Finite Group Databasebeta

← #13105012

Subgroups — #13105012

Order 1600.

576 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C42×C5⋊D5C211yesmaximal, characteristic, complemented
2C5×#4022C212yesmaximal, complemented
3#11717033C211yesmaximal, characteristic, complemented
4#11717030C211yesmaximal, characteristic
5#11717020C211yesmaximal, characteristic, complemented
6#11716999C212yesmaximal, complemented
7#11716996C212yesmaximal, complemented
8#11716989C211yesmaximal, characteristic, complemented
9#11716984C211yesmaximal, characteristic, complemented
10#11716971C212yesmaximal, complemented
11#11716956C211yesmaximal, characteristic, complemented
12C2×C4×C5⋊D5C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C2×#6039C2211yescharacteristic
16C2×#6038C2212yes—
17C2×#6037C2211yescharacteristic
18C2×D5×Dic5C2212yes—
19C42×C52C2211yescharacteristic
20C4×#1039C2211yescharacteristic
21C4×#1039C2211yescharacteristic
22C4×#1039C2211yescharacteristic
23C5×#685C2212yes—
24C5×#685C2212yes—
25C5×#685C2212yes—
26C5×#683C2212yes—
27C5×#683C2212yes—
28C5×#683C2212yes—
29#87781C2211yescharacteristic
30#87781C2211yescharacteristic
31#87780C2212yes—
32#87779C2212yes—
33#87778C2212yes—
34Dic52C2212yes—
35#66040—52nomaximal
36C22×C5⋊D5C2311yescharacteristic
37C22×C5×D5C2312yes—
38C2×C4×C52C2311yescharacteristic
39C2×C4×C52C2311yescharacteristic
40C2×C4×C52C2311yescharacteristic
41C2×#1039C2311yescharacteristic
42C2×#1039C2311yescharacteristic
43C2×#1039C2311yescharacteristic
44C4×C5⋊D5—22no—
45C4×C5⋊D5—22no—
46C4×C5⋊D5—22no—
47C2×C5×Dic5C2312yes—
48C2×C5×Dic5C2312yes—
49C2×C5×Dic5C2312yes—
50#6039—22no—
51#6038—24no—
52#6037—22no—
53D5×Dic5—24no—
54C5×#126—52no—
55#4042—52no—
56#4040—52no—
57#4039—52no—
58#4038—52no—
59#4035—52no—
60#4033—52no—
61#4031—52no—
62#4030—52no—
63#4029—52no—
64#4027—52no—
65#4027—52no—
66#4023—52no—
67#4022D512yescomplemented
68C42×D5—52no—
69C42×D5—102no—
70C22×C52C2411yescharacteristic
71C2×C5⋊D5—21no—
72C2×C5⋊D5—21no—
73C2×C5⋊D5#4112yes—
74C2×C5×D5—22no—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C4×C52—21no—
78C4×C52—21no—
79C4×C52—21no—
80#1039—21no—
81#1039#4112yes—
82#1039#4112yes—
83C5×Dic5—22no—
84C5×Dic5—22no—
85C5×Dic5—22no—
86C2×C22⋊D5—52no—
87C2×C22⋊D5—52no—
88C22×Dic5—52no—
89C2×D20—52no—
90C2×C4×D5—52no—
91C2×C4×D5—52no—
92C2×C4×D5—52no—
93C2×C4×D5—52no—
94C2×C4×D5—102no—
95C2×C4×D5—102no—
96C2×C4×D5—102no—
97C2×Dic10—52no—
98C5×C4⋊C4—52no—
99C5×C4⋊C4—52no—
100C5×C4⋊C4—52no—
101C5×C22⋊C4—52no—
102C5×C22⋊C4—52no—
103C5×C22⋊C4—52no—
104C42×C5—22no—
105C42×C5C2×D512yes—
106C22⋊Dic5—52no—
107#685—52no—
108#685—52no—
109#685—52no—
110#685C2×D512yes—
111#685C2×D512yes—
112#685C2×D512yes—
113C4⋊Dic5—52no—
114#683—52no—
115#683—52no—
116#683—52no—
117#683C2×D512yes—
118#683C2×D512yes—
119#683C2×D512yes—
120C4×Dic5—52no—
121C4×Dic5—52no—
122C4×Dic5—52no—
123C4×Dic5—52no—
124C4×Dic5—52no—
125C4×Dic5—102no—
126C4×Dic5—102no—
127C4×Dic5—102no—
128#529—251noSylow
129C2×C52C2×#4111yescharacteristic
130C2×C52C2×#4111yescharacteristic
131C2×C52C2×#4111yescharacteristic
132C5⋊D5—22no—
133C5×D5—42no—
134C23×C5—52no—
135C22×D5—52no—
136C22×D5—102no—
137C22×D5C22×D512yes—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5—22no—
141C2×C4×C5—52no—
142C2×C4×C5—52no—
143C2×C4×C5—52no—
144C2×C4×C5C22×D512yes—
145C2×C4×C5C22×D512yes—
146C2×C4×C5C22×D512yes—
147C22⋊D5—104no—
148C22⋊D5—104no—
149C2×Dic5—52no—
150C2×Dic5—52no—
151C2×Dic5—52no—
152C2×Dic5—102no—
153C2×Dic5—102no—
154C2×Dic5—102no—
155C2×Dic5—104no—
156C2×Dic5C22×D512yes—
157C2×Dic5C22×D512yes—
158C2×Dic5C22×D512yes—
159D20—104no—
160C4×D5—104no—
161C4×D5—104no—
162C4×D5—104no—
163C4×D5—104no—
164C4×D5—108no—
165C4×D5—108no—
166C4×D5—108no—
167Dic10—104no—
168#126—52no—
169#125—251no—
170#124—251no—
171#123—252no—
172C22⋊Q8—251no—
173#121—251no—
174C4×Q8—251no—
175C4×D4—251no—
176C4×D4—252no—
177#117—252no—
178C2×C42—251no—
179C52#52911yescharacteristic, complemented, Sylow
180C22×C5—22no—
181C22×C5—102no—
182C22×C5—102no—
183C22×C5—102no—
184C22×C5C23×D512yes—
185C2×D5—22no—
186C2×D5—22no—
187C2×D5—22no—
188C2×D5—54no—
189C2×D5—104no—
190C2×D5—104no—
191C2×D5—104no—
192C2×D5—102no—
193C2×D5—102no—
194C4×C5—24no—
195C4×C5—24no—
196C4×C5—24no—
197C4×C5—22no—
198C4×C5—22no—
199C4×C5—22no—
200C4×C5—102no—
201C4×C5—102no—
202C4×C5—102no—
203Dic5—22no—
204Dic5—22no—
205Dic5—22no—
206Dic5—54no—
207Dic5—54no—
208Dic5—104no—
209Dic5—104no—
210Dic5—104no—
211Dic5—102no—
212C2×Q8—251no—
213C2×D4—252no—
214C2×D4—251no—
215C22×C4—252no—
216C22×C4—251no—
217C22×C4—251no—
218C22×C4—251no—
219C4⋊C4—52no—
220C4⋊C4—52no—
221C4⋊C4—52no—
222C4⋊C4—252no—
223C4⋊C4—251no—
224C4⋊C4—251no—
225C22⋊C4—52no—
226C22⋊C4—52no—
227C22⋊C4—52no—
228C22⋊C4—252no—
229C22⋊C4—252no—
230C42—252no—
231C42—251no—
232C42—251no—
233C42—251no—
234C42D5211yescharacteristic
235C2×C5—22no—
236C2×C5—22no—
237C2×C5—22no—
238C2×C5—202no—
239C2×C5D5×#4112yes—
240C2×C5D5×#4112yes—
241C2×C5D5×#4112yes—
242D5—42no—
243D5—108no—
244D5—104no—
245C23—52no—
246C23—251no—
247Q8—502no—
248D4—504no—
249D4—502no—
250C2×C4—52no—
251C2×C4—52no—
252C2×C4—52no—
253C2×C4—251no—
254C2×C4—251no—
255C2×C4—251no—
256C2×C4—502no—
257C2×C4—502no—
258C2×C4—502no—
259C2×C4—504no—
260C2×C4C2×D5211yescharacteristic
261C2×C4C2×D5211yescharacteristic
262C2×C4C2×D5211yescharacteristic
263C5—22no—
264C5#6604012yescomplemented
265C22—102no—
266C22—102no—
267C22—102no—
268C22—252no—
269C22—501no—
270C22—501no—
271C22C22×D5211yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—102no—
276C4—102no—
277C4—102no—
278C4—252no—
279C4—252no—
280C4—501no—
281C2—202no—
282C2—502no—
283C2#1171747111yescharacteristic
284C2#1171747111yescharacteristic
285C2#1171747111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radical#11717030C2
hypercenterC42D52
hypocenterC52#529
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C42×C5⋊D5C2semidirect
2. C5×#4022C2semidirect
3. #11717033C2semidirect
5. #11717020C2semidirect
6. #11716999C2semidirect
7. #11716996C2semidirect
8. #11716989C2semidirect
9. #11716984C2semidirect
10. #11716971C2semidirect
11. #11716956C2semidirect
67. #4022D5semidirect
179. C52#529semidirect
264. C5#66040semidirect

Maximal subgroups

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

subgroupindex
C42×C5⋊D52
C5×#40222
#117170332
#117170302
#117170202
#117169992
#117169962
#117169892
#117169842
#117169712
#117169562
#660405