Finite Group Databasebeta

← #13105098

Subgroups — #13105098

Order 1600.

762 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×D20C211yesmaximal, characteristic, complemented
2C2×#87874C211yesmaximal, characteristic, complemented
3C4×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
4C5×#4021C211yesmaximal, characteristic, complemented
5C4×C5×D20C211yesmaximal, characteristic, complemented
6#11717040C212yesmaximal, complemented
7#11717039C212yesmaximal, complemented
8#11717030C212yesmaximal, complemented
9#11717026C211yesmaximal, characteristic, complemented
10#11716974C211yesmaximal, characteristic, complemented
11#11716972C212yesmaximal, complemented
12C22×D52C2212yes—
13C2×C4⋊(C5⋊D5)C2211yescharacteristic
14C2×C4×C5⋊D5C2212yescomplemented
15C2×C5×D20C2211yescharacteristic
16C2×C5×D20C2211yescharacteristic
17C2×C4×C5×D5C2211yescharacteristic
18C2×C4×C5×D5C2211yescharacteristic, complemented
19C2×C5×Dic10C2211yescharacteristic, complemented
20C2×#6038C2212yes—
21C2×#6038C2212yescomplemented
22C2×#6037C2212yescomplemented
23C2×#6036C2212yescomplemented
24D5×D20—24no—
25#87874—24no—
26C42×C52C2211yescharacteristic, complemented
27#87808C2212yescomplemented
28C4⋊#1039C2211yescharacteristic, complemented
29C5×#685C2211yescharacteristic
30C5×#685C2212yescomplemented
31C5×#685C2211yescharacteristic, complemented
32C5×#685C2212yescomplemented
33C5×C4⋊Dic5C2211yescharacteristic, complemented
34#87781C2212yescomplemented
35#87778C2212yes—
36#87778C2212yescomplemented
37#66010—51nomaximal
38#65884—51nomaximal
39C22×C5⋊D5C2312yes—
40C22×C5×D5C2311yescharacteristic
41C22×C5×D5C2312yes—
42C22×C5×D5C2311yescharacteristic
43C2×D52—24no—
44C2×D52—24no—
45C2×D52—42no—
46C2×C4×C52C2311yescharacteristic
47C2×C4×C52C2311yescharacteristic
48C2×C4×C52C2311yescharacteristic
49C2×#1039C2312yes—
50C4⋊(C5⋊D5)D414yescomplemented
51C4×C5⋊D5—42no—
52C2×C5×Dic5C2311yescharacteristic
53C2×C5×Dic5C2312yes—
54C2×C5×Dic5C2311yescharacteristic
55C5×D20—22no—
56C5×D20—22no—
57C4×C5×D5—22no—
58C4×C5×D5—22no—
59C5×Dic10—22no—
60#6038—24no—
61#6038—24no—
62#6037—42no—
63#6036—24no—
64C2×#713—51no—
65C2×D4×D5—51no—
66C2×#709—51no—
67C22×D20—51no—
68C5×#124—51no—
69C4×C5×D4—51no—
70#4092—51no—
71#4084—51no—
72#4083—52no—
73#4076—51no—
74#4070—51no—
75#4040—52no—
76#4030—52no—
77#4030—52no—
78#4029—52no—
79#4028—52no—
80D5×C22⋊C4—52no—
81D5×C22⋊C4—52no—
82#4021D511yescharacteristic, complemented
83C4×D20—51no—
84C4×D20—51no—
85C4×D20—102no—
86C4×D20D511yescharacteristic, complemented
87C22×C52C2411yescharacteristic
88C2×C5⋊D5—22no—
89C2×C5⋊D5—22no—
90C2×C5⋊D5C2×D414yes—
91C2×C5×D5—21no—
92C2×C5×D5—21no—
93C2×C5×D5—22no—
94C2×C5×D5—21no—
95C2×C5×D5—22no—
96C2×C5×D5—21no—
97C2×C5×D5—21no—
98C2×C5×D5—22no—
99C2×C5×D5#4112yescomplemented
100D52—48no—
101C4×C52—21no—
102C4×C52—21no—
103C4×C52C2×D412yes—
104#1039—22no—
105C5×Dic5—21no—
106C5×Dic5—22no—
107C5×Dic5—21no—
108C23×D5—52no—
109C23×D5—52no—
110C2×C5×Q8—51no—
111C2×C5×D4—51no—
112C2×C5×D4—51no—
113C22×C4×C5—51no—
114C22×C4×C5—51no—
115C2×C22⋊D5—52no—
116C2×C22⋊D5—52no—
117C2×C22⋊D5—52no—
118C2×C22⋊D5—52no—
119#713—104no—
120D4×D5—104no—
121#709—104no—
122C2×D20—51no—
123C2×D20—51no—
124C2×D20—52no—
125C2×D20—52no—
126C2×D20—102no—
127C2×D20—104no—
128C2×D20C2×D511yescharacteristic
129C2×D20C2×D511yescharacteristic
130C2×C4×D5—52no—
131C2×C4×D5—52no—
132C2×C4×D5—52no—
133C2×C4×D5—52no—
134C2×C4×D5—104no—
135C2×C4×D5C2×D511yescharacteristic
136C2×C4×D5C2×D511yescharacteristic, complemented
137C2×Dic10C2×D511yescharacteristic, complemented
138C5×C4⋊C4—51no—
139C5×C22⋊C4—51no—
140C5×C22⋊C4—51no—
141C5×C22⋊C4—52no—
142C5×C22⋊C4—52no—
143C42×C5—22no—
144C42×C5C2×D511yescharacteristic, complemented
145C42×C5C2×D511yescharacteristic, complemented
146C22⋊Dic5—52no—
147C22⋊Dic5—52no—
148#685—52no—
149#685—52no—
150#685—52no—
151#685—52no—
152#685—104no—
153#685C2×D511yescharacteristic
154#685C2×D511yescharacteristic, complemented
155#685C2×D512yescomplemented
156#685C2×D512yescomplemented
157C4⋊Dic5—51no—
158C4⋊Dic5—51no—
159C4⋊Dic5—102no—
160C4⋊Dic5C2×D511yescharacteristic, complemented
161#683—52no—
162#683—52no—
163#546—251noSylow
164C2×C52C22×D411yescharacteristic
165C2×C52C2×#4111yescharacteristic
166C2×C52H(5,𝔽2)11yescharacteristic
167C5⋊D5—24no—
168C5×D5—22no—
169C5×D5—42no—
170C5×D5—41no—
171C23×C5—51no—
172C23×C5—52no—
173C23×C5—51no—
174C22×D5—52no—
175C22×D5—52no—
176C22×D5—104no—
177C22×D5—104no—
178C22×D5—104no—
179C22×D5—104no—
180C22×D5—104no—
181C22×D5—202no—
182C22×D5—202no—
183C22×D5C22×D512yes—
184C22×D5C22×D511yescharacteristic
185C22×D5C22×D511yescharacteristic
186C5×Q8—102no—
187C5×D4—102no—
188C5×D4—102no—
189C2×C4×C5—22no—
190C2×C4×C5—22no—
191C2×C4×C5—22no—
192C2×C4×C5—51no—
193C2×C4×C5—52no—
194C2×C4×C5—51no—
195C2×C4×C5—102no—
196C2×C4×C5—102no—
197C2×C4×C5C22×D511yescharacteristic
198C2×C4×C5C22×D511yescharacteristic
199C2×C4×C5C22×D511yescharacteristic
200C2×C4×C5C22×D511yescharacteristic
201C2×C4×C5C22×D511yescharacteristic
202C2×C4×C5C22×D511yescharacteristic
203C22⋊D5—104no—
204C22⋊D5—104no—
205C22⋊D5—202no—
206C22⋊D5—202no—
207C2×Dic5—52no—
208C2×Dic5—52no—
209C2×Dic5—104no—
210C2×Dic5C22×D511yescharacteristic
211C2×Dic5C22×D512yes—
212C2×Dic5C22×D511yescharacteristic
213D20—22no—
214D20—22no—
215D20—54no—
216D20—54no—
217D20—108no—
218D20—104no—
219D20—104no—
220C4×D5—22no—
221C4×D5—22no—
222C4×D5—104no—
223C4×D5—104no—
224C4×D5—202no—
225C4×D5—202no—
226C4×D5—208no—
227Dic10—22no—
228C2×#41—251no—
229C22×D4—251no—
230#124—51no—
231#123—252no—
232C22⋊Q8—251no—
233#121—252no—
234#121—251no—
235#120—252no—
236C4×D4—51no—
237C4×D4—251no—
238C2×C22⋊C4—252no—
239C52#54611yescharacteristic, complemented, Sylow
240C22×C5—22no—
241C22×C5—52no—
242C22×C5—101no—
243C22×C5—101no—
244C22×C5—102no—
245C22×C5—101no—
246C22×C5—102no—
247C22×C5—101no—
248C22×C5—101no—
249C22×C5—102no—
250C22×C5C23×D511yescharacteristic
251C22×C5C23×D511yescharacteristic
252C2×D5—21no—
253C2×D5—21no—
254C2×D5—21no—
255C2×D5—21no—
256C2×D5—22no—
257C2×D5—22no—
258C2×D5—21no—
259C2×D5—22no—
260C2×D5—54no—
261C2×D5—54no—
262C2×D5—108no—
263C2×D5—102no—
264C2×D5—102no—
265C2×D5—102no—
266C2×D5—102no—
267C2×D5—204no—
268C2×D5—204no—
269C2×D5—208no—
270C2×D5—208no—
271C2×D5#70912yescomplemented
272C4×C5—24no—
273C4×C5—21no—
274C4×C5—21no—
275C4×C5—21no—
276C4×C5—21no—
277C4×C5—24no—
278C4×C5—42no—
279C4×C5—101no—
280C4×C5—102no—
281C4×C5—101no—
282C4×C5D4×D512yes—
283C4×C5D4×D512yes—
284Dic5—22no—
285Dic5—21no—
286Dic5—21no—
287Dic5—102no—
288Dic5—102no—
289Dic5—204no—
290C24—252no—
291#41—504no—
292C2×Q8—51no—
293C2×D4—51no—
294C2×D4—51no—
295C2×D4—252no—
296C2×D4—252no—
297C2×D4—252no—
298C2×D4—251no—
299C2×D4—504no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—252no—
303C22×C4—252no—
304C4⋊C4—51no—
305C4⋊C4—252no—
306C4⋊C4—251no—
307C22⋊C4—51no—
308C22⋊C4—52no—
309C22⋊C4—51no—
310C22⋊C4—52no—
311C22⋊C4—252no—
312C22⋊C4—252no—
313C22⋊C4—252no—
314C42D5211yescharacteristic, complemented
315C2×C5—22no—
316C2×C5—22no—
317C2×C5—22no—
318C2×C5—102no—
319C2×C5—202no—
320C2×C5—201no—
321C2×C5C2×#70911yescharacteristic
322C2×C5C2×D4×D511yescharacteristic
323C2×C5C2×D4×D511yescharacteristic
324C2×C5D5×#4111yescharacteristic
325C2×C5#414911yescharacteristic
326C2×C5#414911yescharacteristic
327D5—22no—
328D5—41no—
329D5—42no—
330D5—104no—
331D5—104no—
332D5—208no—
333C23—51no—
334C23—52no—
335C23—51no—
336C23—252no—
337C23—504no—
338C23—504no—
339C23—1002no—
340Q8—102no—
341D4—102no—
342D4—102no—
343D4—254no—
344D4—504no—
345D4—504no—
346D4—1002no—
347C2×C4—52no—
348C2×C4—51no—
349C2×C4—51no—
350C2×C4—102no—
351C2×C4—102no—
352C2×C4—252no—
353C2×C4—504no—
354C2×C4—1002no—
355C2×C4C2×D5211yescharacteristic
356C2×C4C2×D5211yescharacteristic
357C2×C4C2×D5211yescharacteristic
358C5—22no—
359C5#6588411yescharacteristic, complemented
360C5#6601011yescharacteristic, complemented
361C22—52no—
362C22—101no—
363C22—101no—
364C22—101no—
365C22—102no—
366C22—101no—
367C22—102no—
368C22—101no—
369C22—102no—
370C22—254no—
371C22—502no—
372C22—502no—
373C22—1008no—
374C22C22×D5211yescharacteristic
375C4—21no—
376C4—21no—
377C4—101no—
378C4—101no—
379C4—102no—
380C4—502no—
381C4#8787712yes—
382C2—102no—
383C2—202no—
384C2—201no—
385C2—504no—
386C2C2×#8787711yescharacteristic
387C2D5×#70911yescharacteristic
388C2#1171747311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
hypercenterC42D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×D5×D20C2semidirect
2. C2×#87874C2semidirect
3. C4×C4⋊(C5⋊D5)C2semidirect
4. C5×#4021C2semidirect
5. C4×C5×D20C2semidirect
6. #11717040C2semidirect
7. #11717039C2semidirect
8. #11717030C2semidirect
9. #11717026C2semidirect
10. #11716974C2semidirect
11. #11716972C2semidirect
14. C2×C4×C5⋊D5C22semidirect
18. C2×C4×C5×D5C22semidirect
19. C2×C5×Dic10C22semidirect
21. C2×#6038C22semidirect
22. C2×#6037C22semidirect
23. C2×#6036C22semidirect
26. C42×C52C22semidirect
27. #87808C22semidirect
28. C4⋊#1039C22semidirect
30. C5×#685C22semidirect
31. C5×#685C22semidirect
32. C5×#685C22semidirect
33. C5×C4⋊Dic5C22semidirect
34. #87781C22semidirect
36. #87778C22semidirect
50. C4⋊(C5⋊D5)D4semidirect
82. #4021D5semidirect
86. C4×D20D5semidirect
99. C2×C5×D5#41semidirect
136. C2×C4×D5C2×D5semidirect
137. C2×Dic10C2×D5semidirect
144. C42×C5C2×D5semidirect
145. C42×C5C2×D5semidirect
154. #685C2×D5semidirect
155. #685C2×D5semidirect
156. #685C2×D5semidirect
160. C4⋊Dic5C2×D5semidirect
239. C52#546semidirect
271. C2×D5#709semidirect
314. C42D52semidirect
359. C5#65884semidirect
360. C5#66010semidirect

Maximal subgroups

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

subgroupindex
C2×D5×D202
C2×#878742
C4×C4⋊(C5⋊D5)2
C5×#40212
C4×C5×D202
#117170402
#117170392
#117170302
#117170262
#117169742
#117169722
#660105
#658845