Finite Group Databasebeta

← #13106522

Subgroups — #13106522

Order 1600.

832 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 140 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×#87904C211yesmaximal, characteristic, complemented
2C2×D5×D20C211yesmaximal, characteristic, complemented
3C5×#4093C211yesmaximal, characteristic, complemented
4C5×#4092C211yesmaximal, characteristic, complemented
5#11717041C212yesmaximal, complemented
6#11717039C212yesmaximal, complemented
7#11717034C211yesmaximal, characteristic, complemented
8#11717028C212yesmaximal, complemented
9C4×#6038C211yesmaximal, characteristic, complemented
10#11717013C211yesmaximal, characteristic, complemented
11#11716971C212yesmaximal, complemented
12C22×D52C2212yes—
13C2×C52×Q8C2211yescharacteristic, complemented
14#87904—24no—
15C2×C4⋊(C5⋊D5)C2211yescharacteristic
16C2×C4⋊(C5⋊D5)C2212yescomplemented
17C2×C4×C5⋊D5C2212yescomplemented
18C2×C4×C5⋊D5C2211yescharacteristic, complemented
19C2×C5×D20C2211yescharacteristic
20C2×C4×C5×D5C2211yescharacteristic
21C2×#6038C2212yes—
22C2×#6038C2211yescharacteristic
23C2×#6038C2212yescomplemented
24C2×#6036C2212yescomplemented
25D5×D20—24no—
26C5×#685C2212yes—
27C5×#685C2212yescomplemented
28C5×#685C2212yescomplemented
29C5×C4⋊Dic5C2211yescharacteristic, complemented
30C5×#683C2212yescomplemented
31C4×C5×Dic5C2211yescharacteristic, complemented
32#87780C2211yescharacteristic, complemented
33#87779C2212yescomplemented
34#87779C2211yescharacteristic, complemented
35#87778C2211yescharacteristic
36#87778C2212yescomplemented
37#66010—51nomaximal
38#65964—51nomaximal
39C22×C5⋊D5C2311yescharacteristic
40C22×C5⋊D5C2312yes—
41C22×C5×D5C2312yes—
42C22×C5×D5C2311yescharacteristic
43C2×D52—24no—
44C2×D52—24no—
45C2×D52—42no—
46C52×Q8—22no—
47C2×C4×C52C2312yes—
48C2×C4×C52C2311yescharacteristic
49C2×#1039C2311yescharacteristic
50C4⋊(C5⋊D5)—24no—
51C4⋊(C5⋊D5)D414yescomplemented
52C4×C5⋊D5—24no—
53C4×C5⋊D5—22no—
54C2×C5×Dic5C2311yescharacteristic
55C2×C5×Dic5C2312yes—
56C2×C5×Dic5C2311yescharacteristic
57C2×C5×Dic5C2311yescharacteristic
58C5×D20—22no—
59C4×C5×D5—22no—
60#6038—24no—
61#6038—22no—
62#6038—42no—
63#6036—42no—
64C2×#713—51no—
65C2×#713—51no—
66C2×#713—102no—
67C2×D4×D5—51no—
68C22×D20—51no—
69C5×#124—51no—
70C5×C22⋊Q8—51no—
71#4093D511yescharacteristic, complemented
72#4092D511yescharacteristic, complemented
73#4084—51no—
74C4×C22⋊D5—51no—
75#4041—51no—
76#4041—52no—
77#4040—52no—
78#4039—51no—
79#4030—52no—
80#4029—52no—
81#4028—52no—
82#4028—52no—
83D5×C22⋊C4—52no—
84D5×C22⋊C4—52no—
85C4×D20—51no—
86C4×D20—51no—
87C22×C52C2411yescharacteristic
88C2×C5⋊D5—22no—
89C2×C5⋊D5—22no—
90C2×C5⋊D5—21no—
91C2×C5⋊D5—21no—
92C2×C5⋊D5—21no—
93C2×C5⋊D5C2×D414yes—
94C2×C5×D5—21no—
95C2×C5×D5—22no—
96C2×C5×D5—22no—
97C2×C5×D5—21no—
98C2×C5×D5—22no—
99C2×C5×D5#4112yescomplemented
100D52—48no—
101C4×C52—22no—
102C4×C52C2×D412yes—
103#1039—21no—
104C5×Dic5—21no—
105C5×Dic5—21no—
106C5×Dic5—21no—
107C5×Dic5—22no—
108C23×D5—52no—
109C23×D5—52no—
110C2×C5×Q8—22no—
111C2×C5×Q8C2×D511yescharacteristic, complemented
112C2×C5×Q8C2×D511yescharacteristic, complemented
113C2×C5×D4—51no—
114C22×C4×C5—51no—
115C2×C22⋊D5—52no—
116C2×C22⋊D5—51no—
117C2×C22⋊D5—52no—
118#713—104no—
119#713—104no—
120#713—1016no—
121D4×D5—104no—
122C2×D20—51no—
123C2×D20—51no—
124C2×D20—52no—
125C2×D20—51no—
126C2×D20—52no—
127C2×D20—52no—
128C2×D20—52no—
129C2×D20—102no—
130C2×D20—104no—
131C2×D20—104no—
132C2×D20C2×D511yescharacteristic
133C2×C4×D5—51no—
134C2×C4×D5—52no—
135C2×C4×D5—52no—
136C2×C4×D5—51no—
137C2×C4×D5—52no—
138C2×C4×D5—52no—
139C2×C4×D5—104no—
140C2×C4×D5—102no—
141C2×C4×D5C2×D511yescharacteristic
142C5×C4⋊C4—51no—
143C5×C4⋊C4—52no—
144C5×C22⋊C4—52no—
145C5×C22⋊C4—52no—
146C5×C22⋊C4—52no—
147C42×C5—51no—
148C22⋊Dic5—51no—
149C22⋊Dic5—52no—
150#685—51no—
151#685—52no—
152#685—52no—
153#685—51no—
154#685—52no—
155#685—51no—
156#685C2×D512yes—
157#685C2×D512yescomplemented
158#685C2×D512yescomplemented
159C4⋊Dic5—51no—
160C4⋊Dic5C2×D511yescharacteristic, complemented
161#683—51no—
162#683C2×D512yescomplemented
163C4×Dic5C2×D511yescharacteristic, complemented
164#546—251noSylow
165C2×C52C22×D411yescharacteristic
166C2×C52C2×#4111yescharacteristic
167C2×C52H(5,𝔽2)11yescharacteristic
168C5⋊D5—24no—
169C5⋊D5—41no—
170C5×D5—22no—
171C5×D5—42no—
172C23×C5—51no—
173C23×C5—52no—
174C22×D5—52no—
175C22×D5—52no—
176C22×D5—51no—
177C22×D5—51no—
178C22×D5—104no—
179C22×D5—102no—
180C22×D5—104no—
181C22×D5—104no—
182C22×D5—104no—
183C22×D5—104no—
184C22×D5—202no—
185C22×D5—202no—
186C22×D5C22×D512yes—
187C22×D5C22×D511yescharacteristic
188C5×Q8—28no—
189C5×Q8—22no—
190C5×Q8—22no—
191C5×D4—102no—
192C2×C4×C5—24no—
193C2×C4×C5—22no—
194C2×C4×C5—51no—
195C2×C4×C5—51no—
196C2×C4×C5—51no—
197C2×C4×C5—52no—
198C2×C4×C5—102no—
199C2×C4×C5C22×D512yes—
200C2×C4×C5C22×D512yes—
201C2×C4×C5C22×D511yescharacteristic
202C2×C4×C5C22×D511yescharacteristic
203C22⋊D5—104no—
204C22⋊D5—102no—
205C22⋊D5—202no—
206C2×Dic5—51no—
207C2×Dic5—51no—
208C2×Dic5—102no—
209C2×Dic5C22×D511yescharacteristic
210C2×Dic5C22×D511yescharacteristic
211C2×Dic5C22×D511yescharacteristic
212C2×Dic5C22×D512yes—
213D20—22no—
214D20—54no—
215D20—54no—
216D20—108no—
217D20—104no—
218D20—104no—
219D20—1016no—
220D20—104no—
221D20—102no—
222D20—202no—
223C4×D5—22no—
224C4×D5—104no—
225C4×D5—104no—
226C4×D5—1016no—
227C4×D5—108no—
228C4×D5—102no—
229C4×D5—102no—
230C4×D5—202no—
231C4×D5—202no—
232C2×#41—251no—
233C22×D4—251no—
234#124—51no—
235#123—252no—
236C22⋊Q8—51no—
237#121—252no—
238#121—251no—
239#120—252no—
240C4×D4—251no—
241C4×D4—251no—
242C2×C22⋊C4—252no—
243C52#54611yescharacteristic, complemented, Sylow
244C22×C5—22no—
245C22×C5—52no—
246C22×C5—101no—
247C22×C5—101no—
248C22×C5—102no—
249C22×C5—102no—
250C22×C5—102no—
251C22×C5C23×D511yescharacteristic
252C22×C5C23×D511yescharacteristic
253C2×D5—21no—
254C2×D5—21no—
255C2×D5—22no—
256C2×D5—22no—
257C2×D5—22no—
258C2×D5—54no—
259C2×D5—54no—
260C2×D5—108no—
261C2×D5—102no—
262C2×D5—102no—
263C2×D5—102no—
264C2×D5—102no—
265C2×D5—104no—
266C2×D5—101no—
267C2×D5—101no—
268C2×D5—101no—
269C2×D5—101no—
270C2×D5—101no—
271C2×D5—101no—
272C2×D5—204no—
273C2×D5—204no—
274C2×D5—202no—
275C2×D5—202no—
276C2×D5—208no—
277C2×D5—208no—
278C2×D5#71312yescomplemented
279C4×C5—28no—
280C4×C5—22no—
281C4×C5—22no—
282C4×C5—24no—
283C4×C5—101no—
284C4×C5—101no—
285C4×C5—101no—
286C4×C5—102no—
287C4×C5D4×D512yes—
288C4×C5D4×D512yes—
289Dic5—21no—
290Dic5—21no—
291Dic5—21no—
292Dic5—22no—
293Dic5—104no—
294Dic5—101no—
295Dic5—101no—
296C24—252no—
297#41—504no—
298C2×Q8D5211yescharacteristic, complemented
299C2×D4—51no—
300C2×D4—251no—
301C2×D4—252no—
302C2×D4—252no—
303C2×D4—251no—
304C2×D4—252no—
305C2×D4—504no—
306C22×C4—51no—
307C22×C4—252no—
308C22×C4—252no—
309C22×C4—251no—
310C4⋊C4—51no—
311C4⋊C4—52no—
312C4⋊C4—251no—
313C22⋊C4—52no—
314C22⋊C4—52no—
315C22⋊C4—52no—
316C22⋊C4—252no—
317C22⋊C4—251no—
318C22⋊C4—251no—
319C22⋊C4—252no—
320C42—51no—
321C2×C5—22no—
322C2×C5—22no—
323C2×C5—22no—
324C2×C5—102no—
325C2×C5—202no—
326C2×C5C2×D4×D511yescharacteristic
327C2×C5C2×D4×D511yescharacteristic
328C2×C5C2×#71311yescharacteristic
329C2×C5D5×#4111yescharacteristic
330C2×C5#414911yescharacteristic
331C2×C5#414911yescharacteristic
332D5—22no—
333D5—42no—
334D5—104no—
335D5—104no—
336D5—208no—
337D5—204no—
338D5—201no—
339D5—201no—
340C23—51no—
341C23—52no—
342C23—252no—
343C23—251no—
344C23—504no—
345C23—504no—
346C23—1002no—
347Q8—22no—
348D4—102no—
349D4—254no—
350D4—504no—
351D4—502no—
352D4—504no—
353D4—1002no—
354C2×C4—51no—
355C2×C4—51no—
356C2×C4—52no—
357C2×C4—51no—
358C2×C4—102no—
359C2×C4—251no—
360C2×C4—504no—
361C2×C4—502no—
362C2×C4—1002no—
363C2×C4C2×D5212yes—
364C2×C4C2×D5211yescharacteristic
365C5—22no—
366C5#6596411yescharacteristic, complemented
367C5#6601011yescharacteristic, complemented
368C22—52no—
369C22—101no—
370C22—101no—
371C22—102no—
372C22—102no—
373C22—102no—
374C22—254no—
375C22—502no—
376C22—502no—
377C22—501no—
378C22—501no—
379C22—501no—
380C22—1008no—
381C22C22×D5211yescharacteristic
382C4—22no—
383C4—101no—
384C4—101no—
385C4—102no—
386C4—101no—
387C4—501no—
388C4#8788612yes—
389C2—102no—
390C2—202no—
391C2—504no—
392C2—1001no—
393C2D5×#71311yescharacteristic
394C2#1171748511yescharacteristic
395C2C2×#8788611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
hypercenterC2×Q8D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#87904C2semidirect
2. C2×D5×D20C2semidirect
3. C5×#4093C2semidirect
4. C5×#4092C2semidirect
5. #11717041C2semidirect
6. #11717039C2semidirect
7. #11717034C2semidirect
8. #11717028C2semidirect
9. C4×#6038C2semidirect
10. #11717013C2semidirect
11. #11716971C2semidirect
13. C2×C52×Q8C22semidirect
16. C2×C4⋊(C5⋊D5)C22semidirect
17. C2×C4×C5⋊D5C22semidirect
18. C2×C4×C5⋊D5C22semidirect
23. C2×#6038C22semidirect
24. C2×#6036C22semidirect
27. C5×#685C22semidirect
28. C5×#685C22semidirect
29. C5×C4⋊Dic5C22semidirect
30. C5×#683C22semidirect
31. C4×C5×Dic5C22semidirect
32. #87780C22semidirect
33. #87779C22semidirect
34. #87779C22semidirect
36. #87778C22semidirect
51. C4⋊(C5⋊D5)D4semidirect
71. #4093D5semidirect
72. #4092D5semidirect
99. C2×C5×D5#41semidirect
111. C2×C5×Q8C2×D5semidirect
112. C2×C5×Q8C2×D5semidirect
157. #685C2×D5semidirect
158. #685C2×D5semidirect
160. C4⋊Dic5C2×D5semidirect
162. #683C2×D5semidirect
163. C4×Dic5C2×D5semidirect
243. C52#546semidirect
278. C2×D5#713semidirect
298. C2×Q8D52semidirect
366. C5#65964semidirect
367. C5#66010semidirect

Maximal subgroups

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

subgroupindex
C2×#879042
C2×D5×D202
C5×#40932
C5×#40922
#117170412
#117170392
#117170342
#117170282
C4×#60382
#117170132
#117169712
#660105
#659645