Finite Group Databasebeta

← #13106083

Subgroups — #13106083

Order 1600.

896 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 146 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×#87873C211yesmaximal, characteristic, complemented
3C4×C5×C22⋊D5C212yesmaximal, complemented
4#11717100C211yesmaximal, characteristic, complemented
5#11717099C212yesmaximal, complemented
6#11717098C211yesmaximal, characteristic, complemented
7#11717030C211yesmaximal, characteristic, complemented
8C4×#6038C212yesmaximal, complemented
9#11716992C211yesmaximal, characteristic, complemented
10#11716984C211yesmaximal, characteristic, complemented
11#11716971C212yesmaximal, complemented
12C23×C5⋊D5C2211yescharacteristic
13C22×C4×C52C2211yescharacteristic
14C22×#1039C2211yescharacteristic
15C2×C4×C5⋊D5—24no—
16C2×C4×C5⋊D5C2211yescharacteristic
17C2×C4×C5⋊D5C2211yescharacteristic
18C2×C4×C5⋊D5C2211yescharacteristic
19C2×C4×C5⋊D5C2211yescharacteristic
20C2×C5×C22⋊D5C2212yes—
21C2×C4×C5×D5C2212yescomplemented
22C2×#6039C2211yescharacteristic, complemented
23C2×#6038C2212yes—
24C2×#6038C2212yes—
25C2×#6037C2211yescharacteristic, complemented
26#87873—24no—
27C5×C22⋊Dic5C2212yes—
28C5×#685C2212yescomplemented
29C5×#683C2212yescomplemented
30C4×C5×Dic5C2212yescomplemented
31#87781C2211yescharacteristic, complemented
32#87781C2211yescharacteristic, complemented
33#87780C2212yescomplemented
34#87779C2212yes—
35#87779C2212yes—
36#87778C2212yescomplemented
37#65881—52nomaximal
38C23×C52C2311yescharacteristic
39C22×C5⋊D5—22no—
40C22×C5⋊D5—22no—
41C22×C5⋊D5—22no—
42C22×C5⋊D5C2311yescharacteristic
43C22×C5⋊D5C2311yescharacteristic
44C22×C5×D5C2312yes—
45C2×C4×C52—22no—
46C2×C4×C52C2311yescharacteristic
47C2×C4×C52C2311yescharacteristic
48C2×#1039—22no—
49C2×#1039C2311yescharacteristic
50C2×#1039C2311yescharacteristic
51C4×C5⋊D5—22no—
52C4×C5⋊D5—22no—
53C4×C5⋊D5—22no—
54C4×C5⋊D5D414yescomplemented
55C5×C22⋊D5—24no—
56C2×C5×Dic5C2312yes—
57C2×C5×Dic5C2312yes—
58C2×C5×Dic5C2312yes—
59C4×C5×D5—24no—
60#6039—22no—
61#6038—24no—
62#6038—24no—
63#6037—22no—
64C2×#709—52no—
65C22×C4×D5—52no—
66C22×C4×D5—102no—
67C4×C5×D4—52no—
68#4085—52no—
69#4083—52no—
70C4×C22⋊D5—52no—
71C4×C22⋊D5D512yescomplemented
72#4042—52no—
73#4040—52no—
74#4030—52no—
75#4029—52no—
76#4028—52no—
77#4024—52no—
78C4×D20—52no—
79#4018—52no—
80C22×C52—21no—
81C22×C52—21no—
82C22×C52#4112yescomplemented
83C22×C52C2411yescharacteristic
84C2×C5⋊D5—28no—
85C2×C5⋊D5—21no—
86C2×C5⋊D5—21no—
87C2×C5⋊D5—21no—
88C2×C5⋊D5—21no—
89C2×C5⋊D5C2×D412yes—
90C2×C5⋊D5#4112yescomplemented
91C2×C5×D5—22no—
92C2×C5×D5—22no—
93C2×C5×D5—22no—
94C4×C52—21no—
95C4×C52C2×D412yes—
96#1039—21no—
97#1039C2×D412yes—
98C5×Dic5—22no—
99C5×Dic5—22no—
100C5×Dic5—22no—
101C23×D5—52no—
102C23×D5—102no—
103C2×C5×D4—52no—
104C22×C4×C5—22no—
105C22×C4×C5—52no—
106C22×C4×C5C2×D512yes—
107C2×C22⋊D5—52no—
108C2×C22⋊D5—52no—
109C2×C22⋊D5—52no—
110C2×C22⋊D5C2×D512yes—
111C22×Dic5—52no—
112C22×Dic5—102no—
113#709—108no—
114C2×D20—52no—
115C2×D20—52no—
116C2×C4×D5—52no—
117C2×C4×D5—52no—
118C2×C4×D5—52no—
119C2×C4×D5—52no—
120C2×C4×D5—102no—
121C2×C4×D5—102no—
122C2×C4×D5—102no—
123C2×C4×D5—102no—
124C2×C4×D5—108no—
125C2×C4×D5—1016no—
126C2×C4×D5C2×D512yescomplemented
127C2×Dic10—52no—
128C5×C4⋊C4—52no—
129C5×C22⋊C4—52no—
130C5×C22⋊C4—52no—
131C42×C5—52no—
132C22⋊Dic5—52no—
133C22⋊Dic5C2×D512yes—
134#685—52no—
135#685—52no—
136#685—52no—
137#685—52no—
138#685—52no—
139#685C2×D512yescomplemented
140C4⋊Dic5—52no—
141#683—52no—
142#683—52no—
143#683—52no—
144#683C2×D512yescomplemented
145C4×Dic5C2×D512yescomplemented
146#525—251noSylow
147C2×C52—22no—
148C2×C52C22×D411yescharacteristic
149C2×C52C2×#4111yescharacteristic
150C2×C52C2×#4111yescharacteristic
151C5⋊D5—22no—
152C5⋊D5—22no—
153C5×D5—42no—
154C23×C5—22no—
155C23×C5—52no—
156C23×C5C22×D512yes—
157C22×D5—52no—
158C22×D5—52no—
159C22×D5—102no—
160C22×D5—102no—
161C22×D5—104no—
162C22×D5—104no—
163C22×D5—108no—
164C22×D5—108no—
165C22×D5—108no—
166C22×D5—104no—
167C22×D5C22×D512yes—
168C5×D4—104no—
169C2×C4×C5—28no—
170C2×C4×C5—24no—
171C2×C4×C5—22no—
172C2×C4×C5—22no—
173C2×C4×C5—52no—
174C2×C4×C5—52no—
175C2×C4×C5—52no—
176C2×C4×C5—104no—
177C2×C4×C5C22×D512yes—
178C2×C4×C5C22×D512yes—
179C22⋊D5—24no—
180C22⋊D5—104no—
181C22⋊D5—104no—
182C22⋊D5—104no—
183C2×Dic5—52no—
184C2×Dic5—52no—
185C2×Dic5—102no—
186C2×Dic5—102no—
187C2×Dic5—104no—
188C2×Dic5—108no—
189C2×Dic5C22×D512yes—
190C2×Dic5C22×D512yes—
191C2×Dic5C22×D512yes—
192D20—104no—
193D20—104no—
194C4×D5—24no—
195C4×D5—58no—
196C4×D5—108no—
197C4×D5—104no—
198C4×D5—104no—
199C4×D5—108no—
200C4×D5—108no—
201C4×D5—108no—
202C4×D5—104no—
203Dic10—104no—
204C2×#41—251no—
205C23×C4—251no—
206#123—252no—
207C22⋊Q8—251no—
208C22⋊Q8—251no—
209#121—251no—
210#121—251no—
211#120—252no—
212C4×D4—52no—
213C4×D4—252no—
214#117—251no—
215C52#52511yescharacteristic, complemented, Sylow
216C22×C5—24no—
217C22×C5—24no—
218C22×C5—22no—
219C22×C5—22no—
220C22×C5—24no—
221C22×C5—22no—
222C22×C5—102no—
223C22×C5—102no—
224C22×C5—102no—
225C22×C5#70914yescomplemented
226C22×C5C23×D512yes—
227C2×D5—22no—
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—54no—
231C2×D5—54no—
232C2×D5—104no—
233C2×D5—104no—
234C2×D5—104no—
235C2×D5—1032no—
236C2×D5—1016no—
237C2×D5—104no—
238C2×D5—104no—
239C2×D5—104no—
240C2×D5—102no—
241C2×D5—102no—
242C2×D5—102no—
243C2×D5—102no—
244C4×C5—24no—
245C4×C5—22no—
246C4×C5—24no—
247C4×C5—102no—
248C4×C5—102no—
249C4×C5—102no—
250C4×C5D4×D514yes—
251Dic5—22no—
252Dic5—22no—
253Dic5—22no—
254Dic5—54no—
255Dic5—104no—
256Dic5—104no—
257Dic5—102no—
258C24—251no—
259#41—504no—
260C2×Q8—251no—
261C2×D4—52no—
262C2×D4—252no—
263C2×D4—252no—
264C2×D4—251no—
265C22×C4—52no—
266C22×C4—251no—
267C22×C4—251no—
268C22×C4—251no—
269C22×C4—251no—
270C22×C4—251no—
271C22×C4—504no—
272C22×C4D5211yescharacteristic
273C4⋊C4—52no—
274C4⋊C4—251no—
275C4⋊C4—251no—
276C4⋊C4—252no—
277C22⋊C4—52no—
278C22⋊C4—52no—
279C22⋊C4—252no—
280C22⋊C4—252no—
281C22⋊C4—252no—
282C42—52no—
283C2×C5—28no—
284C2×C5—24no—
285C2×C5—22no—
286C2×C5—22no—
287C2×C5—22no—
288C2×C5—202no—
289C2×C5C2×#70912yes—
290C2×C5C2×D4×D512yes—
291C2×C5D5×#4112yes—
292D5—42no—
293D5—108no—
294D5—108no—
295D5—104no—
296D5—104no—
297C23—52no—
298C23—251no—
299C23—251no—
300C23—502no—
301C23—502no—
302C23—502no—
303C23C2×D5211yescharacteristic
304Q8—502no—
305D4—104no—
306D4—504no—
307D4—504no—
308D4—502no—
309C2×C4—22no—
310C2×C4—52no—
311C2×C4—52no—
312C2×C4—52no—
313C2×C4—104no—
314C2×C4—251no—
315C2×C4—254no—
316C2×C4—251no—
317C2×C4—502no—
318C2×C4—502no—
319C2×C4—502no—
320C2×C4—502no—
321C2×C4C2×D5211yescharacteristic
322C2×C4C2×D5211yescharacteristic
323C5—22no—
324C5#6588112yescomplemented
325C22—21no—
326C22—21no—
327C22—102no—
328C22—102no—
329C22—102no—
330C22—252no—
331C22—252no—
332C22—508no—
333C22—501no—
334C22—501no—
335C22—501no—
336C22—501no—
337C22#8787312yescomplemented
338C22C22×D5211yescharacteristic
339C4—21no—
340C4—102no—
341C4—102no—
342C4—102no—
343C4—252no—
344C4—501no—
345C4#8788612yes—
346C2—22no—
347C2—202no—
348C2—502no—
349C2—502no—
350C2C2×#8787311yescharacteristic
351C2#1171747111yescharacteristic
352C2C2×#8788611yescharacteristic
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×#87873C2semidirect
3. C4×C5×C22⋊D5C2semidirect
4. #11717100C2semidirect
5. #11717099C2semidirect
6. #11717098C2semidirect
7. #11717030C2semidirect
8. C4×#6038C2semidirect
9. #11716992C2semidirect
10. #11716984C2semidirect
11. #11716971C2semidirect
21. C2×C4×C5×D5C22semidirect
22. C2×#6039C22semidirect
25. C2×#6037C22semidirect
28. C5×#685C22semidirect
29. C5×#683C22semidirect
30. C4×C5×Dic5C22semidirect
31. #87781C22semidirect
32. #87781C22semidirect
33. #87780C22semidirect
36. #87778C22semidirect
54. C4×C5⋊D5D4semidirect
71. C4×C22⋊D5D5semidirect
82. C22×C52#41semidirect
90. C2×C5⋊D5#41semidirect
126. C2×C4×D5C2×D5semidirect
139. #685C2×D5semidirect
144. #683C2×D5semidirect
145. C4×Dic5C2×D5semidirect
215. C52#525semidirect
225. C22×C5#709semidirect
324. C5#65881semidirect
337. C22#87873semidirect

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×#878732
C4×C5×C22⋊D52
#117171002
#117170992
#117170982
#117170302
C4×#60382
#117169922
#117169842
#117169712
#658815