Finite Group Databasebeta

← #13105136

Subgroups — #13105136

Order 1600.

776 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 156 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×#6036C211yesmaximal, characteristic
2C2×#87872C211yesmaximal, characteristic, complemented
3C22⋊C4×C5⋊D5C211yesmaximal, characteristic
4C5×#4024C212yesmaximal, complemented
5#11717092C211yesmaximal, characteristic
6#11717082C212yesmaximal, complemented
7#11717018C211yesmaximal, characteristic, complemented
8#11717009C212yesmaximal, complemented
9#11717008C212yesmaximal, complemented
10#11717001C211yesmaximal, characteristic, complemented
11#11717001C211yesmaximal, characteristic, complemented
12C23×C5⋊D5C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C22×C5×Dic5C2212yes—
16C2×C5×Dic10C2212yescomplemented
17C2×#6039C2212yescomplemented
18C2×#6036—24no—
19C2×#6036C2211yescharacteristic
20C2×#6036C2211yescharacteristic
21C2×#6036C2212yes—
22C2×#6036C2211yescharacteristic
23C2×#6036C2211yescharacteristic
24#87872—24no—
25C52×C22⋊C4C2211yescharacteristic
26C22⋊#1039C2211yescharacteristic
27#87808C2211yescharacteristic
28#87808C2211yescharacteristic
29C5×C22⋊Dic5C2212yes—
30C5×C4⋊Dic5C2212yescomplemented
31C5×#683C2212yescomplemented
32C5×#683C2212yescomplemented
33#87781C2212yescomplemented
34#87780C2212yescomplemented
35#87780C2212yescomplemented
36#87779C2212yes—
37D5×C22⋊Q8—52nomaximal
38C23×C52C2311yescharacteristic
39C22×C5⋊D5—22no—
40C22×C5⋊D5—22no—
41C22×C5⋊D5Q814yescomplemented
42C22×C5⋊D5C2311yescharacteristic
43C22×C5⋊D5C2311yescharacteristic
44C2×C4×C52C2311yescharacteristic
45C2×C4×C52C2311yescharacteristic
46C2×#1039C2311yescharacteristic
47C2×#1039C2311yescharacteristic
48C4×C5⋊D5—22no—
49C4×C5⋊D5—22no—
50C2×C5×Dic5—24no—
51C2×C5×Dic5C2312yes—
52C2×C5×Dic5C2312yes—
53C2×C5×Dic5C2312yes—
54C2×C5×Dic5C2312yes—
55C5×Dic10—24no—
56#6039—24no—
57#6036—22no—
58#6036—22no—
59#6036—22no—
60#6036—24no—
61#6036D414yes—
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#4042—52no—
70D5×C4⋊C4—52no—
71D5×C4⋊C4—52no—
72D5×C4⋊C4—52no—
73D5×C22⋊C4—52no—
74D5×C22⋊C4—52no—
75D5×C22⋊C4—102no—
76#4024—52no—
77#4024D512yescomplemented
78C22×C52—21no—
79C22×C52—21no—
80C22×C52C2×Q812yescomplemented
81C22×C52C2411yescharacteristic
82C2×C5⋊D5—21no—
83C2×C5⋊D5—21no—
84C2×C5⋊D5—28no—
85C2×C5⋊D5C2×D412yes—
86C2×C5⋊D5C2×Q812yes—
87C2×C5⋊D5C2×Q812yescomplemented
88C2×C5⋊D5#4112yes—
89C4×C52—21no—
90C4×C52—21no—
91#1039—21no—
92#1039—21no—
93C5×Dic5—22no—
94C5×Dic5—22no—
95C5×Dic5—22no—
96C5×Dic5C2×D414yes—
97C23×D5—52no—
98C23×D5—102no—
99C2×C5×Q8—52no—
100C22×C4×C5—52no—
101C22×Dic5C2×D512yes—
102Q8×D5—108no—
103C2×C4×D5—52no—
104C2×C4×D5—52no—
105C2×C4×D5—52no—
106C2×C4×D5—52no—
107C2×C4×D5—52no—
108C2×C4×D5—52no—
109C2×C4×D5—52no—
110C2×C4×D5—52no—
111C2×C4×D5—102no—
112C2×C4×D5—102no—
113C2×C4×D5—108no—
114C2×Dic10—52no—
115C2×Dic10—52no—
116C2×Dic10C2×D512yescomplemented
117C5×C4⋊C4—52no—
118C5×C4⋊C4—52no—
119C5×C4⋊C4—52no—
120C5×C22⋊C4—22no—
121C5×C22⋊C4—52no—
122C5×C22⋊C4C2×D512yes—
123C22⋊Dic5—52no—
124C22⋊Dic5—102no—
125C22⋊Dic5C2×D512yes—
126#685—52no—
127#685—52no—
128#685—52no—
129#685—52no—
130#685—102no—
131#685—102no—
132C4⋊Dic5—52no—
133C4⋊Dic5—52no—
134C4⋊Dic5C2×D512yescomplemented
135#683—52no—
136#683—52no—
137#683—52no—
138#683—52no—
139#683C2×D512yescomplemented
140#683C2×D512yescomplemented
141C2×C22⋊Q8—251noSylow
142C2×C52—22no—
143C2×C52C22×D411yescharacteristic
144C2×C52C22×Q811yescharacteristic
145C2×C52C2×#4111yescharacteristic
146C5⋊D5—22no—
147C5⋊D5C22⋊Q814yescomplemented
148C23×C5—22no—
149C23×C5C22×D512yes—
150C22×D5—52no—
151C22×D5—58no—
152C22×D5—52no—
153C22×D5—102no—
154C22×D5—102no—
155C22×D5—108no—
156C22×D5—104no—
157C22×D5—104no—
158C22×D5—204no—
159C22×D5—204no—
160C5×Q8—104no—
161C2×C4×C5—22no—
162C2×C4×C5—22no—
163C2×C4×C5—52no—
164C2×C4×C5—52no—
165C2×C4×C5—52no—
166C2×C4×C5—52no—
167C2×C4×C5—104no—
168C2×C4×C5C22×D512yes—
169C2×C4×C5C22×D512yes—
170C2×Dic5—24no—
171C2×Dic5—52no—
172C2×Dic5—52no—
173C2×Dic5—102no—
174C2×Dic5—102no—
175C2×Dic5C22×D512yes—
176C2×Dic5C22×D512yes—
177C2×Dic5C22×D512yes—
178C2×Dic5C22×D512yes—
179C4×D5—58no—
180C4×D5—104no—
181C4×D5—104no—
182C4×D5—104no—
183C4×D5—104no—
184C4×D5—104no—
185C4×D5—104no—
186C4×D5—104no—
187C4×D5—204no—
188C4×D5—204no—
189Dic10—24no—
190Dic10—104no—
191Dic10—104no—
192C22×Q8—251no—
193C23×C4—251no—
194C22⋊Q8—52no—
195C22⋊Q8—252no—
196C22⋊Q8—252no—
197C22⋊Q8—252no—
198C2×C4⋊C4—251no—
199C2×C4⋊C4—251no—
200C2×C4⋊C4—251no—
201C2×C22⋊C4—251no—
202C2×C22⋊C4—251no—
203C52C2×C22⋊Q811yescharacteristic, complemented, Sylow
204C22×C5—22no—
205C22×C5—22no—
206C22×C5—24no—
207C22×C5—22no—
208C22×C5—42no—
209C22×C5—42no—
210C22×C5Q8×D514yescomplemented
211C22×C5C23×D512yes—
212C2×D5—54no—
213C2×D5—54no—
214C2×D5—54no—
215C2×D5—54no—
216C2×D5—104no—
217C2×D5—104no—
218C2×D5—104no—
219C2×D5—104no—
220C2×D5—1016no—
221C2×D5—102no—
222C2×D5—102no—
223C2×D5—2016no—
224C2×D5—202no—
225C2×D5—202no—
226C4×C5—22no—
227C4×C5—22no—
228C4×C5—42no—
229C4×C5—42no—
230C4×C5—54no—
231C4×C5—102no—
232C4×C5—102no—
233C4×C5—102no—
234Dic5—22no—
235Dic5—22no—
236Dic5—22no—
237Dic5—102no—
238Dic5—102no—
239Dic5—202no—
240Dic5—202no—
241Dic5D4×D514yes—
242C24—251no—
243C2×Q8—52no—
244C2×Q8—252no—
245C2×Q8—504no—
246C22×C4—52no—
247C22×C4—251no—
248C22×C4—251no—
249C22×C4—251no—
250C22×C4—251no—
251C22×C4—252no—
252C22×C4—251no—
253C22×C4—251no—
254C22×C4—504no—
255C4⋊C4—52no—
256C4⋊C4—52no—
257C4⋊C4—52no—
258C4⋊C4—252no—
259C4⋊C4—252no—
260C4⋊C4—252no—
261C22⋊C4—52no—
262C22⋊C4—252no—
263C22⋊C4—251no—
264C22⋊C4—251no—
265C22⋊C4—251no—
266C22⋊C4D5211yescharacteristic
267C2×C5—24no—
268C2×C5—22no—
269C2×C5—22no—
270C2×C5—22no—
271C2×C5—44no—
272C2×C5C2×D4×D512yes—
273C2×C5C2×Q8×D512yes—
274C2×C5D5×#4112yes—
275D5—58no—
276D5—108no—
277D5—104no—
278D5—204no—
279C23—251no—
280C23—254no—
281C23—251no—
282C23—502no—
283C23—502no—
284C23C2×D5211yescharacteristic
285Q8—104no—
286Q8—504no—
287C2×C4—52no—
288C2×C4—52no—
289C2×C4—52no—
290C2×C4—52no—
291C2×C4—104no—
292C2×C4—254no—
293C2×C4—251no—
294C2×C4—251no—
295C2×C4—502no—
296C2×C4—502no—
297C2×C4—504no—
298C2×C4—502no—
299C2×C4—502no—
300C2×C4—502no—
301C2×C4C2×D5211yescharacteristic
302C2×C4C2×D5211yescharacteristic
303C5—22no—
304C5D5×C22⋊Q812yescomplemented
305C22—21no—
306C22—21no—
307C22—252no—
308C22—252no—
309C22—252no—
310C22—252no—
311C22—508no—
312C22—501no—
313C22—501no—
314C22#8787212yescomplemented
315C22C22×D5211yescharacteristic
316C4—21no—
317C4—21no—
318C4—54no—
319C4—102no—
320C4—102no—
321C4—102no—
322C4—501no—
323C4—501no—
324C2—22no—
325C2—254no—
326C2—502no—
327C2C2×#8787211yescharacteristic
328C2D4×D5211yescharacteristic
329C2#1171747711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC23×C5⋊D5C22
hypercenterC22⋊C4D52
hypocenterC52C2×C22⋊Q8
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#87872C2semidirect
4. C5×#4024C2semidirect
6. #11717082C2semidirect
7. #11717018C2semidirect
8. #11717009C2semidirect
9. #11717008C2semidirect
10. #11717001C2semidirect
11. #11717001C2semidirect
16. C2×C5×Dic10C22semidirect
17. C2×#6039C22semidirect
30. C5×C4⋊Dic5C22semidirect
31. C5×#683C22semidirect
32. C5×#683C22semidirect
33. #87781C22semidirect
34. #87780C22semidirect
35. #87780C22semidirect
41. C22×C5⋊D5Q8semidirect
77. #4024D5semidirect
80. C22×C52C2×Q8semidirect
87. C2×C5⋊D5C2×Q8semidirect
116. C2×Dic10C2×D5semidirect
134. C4⋊Dic5C2×D5semidirect
139. #683C2×D5semidirect
140. #683C2×D5semidirect
147. C5⋊D5C22⋊Q8semidirect
203. C52C2×C22⋊Q8semidirect
210. C22×C5Q8×D5semidirect
304. C5D5×C22⋊Q8semidirect
314. C22#87872semidirect

Maximal subgroups

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

subgroupindex
C22×#60362
C2×#878722
C22⋊C4×C5⋊D52
C5×#40242
#117170922
#117170822
#117170182
#117170092
#117170082
#117170012
D5×C22⋊Q85