C2×C4×C5—52no—
140C2×C4×C5—52no—
141C2×C4×C5—52no—
142C2×C4×C5—52no—
143C2×C4×C5C22×D512yes—
144C2×C4×C5C22×D512yes—
145C22⋊D5—202no—
146C22⋊D5—204no—
147C2×Dic5—52no—
148C2×Dic5—102no—
149C2×Dic5C22×D512yes—
150C2×Dic5C22×D512yes—
151C2×Dic5C22×D512yes—
152C2×Dic5C22×D512yes—
153D20—104no—
154D20—104no—
155D20—208no—
156C4×D5—104no—
157C4×D5—104no—
158C4×D5—202no—
159C4×D5—202no—
160C4×D5—202no—
161Dic10—202no—
162#126—52no—
163#124—251no—
164#123—251no—
165#123—252no—
166C22⋊Q8—252no—
167#120—251no—
168C4×D4—252no—
169#117—252no—
170C2×C22⋊C4—252no—
171C52#55111yescharacteristic, complemented, Sylow
172C22×C5—24no—
173C22×C5—22no—
174C22×C5—22no—
175C22×C5—22no—
176C22×C5—22no—
177C22×C5—44no—
178C22×C5C23×D512yes—
179C2×D5—54no—
180C2×D5—54no—
181C2×D5—108no—
182C2×D5—102no—
183C2×D5—102no—
184C2×D5—102no—
185C2×D5—102no—
186C2×D5—102no—
187C2×D5—102no—
188C2×D5—102no—
189C2×D5—204no—
190C2×D5—204no—
191C2×D5—204no—
192C2×D5—202no—
193C2×D5—208no—
194C2×D5—2016no—
195C4×C5—22no—
196C4×C5—22no—
197C4×C5—44no—
198C4×C5—102no—
199C4×C5—102no—
200C4×C5—102no—
201C4×C5—102no—
202Dic5—22no—
203Dic5—22no—
204Dic5—22no—
205Dic5—22no—
206Dic5—102no—
207Dic5—202no—
208C24—251no—
209C2×Q8—251no—
210C2×D4—251no—
211C2×D4—252no—
212C22×C4—252no—
213C22×C4—252no—
214C22×C4—251no—
215C4⋊C4—52no—
216C4⋊C4—52no—
217C4⋊C4—52no—
218C4⋊C4—252no—
219C22⋊C4—52no—
220C22⋊C4—52no—
221C22⋊C4—252no—
222C22⋊C4—252no—
223C22⋊C4—252no—
224C22⋊C4—251no—
225C22⋊C4—252no—
226C22⋊C4D5211yescharacteristic
227C42—52no—
228C42—251no—
229C2×C5—22no—
230C2×C5—24no—
231C2×C5—44no—
232C2×C5—42no—
233C2×C5D5×#4112yes—
234C2×C5D5×#4112yes—
235C2×C5#414912yes—
236D5—104no—
237D5—104no—
238D5—208no—
239D5—202no—
240D5—402no—
241C23—252no—
242C23—251no—
243C23—504no—
244C23—1001no—
245C23C2×D5211yescharacteristic
246Q8—1001no—
247D4—504no—
248D4—1001no—
249C2×C4—52no—
250C2×C4—52no—
251C2×C4—52no—
252C2×C4—52no—
253C2×C4—251no—
254C2×C4—504no—
255C2×C4—1001no—
256C2×C4—1002no—
257C2×C4C2×D5212yes—
258C5—22no—
259C5#6603912yescomplemented
260C22—21no—
261C22—22no—
262C22—254no—
263C22—502no—
264C22—502no—
265C22—501no—
266C22—502no—
267C22—1004no—
268C22C22×D5211yescharacteristic
269C4—22no—
270C4—102no—
271C4—102no—
272C4—102no—
273C4—102no—
274C4—501no—
275C2—41no—
276C2—504no—
277C2—1001no—
278C2#1171747311yescharacteristic
279C2#1171747712yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radical#11717202C2
hypercenterC22⋊C4D52
hypocenterC52#551
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#4025C2semidirect
3. #11717093C2semidirect
4. #11717092C2semidirect
5. #11717071C2semidirect
6. #11717016C2semidirect
7. #11717013C2semidirect
8. #11717009C2semidirect
9. #11716954C2semidirect
13. C2×#6039C22semidirect
16. C2×#6036C22semidirect
21. C5×C4⋊Dic5C22semidirect
22. C5×#683C22semidirect
23. C5×#683C22semidirect
24. C4×C5×Dic5C22semidirect
25. #87780C22semidirect
28. #87779C22semidirect
29. #87779C22semidirect
30. Dic52C22semidirect
62. #4025D5semidirect
114. C4⋊Dic5C2×D5semidirect
116. #683C2×D5semidirect
117. #683C2×D5semidirect
119. C4×Dic5C2×D5semidirect
171. C52#551semidirect
259. C5#66039semidirect

Maximal subgroups

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

subgroupindex
#117172022
C5×#40252
#117170932
#117170922
#117170712
#117170162
#117170132
#117170092
#117169542
#660395