Finite Group Databasebeta

← #13105245

Subgroups — #13105245

Order 1600.

762 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 136 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×#87880C211yesmaximal, characteristic, complemented
2C2×#87877C211yesmaximal, characteristic, complemented
3#11717201C211yesmaximal, characteristic, complemented
4C5×#4030C212yesmaximal, complemented
5#11717101C212yesmaximal, complemented
6#11717098C211yesmaximal, characteristic, complemented
7D5×#685C212yesmaximal, complemented
8#11717033C211yesmaximal, characteristic, complemented
9C4×#6037C211yesmaximal, characteristic, complemented
10#11717002C212yesmaximal, complemented
11#11716984C211yesmaximal, characteristic, complemented
12C22×D52C2212yes—
13C2×C22⋊(C5⋊D5)C2211yescharacteristic
14C22×#1039C2211yescharacteristic, complemented
15C2×C4×C5⋊D5C2211yescharacteristic
16C2×C5×C22⋊D5C2212yescomplemented
17C2×C5×C22⋊D5C2212yescomplemented
18C2×C5×D20C2212yes—
19C2×C4×C5×D5C2212yescomplemented
20C2×#6039C2211yescharacteristic, complemented
21C2×#6037C2211yescharacteristic
22C2×#6037C2211yescharacteristic
23#87880—24no—
24C2×D5×Dic5C2212yescomplemented
25#87877—24no—
26C52×C22⋊C4C2211yescharacteristic, complemented
27#87808C2211yescharacteristic
28#87806C2211yescharacteristic, complemented
29C4×#1039C2211yescharacteristic, complemented
30C5×#685C2212yescomplemented
31C5×#683C2212yescomplemented
32#87781C2211yescharacteristic, complemented
33#87779C2212yes—
34#87778C2212yescomplemented
35#87778C2212yescomplemented
36#87778C2212yescomplemented
37#65944—52nomaximal
38C23×C52C2311yescharacteristic
39C22×C5⋊D5C2311yescharacteristic
40C22×C5×D5C2312yes—
41C22×C5×D5C2312yes—
42C2×D52—24no—
43C2×D52—24no—
44C2×D52—42no—
45C2×C4×C52C2311yescharacteristic
46C2×C4×C52C2311yescharacteristic
47C22⋊(C5⋊D5)—22no—
48C2×#1039—22no—
49C2×#1039C2311yescharacteristic
50C2×#1039C2311yescharacteristic
51C2×#1039C2311yescharacteristic
52C4×C5⋊D5—22no—
53C5×C22⋊D5—24no—
54C5×C22⋊D5—42no—
55C2×C5×Dic5C2312yes—
56C2×C5×Dic5C2312yes—
57C5×D20—24no—
58C4×C5×D5—42no—
59#6039—22no—
60#6037—22no—
61#6037D414yescomplemented
62D5×Dic5—24no—
63C2×#711—52no—
64C2×D4×D5—52no—
65C5×#121—52no—
66#4085—52no—
67#4083—52no—
68#4083—52no—
69#4081—52no—
70C4×C22⋊D5—52no—
71C2×#685—52no—
72#4042—52no—
73#4040—52no—
74#4031—52no—
75#4030D512yescomplemented
76#4027—52no—
77#4027—102no—
78D5×C22⋊C4—52no—
79C22×C52—21no—
80C22×C52—21no—
81C22×C52—21no—
82C22×C52C2411yescharacteristic
83C2×C5⋊D5—21no—
84C2×C5⋊D5—21no—
85C2×C5⋊D5#4112yescomplemented
86C2×C5×D5—22no—
87C2×C5×D5—22no—
88C2×C5×D5—22no—
89C2×C5×D5—22no—
90C2×C5×D5—22no—
91C2×C5×D5C2×D414yes—
92D52—48no—
93C4×C52—21no—
94C4×C52—21no—
95#1039—21no—
96#1039—21no—
97#1039C2×D412yes—
98C5×Dic5—22no—
99C5×Dic5—22no—
100C23×D5—52no—
101C23×D5—52no—
102C2×C5×D4—52no—
103C2×C5×D4—52no—
104C2×C5×D4—52no—
105C22×C4×C5—52no—
106C2×C22⋊D5—52no—
107C2×C22⋊D5—52no—
108C2×C22⋊D5—52no—
109C2×C22⋊D5—102no—
110C2×C22⋊D5C2×D512yescomplemented
111C2×C22⋊D5C2×D512yescomplemented
112C22×Dic5—52no—
113C22×Dic5—52no—
114C22×Dic5—102no—
115#711—108no—
116D4×D5—108no—
117C2×D20C2×D512yes—
118C2×C4×D5—52no—
119C2×C4×D5—52no—
120C2×C4×D5—102no—
121C2×C4×D5C2×D512yescomplemented
122C2×Dic10—52no—
123C5×C4⋊C4—52no—
124C5×C22⋊C4—22no—
125C5×C22⋊C4—52no—
126C5×C22⋊C4C2×D512yescomplemented
127C22⋊Dic5—52no—
128C22⋊Dic5—52no—
129C22⋊Dic5—52no—
130#685—52no—
131#685—52no—
132#685—52no—
133#685—52no—
134#685—52no—
135#685—52no—
136#685—102no—
137#685C2×D512yescomplemented
138#683—52no—
139#683—52no—
140#683—102no—
141#683C2×D512yescomplemented
142C4×Dic5—52no—
143C4×Dic5—102no—
144#546—251noSylow
145C2×C52—41no—
146C2×C52C22×D411yescharacteristic
147C2×C52C2×#4111yescharacteristic
148C2×C52H(5,𝔽2)11yescharacteristic
149C5⋊D5—22no—
150C5×D5—24no—
151C5×D5—42no—
152C23×C5—22no—
153C23×C5—52no—
154C23×C5—52no—
155C23×C5C22×D512yes—
156C22×D5—52no—
157C22×D5—102no—
158C22×D5—104no—
159C22×D5—104no—
160C22×D5—104no—
161C22×D5—104no—
162C22×D5—202no—
163C22×D5—202no—
164C22×D5C22×D512yes—
165C22×D5C22×D512yes—
166C5×D4—104no—
167C5×D4—104no—
168C5×D4—202no—
169C2×C4×C5—22no—
170C2×C4×C5—22no—
171C2×C4×C5—52no—
172C2×C4×C5—52no—
173C2×C4×C5—202no—
174C2×C4×C5C22×D512yes—
175C2×C4×C5C22×D512yes—
176C22⋊D5—24no—
177C22⋊D5—42no—
178C22⋊D5—58no—
179C22⋊D5—104no—
180C22⋊D5—104no—
181C22⋊D5—108no—
182C2×Dic5—52no—
183C2×Dic5—52no—
184C2×Dic5—52no—
185C2×Dic5—102no—
186C2×Dic5—102no—
187C2×Dic5—102no—
188C2×Dic5—104no—
189C2×Dic5—104no—
190C2×Dic5—204no—
191C2×Dic5C22×D512yes—
192C2×Dic5C22×D512yes—
193D20—24no—
194C4×D5—42no—
195C4×D5—104no—
196C4×D5—104no—
197C4×D5—204no—
198Dic10—104no—
199C2×#41—251no—
200C22×D4—251no—
201#124—251no—
202#123—252no—
203C22⋊Q8—251no—
204#121—52no—
205#121—251no—
206#120—252no—
207C4×D4—251no—
208C4×D4—251no—
209C2×C22⋊C4—252no—
210C52#54611yescharacteristic, complemented, Sylow
211C22×C5—24no—
212C22×C5—22no—
213C22×C5—22no—
214C22×C5—22no—
215C22×C5—22no—
216C22×C5—42no—
217C22×C5—42no—
218C22×C5—54no—
219C22×C5—102no—
220C22×C5—102no—
221C22×C5—102no—
222C22×C5—102no—
223C22×C5—102no—
224C22×C5C23×D512yes—
225C2×D5—22no—
226C2×D5—22no—
227C2×D5—22no—
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—54no—
231C2×D5—104no—
232C2×D5—102no—
233C2×D5—102no—
234C2×D5—202no—
235C2×D5—202no—
236C2×D5—208no—
237C2×D5—208no—
238C2×D5D4×D514yes—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—42no—
242C4×C5—42no—
243C4×C5—102no—
244C4×C5—102no—
245Dic5—22no—
246Dic5—22no—
247Dic5—54no—
248Dic5—104no—
249Dic5—104no—
250Dic5—102no—
251Dic5—102no—
252Dic5—202no—
253C24—252no—
254#41—504no—
255C2×Q8—251no—
256C2×D4—52no—
257C2×D4—52no—
258C2×D4—52no—
259C2×D4—251no—
260C2×D4—251no—
261C2×D4—251no—
262C2×D4—504no—
263C22×C4—52no—
264C22×C4—252no—
265C22×C4—251no—
266C22×C4—251no—
267C4⋊C4—52no—
268C4⋊C4—251no—
269C4⋊C4—251no—
270C22⋊C4—52no—
271C22⋊C4—252no—
272C22⋊C4—252no—
273C22⋊C4—252no—
274C22⋊C4—252no—
275C22⋊C4—251no—
276C22⋊C4D5211yescharacteristic, complemented
277C42—251no—
278C2×C5—22no—
279C2×C5—22no—
280C2×C5—22no—
281C2×C5—44no—
282C2×C5—42no—
283C2×C5—104no—
284C2×C5—202no—
285C2×C5C2×D4×D512yes—
286C2×C5#414412yes—
287C2×C5D5×#4112yes—
288D5—24no—
289D5—42no—
290D5—104no—
291D5—204no—
292C23—52no—
293C23—52no—
294C23—251no—
295C23—504no—
296C23—504no—
297C23—1002no—
298C23C2×D5211yescharacteristic
299Q8—502no—
300D4—104no—
301D4—104no—
302D4—202no—
303D4—254no—
304D4—502no—
305D4—502no—
306C2×C4—52no—
307C2×C4—52no—
308C2×C4—202no—
309C2×C4—251no—
310C2×C4—251no—
311C2×C4—251no—
312C2×C4—502no—
313C2×C4—504no—
314C2×C4—502no—
315C2×C4C2×D5211yescharacteristic
316C2×C4C2×D5211yescharacteristic
317C5—22no—
318C5#6594412yescomplemented
319C22—21no—
320C22—21no—
321C22—21no—
322C22—54no—
323C22—102no—
324C22—102no—
325C22—102no—
326C22—102no—
327C22—102no—
328C22—252no—
329C22—501no—
330C22—501no—
331C22—1008no—
332C22C22×D5211yescharacteristic
333C4—21no—
334C4—21no—
335C4—102no—
336C4—102no—
337C4—252no—
338C4—501no—
339C4—501no—
340C2—41no—
341C2—104no—
342C2—202no—
343C2—502no—
344C2#1171747111yescharacteristic
345C2D4×D5211yescharacteristic
346C2#1171747811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#87880C2semidirect
2. C2×#87877C2semidirect
3. #11717201C2semidirect
4. C5×#4030C2semidirect
5. #11717101C2semidirect
6. #11717098C2semidirect
7. D5×#685C2semidirect
8. #11717033C2semidirect
9. C4×#6037C2semidirect
10. #11717002C2semidirect
11. #11716984C2semidirect
14. C22×#1039C22semidirect
16. C2×C5×C22⋊D5C22semidirect
17. C2×C5×C22⋊D5C22semidirect
19. C2×C4×C5×D5C22semidirect
20. C2×#6039C22semidirect
24. C2×D5×Dic5C22semidirect
26. C52×C22⋊C4C22semidirect
28. #87806C22semidirect
29. C4×#1039C22semidirect
30. C5×#685C22semidirect
31. C5×#683C22semidirect
32. #87781C22semidirect
34. #87778C22semidirect
35. #87778C22semidirect
36. #87778C22semidirect
61. #6037D4semidirect
75. #4030D5semidirect
85. C2×C5⋊D5#41semidirect
110. C2×C22⋊D5C2×D5semidirect
111. C2×C22⋊D5C2×D5semidirect
121. C2×C4×D5C2×D5semidirect
126. C5×C22⋊C4C2×D5semidirect
137. #685C2×D5semidirect
141. #683C2×D5semidirect
210. C52#546semidirect
276. C22⋊C4D52semidirect
318. C5#65944semidirect

Maximal subgroups

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

subgroupindex
C2×#878802
C2×#878772
#117172012
C5×#40302
#117171012
#117170982
D5×#6852
#117170332
C4×#60372
#117170022
#117169842
#659445