Finite Group Databasebeta

← #13105426

Subgroups — #13105426

Order 1600.

736 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C2×C4×D52C211yesmaximal, characteristic
2C2×#87872C211yesmaximal, characteristic, complemented
3C4⋊C4×C5⋊D5C211yesmaximal, characteristic, complemented
4C5×#4042C212yesmaximal, complemented
5#11717039C211yesmaximal, characteristic
6#11717039C211yesmaximal, characteristic
7#11717007C212yesmaximal, complemented
8#11717006C212yesmaximal, complemented
9#11717001C211yesmaximal, characteristic, complemented
10#11717001C211yesmaximal, characteristic, complemented
11#11716969C212yesmaximal, complemented
12C22×D52C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C2×C4×C5⋊D5C2211yescharacteristic
16C2×C4×C5×D5C2212yes—
17C2×C5×Dic10C2212yescomplemented
18C2×#6039C2212yescomplemented
19C2×#6036C2211yescharacteristic
20C2×#6036C2211yescharacteristic
21C2×#6036C2211yescharacteristic
22C2×D5×Dic5C2212yes—
23C4×D52—24no—
24#87872—24no—
25C52×C4⋊C4C2211yescharacteristic, complemented
26C4⋊#1039C2211yescharacteristic, complemented
27#87806C2211yescharacteristic, complemented
28#87806C2211yescharacteristic, complemented
29C5×#685C2212yes—
30C5×#685C2212yes—
31C5×#683C2212yescomplemented
32C5×#683C2212yescomplemented
33#87780C2212yescomplemented
34#87780C2212yescomplemented
35#87778C2212yes—
36#87778C2212yes—
37D5×C22⋊Q8—52nomaximal
38C22×C5⋊D5C2311yescharacteristic
39C22×C5×D5C2312yes—
40C2×D52—22no—
41C2×D52—22no—
42C2×D52Q814yescomplemented
43C2×C4×C52C2311yescharacteristic
44C2×C4×C52C2311yescharacteristic
45C2×C4×C52C2311yescharacteristic
46C2×#1039C2311yescharacteristic
47C2×#1039C2311yescharacteristic
48C2×#1039C2311yescharacteristic
49C4×C5⋊D5—22no—
50C4×C5⋊D5—22no—
51C4×C5⋊D5—22no—
52C2×C5×Dic5C2312yes—
53C2×C5×Dic5C2312yes—
54C2×C5×Dic5C2312yes—
55C4×C5×D5—24no—
56C5×Dic10—24no—
57#6039—24no—
58#6036—22no—
59#6036—22no—
60#6036D414yes—
61D5×Dic5—24no—
62C2×Q8×D5—52no—
63C22×C4×D5—52no—
64C5×C22⋊Q8—52no—
65#4092—52no—
66#4070—52no—
67#4043—52no—
68#4042—52no—
69#4042D512yescomplemented
70D5×C4⋊C4—52no—
71D5×C4⋊C4—52no—
72D5×C4⋊C4—52no—
73D5×C4⋊C4—102no—
74D5×C22⋊C4—52no—
75D5×C22⋊C4—52no—
76#4024—52no—
77#4024—52no—
78C22×C52C2411yescharacteristic
79C2×C5⋊D5C2×D412yes—
80C2×C5⋊D5C2×Q812yes—
81C2×C5⋊D5#4112yes—
82C2×C5×D5—22no—
83C2×C5×D5—22no—
84C2×C5×D5C2×Q814yescomplemented
85D52—28no—
86C4×C52—21no—
87C4×C52—21no—
88C4×C52—21no—
89#1039—21no—
90#1039—21no—
91#1039—21no—
92C5×Dic5—22no—
93C5×Dic5—22no—
94C5×Dic5C2×D414yes—
95C23×D5—52no—
96C2×C5×Q8—52no—
97C22×C4×C5—52no—
98C22×Dic5—52no—
99Q8×D5—108no—
100C2×C4×D5—52no—
101C2×C4×D5—52no—
102C2×C4×D5—52no—
103C2×C4×D5—52no—
104C2×C4×D5—52no—
105C2×C4×D5—52no—
106C2×C4×D5—52no—
107C2×C4×D5—102no—
108C2×C4×D5—102no—
109C2×C4×D5—102no—
110C2×C4×D5—108no—
111C2×C4×D5C2×D512yes—
112C2×Dic10—52no—
113C2×Dic10—52no—
114C2×Dic10C2×D512yescomplemented
115C5×C4⋊C4—22no—
116C5×C4⋊C4—52no—
117C5×C4⋊C4—52no—
118C5×C4⋊C4C2×D512yescomplemented
119C5×C22⋊C4—52no—
120C5×C22⋊C4—52no—
121C22⋊Dic5—52no—
122C22⋊Dic5—52no—
123#685—52no—
124#685—52no—
125#685C2×D512yes—
126#685C2×D512yes—
127C4⋊Dic5—52no—
128C4⋊Dic5—52no—
129C4⋊Dic5—52no—
130C4⋊Dic5—102no—
131#683—52no—
132#683—52no—
133#683—52no—
134#683—52no—
135#683—102no—
136#683—102no—
137#683C2×D512yescomplemented
138#683C2×D512yescomplemented
139C2×C22⋊Q8—251noSylow
140C2×C52C22×D411yescharacteristic
141C2×C52C22×Q811yescharacteristic
142C2×C52C2×#4111yescharacteristic
143C5⋊D5C22⋊Q814yescomplemented
144C5×D5—24no—
145C23×C5—52no—
146C22×D5—52no—
147C22×D5—58no—
148C22×D5—102no—
149C22×D5—104no—
150C22×D5—104no—
151C22×D5C22×D512yes—
152C5×Q8—104no—
153C2×C4×C5—22no—
154C2×C4×C5—22no—
155C2×C4×C5—22no—
156C2×C4×C5—52no—
157C2×C4×C5—52no—
158C2×C4×C5—52no—
159C2×C4×C5—104no—
160C2×C4×C5C22×D512yes—
161C2×C4×C5C22×D512yes—
162C2×C4×C5C22×D512yes—
163C2×Dic5—52no—
164C2×Dic5—52no—
165C2×Dic5—52no—
166C2×Dic5—102no—
167C2×Dic5—102no—
168C2×Dic5—102no—
169C2×Dic5—104no—
170C2×Dic5C22×D512yes—
171C2×Dic5C22×D512yes—
172C2×Dic5C22×D512yes—
173C4×D5—24no—
174C4×D5—58no—
175C4×D5—104no—
176C4×D5—104no—
177C4×D5—104no—
178C4×D5—104no—
179C4×D5—104no—
180C4×D5—108no—
181C4×D5—104no—
182C4×D5—204no—
183C4×D5—204no—
184Dic10—24no—
185Dic10—104no—
186Dic10—104no—
187C22×Q8—251no—
188C23×C4—251no—
189C22⋊Q8—52no—
190C22⋊Q8—252no—
191C22⋊Q8—252no—
192C22⋊Q8—252no—
193C2×C4⋊C4—251no—
194C2×C4⋊C4—251no—
195C2×C4⋊C4—251no—
196C2×C22⋊C4—251no—
197C2×C22⋊C4—251no—
198C52C2×C22⋊Q811yescharacteristic, complemented, Sylow
199C22×C5—22no—
200C22×C5—54no—
201C22×C5—102no—
202C22×C5—102no—
203C22×C5C23×D512yes—
204C2×D5—22no—
205C2×D5—22no—
206C2×D5—54no—
207C2×D5—54no—
208C2×D5—54no—
209C2×D5—104no—
210C2×D5—104no—
211C2×D5—104no—
212C2×D5—1016no—
213C2×D5Q8×D514yescomplemented
214C4×C5—24no—
215C4×C5—22no—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5—42no—
219C4×C5—42no—
220C4×C5—54no—
221C4×C5—102no—
222C4×C5—102no—
223Dic5—22no—
224Dic5—22no—
225Dic5—104no—
226Dic5—102no—
227Dic5—102no—
228Dic5—102no—
229Dic5—202no—
230Dic5—202no—
231Dic5D4×D514yes—
232C24—251no—
233C2×Q8—52no—
234C2×Q8—252no—
235C2×Q8—504no—
236C22×C4—52no—
237C22×C4—251no—
238C22×C4—251no—
239C22×C4—251no—
240C22×C4—252no—
241C22×C4—251no—
242C22×C4—251no—
243C22×C4—251no—
244C22×C4—504no—
245C4⋊C4—52no—
246C4⋊C4—52no—
247C4⋊C4—252no—
248C4⋊C4—252no—
249C4⋊C4—251no—
250C4⋊C4—251no—
251C4⋊C4—251no—
252C4⋊C4D5211yescharacteristic, complemented
253C22⋊C4—52no—
254C22⋊C4—52no—
255C22⋊C4—252no—
256C22⋊C4—252no—
257C2×C5—22no—
258C2×C5—22no—
259C2×C5—22no—
260C2×C5—104no—
261C2×C5C2×D4×D512yes—
262C2×C5C2×Q8×D512yes—
263C2×C5D5×#4112yes—
264D5—24no—
265D5—58no—
266D5—108no—
267C23—52no—
268C23—254no—
269C23—251no—
270C23—502no—
271C23—502no—
272Q8—104no—
273Q8—504no—
274C2×C4—52no—
275C2×C4—52no—
276C2×C4—52no—
277C2×C4—104no—
278C2×C4—254no—
279C2×C4—251no—
280C2×C4—251no—
281C2×C4—251no—
282C2×C4—502no—
283C2×C4—502no—
284C2×C4—502no—
285C2×C4—504no—
286C2×C4—502no—
287C2×C4—502no—
288C2×C4C2×D5211yescharacteristic
289C2×C4C2×D5211yescharacteristic
290C2×C4C2×D5211yescharacteristic
291C5—22no—
292C5D5×C22⋊Q812yescomplemented
293C22—54no—
294C22—102no—
295C22—102no—
296C22—252no—
297C22—252no—
298C22—252no—
299C22—508no—
300C22C22×D5211yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—54no—
305C4—102no—
306C4—102no—
307C4—501no—
308C4—501no—
309C4—501no—
310C2—104no—
311C2—254no—
312C2#1171747111yescharacteristic
313C2D4×D5211yescharacteristic
314C2Q8×D5211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C4⋊C4C22
prime radicalC22×D52C22
hypercenterC4⋊C4D52
hypocenterC52C2×C22⋊Q8
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#87872C2semidirect
3. C4⋊C4×C5⋊D5C2semidirect
4. C5×#4042C2semidirect
7. #11717007C2semidirect
8. #11717006C2semidirect
9. #11717001C2semidirect
10. #11717001C2semidirect
11. #11716969C2semidirect
17. C2×C5×Dic10C22semidirect
18. C2×#6039C22semidirect
25. C52×C4⋊C4C22semidirect
26. C4⋊#1039C22semidirect
27. #87806C22semidirect
28. #87806C22semidirect
31. C5×#683C22semidirect
32. C5×#683C22semidirect
33. #87780C22semidirect
34. #87780C22semidirect
42. C2×D52Q8semidirect
69. #4042D5semidirect
84. C2×C5×D5C2×Q8semidirect
114. C2×Dic10C2×D5semidirect
118. C5×C4⋊C4C2×D5semidirect
137. #683C2×D5semidirect
138. #683C2×D5semidirect
143. C5⋊D5C22⋊Q8semidirect
198. C52C2×C22⋊Q8semidirect
213. C2×D5Q8×D5semidirect
252. C4⋊C4D52semidirect
292. C5D5×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D522
C2×#878722
C4⋊C4×C5⋊D52
C5×#40422
#117170392
#117170072
#117170062
#117170012
#117169692
D5×C22⋊Q85