Finite Group Databasebeta

← #13106104

Subgroups — #13106104

Order 1600.

896 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 164 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×C4×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×C5×#709C211yesmaximal, characteristic, complemented
3C5×#4072C211yesmaximal, characteristic, complemented
4#11717099C212yesmaximal, complemented
5#11717080C212yesmaximal, complemented
6#11717034C212yesmaximal, complemented
7C4×#6038C214yesmaximal, complemented
8#11716990C212yesmaximal, complemented
9C23×C5⋊D5C2211yescharacteristic
10C22×C4×C52C2211yescharacteristic
11C22×#1039C2211yescharacteristic
12C2×C4×C5⋊D5—24no—
13C2×C4×C5⋊D5C2212yes—
14C2×C4×C5⋊D5C2212yes—
15C2×C5×C22⋊D5C2212yes—
16C5×#709—24no—
17C2×C5×D20C2211yescharacteristic, complemented
18C2×C4×C5×D5C2212yescomplemented
19C2×C5×Dic10C2211yescharacteristic, complemented
20C2×#6038C2214yes—
21C5×C22⋊Dic5C2212yes—
22C5×C4⋊Dic5C2212yescomplemented
23C4×C5×Dic5C2212yescomplemented
24#87780C2214yescomplemented
25#87779C2214yes—
26#87778C2214yescomplemented
27#66162—51nomaximal
28#65858—51nomaximal
29C23×C52C2311yescharacteristic
30C22×C5⋊D5—22no—
31C22×C5⋊D5—24no—
32C22×C5⋊D5C2312yes—
33C22×C5×D5C2312yes—
34C2×C4×C52D414yescomplemented
35C2×C4×C52C2311yescharacteristic
36C2×C4×C52C2311yescharacteristic
37C2×#1039—22no—
38C2×#1039C2312yes—
39C4×C5⋊D5—24no—
40C4×C5⋊D5—24no—
41C5×C22⋊D5—24no—
42C2×C5×Dic5C2314yes—
43C2×C5×Dic5C2312yes—
44C5×D20—22no—
45C4×C5×D5—24no—
46C5×Dic10—22no—
47#6038—28no—
48C2×C5×#41—51no—
49C2×#709D511yescharacteristic, complemented
50C22×C4×D5—51no—
51C22×C4×D5—51no—
52C22×C4×D5—102no—
53C5×#117—51no—
54#4092—52no—
55#4084—52no—
56#4083—52no—
57#4081—52no—
58C4×C22⋊D5—54no—
59#4072D511yescharacteristic, complemented
60#4043—52no—
61#4041—52no—
62#4032—52no—
63#4028—52no—
64C4×D20—54no—
65C22×C52—22no—
66C22×C52C2×D412yes—
67C22×C52C2411yescharacteristic
68C2×C5⋊D5—28no—
69C2×C5⋊D5—22no—
70C2×C5⋊D5—22no—
71C2×C5⋊D5#4114yescomplemented
72C2×C5×D5—22no—
73C2×C5×D5—24no—
74C4×C52C2×D412yes—
75C4×C52C2×D412yes—
76#1039—22no—
77C5×Dic5—22no—
78C5×Dic5—24no—
79C23×D5—51no—
80C23×D5—51no—
81C23×D5—102no—
82C5×#41—104no—
83C2×C5×Q8—51no—
84C2×C5×D4—52no—
85C2×C5×D4—51no—
86C22×C4×C5—22no—
87C22×C4×C5—52no—
88C22×C4×C5C2×D511yescharacteristic
89C22×C4×C5C2×D511yescharacteristic
90C2×C22⋊D5—54no—
91C2×C22⋊D5C2×D512yes—
92C22×Dic5—51no—
93C22×Dic5—51no—
94C22×Dic5—102no—
95#709—24no—
96C2×D20—54no—
97C2×D20C2×D511yescharacteristic, complemented
98C2×C4×D5—52no—
99C2×C4×D5—52no—
100C2×C4×D5—52no—
101C2×C4×D5—52no—
102C2×C4×D5—104no—
103C2×C4×D5—104no—
104C2×C4×D5—1016no—
105C2×C4×D5—104no—
106C2×C4×D5—104no—
107C2×C4×D5C2×D512yescomplemented
108C2×Dic10C2×D511yescharacteristic, complemented
109C5×C4⋊C4—52no—
110C5×C22⋊C4—52no—
111C42×C5—52no—
112C22⋊Dic5—54no—
113C22⋊Dic5C2×D512yes—
114#685—54no—
115#685—54no—
116#685—54no—
117C4⋊Dic5—54no—
118C4⋊Dic5C2×D512yescomplemented
119#683—54no—
120C4×Dic5C2×D512yescomplemented
121#525—251noSylow
122C2×C52—22no—
123C2×C52C22×D411yescharacteristic
124C2×C52C2×#4112yes—
125C5⋊D5—24no—
126C5×D5—42no—
127C23×C5—22no—
128C23×C5—52no—
129C23×C5C22×D511yescharacteristic
130C23×C5C22×D511yescharacteristic
131C22×D5—52no—
132C22×D5—52no—
133C22×D5—104no—
134C22×D5—104no—
135C22×D5—104no—
136C22×D5—108no—
137C22×D5—1016no—
138C22×D5—102no—
139C22×D5—102no—
140C22×D5C22×D512yes—
141C5×Q8—102no—
142C5×D4—104no—
143C5×D4—102no—
144C2×C4×C5—28no—
145C2×C4×C5—22no—
146C2×C4×C5—22no—
147C2×C4×C5—52no—
148C2×C4×C5—54no—
149C2×C4×C5—104no—
150C2×C4×C5D2014yescomplemented
151C2×C4×C5C22⋊D514yescomplemented
152C2×C4×C5C22×D511yescharacteristic
153C2×C4×C5C22×D511yescharacteristic
154C2×C4×C5C22×D511yescharacteristic
155C2×C4×C5C22×D511yescharacteristic
156C22⋊D5—24no—
157C22⋊D5—108no—
158C2×Dic5—52no—
159C2×Dic5—52no—
160C2×Dic5—104no—
161C2×Dic5—108no—
162C2×Dic5—102no—
163C2×Dic5—102no—
164C2×Dic5C22×D514yes—
165C2×Dic5C22×D512yes—
166D20—22no—
167D20—108no—
168C4×D5—24no—
169C4×D5—1016no—
170C4×D5—1016no—
171C4×D5—104no—
172C4×D5—104no—
173C4×D5—104no—
174C4×D5—104no—
175Dic10—22no—
176C2×#41—51no—
177C23×C4—251no—
178#123—252no—
179C22⋊Q8—252no—
180#121—252no—
181#120—252no—
182C4×D4—254no—
183#117—51no—
184C52#52511yescharacteristic, complemented, Sylow
185C22×C5—28no—
186C22×C5—22no—
187C22×C5—22no—
188C22×C5—24no—
189C22×C5—22no—
190C22×C5—102no—
191C22×C5—104no—
192C22×C5C2×D2012yes—
193C22×C5C2×C22⋊D512yes—
194C22×C5C23×D511yescharacteristic
195C22×C5C23×D511yescharacteristic
196C2×D5—24no—
197C2×D5—22no—
198C2×D5—54no—
199C2×D5—54no—
200C2×D5—108no—
201C2×D5—108no—
202C2×D5—1032no—
203C2×D5—108no—
204C2×D5—108no—
205C2×D5—108no—
206C2×D5—102no—
207C2×D5—102no—
208C2×D5—102no—
209C2×D5—102no—
210C4×C5—24no—
211C4×C5—24no—
212C4×C5—102no—
213C4×C5—104no—
214C4×C5C2×D2012yes—
215C4×C5C2×D2012yes—
216C4×C5C2×C22⋊D512yes—
217C4×C5C2×C22⋊D512yes—
218Dic5—24no—
219Dic5—22no—
220Dic5—108no—
221Dic5—102no—
222Dic5—102no—
223C24—251no—
224#41—104no—
225C2×Q8—51no—
226C2×D4—52no—
227C2×D4—51no—
228C2×D4—254no—
229C22×C4—52no—
230C22×C4—251no—
231C22×C4—252no—
232C22×C4—252no—
233C22×C4—504no—
234C22×C4D5211yescharacteristic
235C4⋊C4—52no—
236C4⋊C4—254no—
237C22⋊C4—52no—
238C22⋊C4—254no—
239C22⋊C4—254no—
240C42—52no—
241C2×C5—28no—
242C2×C5—22no—
243C2×C5—22no—
244C2×C5—24no—
245C2×C5—22no—
246C2×C5—202no—
247C2×C5C22×D2011yescharacteristic
248C2×C5D5×#4112yes—
249C2×C5D5×#4112yes—
250C2×C5C22×C22⋊D511yescharacteristic
251D5—42no—
252D5—1016no—
253D5—104no—
254D5—104no—
255C23—52no—
256C23—252no—
257C23—504no—
258C23—502no—
259C23C2×D5211yescharacteristic
260Q8—102no—
261D4—104no—
262D4—102no—
263D4—508no—
264C2×C4—54no—
265C2×C4—52no—
266C2×C4—104no—
267C2×C4—252no—
268C2×C4—502no—
269C2×C4—504no—
270C2×C4—504no—
271C2×C4#603814yescomplemented
272C2×C4C2×D5211yescharacteristic
273C2×C4C2×D5211yescharacteristic
274C5—22no—
275C5#6585811yescharacteristic, complemented
276C5#6616211yescharacteristic, complemented
277C22—22no—
278C22—104no—
279C22—102no—
280C22—254no—
281C22—508no—
282C22—502no—
283C22—502no—
284C22C2×#603812yes—
285C22C22×D5211yescharacteristic
286C4—104no—
287C4—102no—
288C4—502no—
289C4C2×#603812yes—
290C4C2×#603812yes—
291C2—22no—
292C2—202no—
293C2—504no—
294C2#1171747112yes—
295C2C22×#603811yescharacteristic
subgroupG/H
centerC2×C4C2×D52
commutator, socle, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC22×C4×C52C22
hypercenterC22×C4D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C5⋊D5C2semidirect
2. C2×C5×#709C2semidirect
3. C5×#4072C2semidirect
4. #11717099C2semidirect
5. #11717080C2semidirect
6. #11717034C2semidirect
7. C4×#6038C2semidirect
8. #11716990C2semidirect
17. C2×C5×D20C22semidirect
18. C2×C4×C5×D5C22semidirect
19. C2×C5×Dic10C22semidirect
22. C5×C4⋊Dic5C22semidirect
23. C4×C5×Dic5C22semidirect
24. #87780C22semidirect
26. #87778C22semidirect
34. C2×C4×C52D4semidirect
49. C2×#709D5semidirect
59. #4072D5semidirect
71. C2×C5⋊D5#41semidirect
97. C2×D20C2×D5semidirect
107. C2×C4×D5C2×D5semidirect
108. C2×Dic10C2×D5semidirect
118. C4⋊Dic5C2×D5semidirect
120. C4×Dic5C2×D5semidirect
150. C2×C4×C5D20semidirect
151. C2×C4×C5C22⋊D5semidirect
184. C52#525semidirect
271. C2×C4#6038semidirect
275. C5#65858semidirect
276. C5#66162semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5⋊D52
C2×C5×#7092
C5×#40722
#117170992
#117170802
#117170342
C4×#60382
#117169902
#661625
#658585