Finite Group Databasebeta

← #10617718

Subgroups — #10617718

Order 640.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×Q8C211yesmaximal, characteristic, complemented
2#66151C211yesmaximal, characteristic, complemented
3Q8×C22⋊D5C211yesmaximal, characteristic, complemented
4#65915C213yesmaximal, complemented
5#65914C213yesmaximal, complemented
6#65912C213yesmaximal, complemented
7#65910C211yesmaximal, characteristic, complemented
8C4×#713C211yesmaximal, characteristic, complemented
9C4×Q8×D5C211yesmaximal, characteristic, complemented
10#65903C211yesmaximal, characteristic, complemented
11#65902C213yesmaximal, complemented
12#65847C213yesmaximal, complemented
13#65841C213yesmaximal, complemented
14#65813C213yesmaximal, complemented
15C4×#709C213yesmaximal, complemented
16C22×C5×Q8C2211yescharacteristic
17C2×#713C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C2×#709C2213yes—
20C4×C5×Q8C2213yes—
21C4×C5×Q8C2211yescharacteristic
22C4×C5×Q8C2211yescharacteristic
23C4×C5×Q8C2213yes—
24C2×C5×C4⋊C4C2213yes—
25C2×C42×C5C2213yes—
26#4093C2213yes—
27#4092C2213yes—
28#4092C2213yes—
29Q8×Dic5C2211yescharacteristic
30Q8×Dic5C2211yescharacteristic
31#4090C2213yes—
32#4076C2213yes—
33#4075C2216yes—
34C4×C22⋊D5C2211yescharacteristic
35C4×C22⋊D5C2213yes—
36C4×C22⋊D5C2213yes—
37C4×C22⋊D5C2213yes—
38#4072C2213yes—
39#4070C2213yes—
40#4044C2216yes—
41#4043C2213yes—
42#4042C2216yes—
43#4041C2213yes—
44#4040C2216yes—
45#4039C2213yes—
46#4038C2213yes—
47D5×C4⋊C4C2213yes—
48#4036C2213yes—
49#4035C2216yes—
50#4034C2213yes—
51#4033C2213yes—
52#4022C2216yes—
53#4021C2213yes—
54C4×D20C2213yes—
55C4×D20C2213yes—
56C4×D20C2213yes—
57#4018C2213yes—
58#4018C2213yes—
59#4018C2216yes—
60C42×D5C2213yes—
61C42×D5C2213yes—
62#4016C2213yes—
63C4×Dic10C2213yes—
64C4×Dic10C2213yes—
65C4×Dic10C2213yes—
66#3482—51nomaximal, Sylow
67C2×C5×Q8—24no—
68C2×C5×Q8C2313yes—
69C2×C5×Q8C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2313yes—
72C22×C4×C5C2313yes—
73C2×C22⋊D5C2313yes—
74C2×C22⋊D5C2311yescharacteristic
75#713—24no—
76Q8×D5—24no—
77#709—212no—
78C2×D20C2313yes—
79C2×D20C2313yes—
80C2×C4×D5C2313yes—
81C2×C4×D5C2313yes—
82C2×C4×D5C2313yes—
83C2×C4×D5C2313yes—
84C2×C4×D5C2311yescharacteristic
85C2×C4×D5C2313yes—
86C2×Dic10C2313yes—
87C2×Dic10C2313yes—
88C5×C4⋊C4C2313yes—
89C5×C4⋊C4C2313yes—
90C5×C4⋊C4C2316yes—
91C42×C5C2313yes—
92C42×C5C2313yes—
93C42×C5C2313yes—
94C42×C5C2313yes—
95C22⋊Dic5C2313yes—
96C22⋊Dic5C2311yescharacteristic
97#685C2311yescharacteristic
98#685C2313yes—
99#685C2313yes—
100#685C2313yes—
101#685C2316yes—
102#685C2316yes—
103#685C2316yes—
104C4⋊Dic5C2313yes—
105C4⋊Dic5C2313yes—
106C4⋊Dic5C2313yes—
107C4⋊Dic5C2313yes—
108#683C2311yescharacteristic
109#683C2313yes—
110#683C2313yes—
111#683C2313yes—
112#683C2316yes—
113#683C2316yes—
114#683C2316yes—
115C4×Dic5C2311yescharacteristic
116C4×Dic5C2313yes—
117C4×Dic5C2313yes—
118C4×Dic5C2313yes—
119C4×Dic5C2313yes—
120C4×Dic5C2313yes—
121#557—51no—
122#556—53no—
123#552—53no—
124#552—53no—
125D4×Q8—51no—
126#548—51no—
127#548—51no—
128#547—53no—
129#533—53no—
130#529—53no—
131#529—53no—
132C4×#41—51no—
133C4×#41—53no—
134C2×C4×Q8—51no—
135C2×C4×Q8D511yescharacteristic, complemented
136C23×C5C2411yescharacteristic
137C22×D5C2413yes—
138C22×D5C2411yescharacteristic
139C5×Q8—26no—
140C5×Q8#4114yescomplemented
141C2×C4×C5—26no—
142C2×C4×C5—26no—
143C2×C4×C5—22no—
144C2×C4×C5C2413yes—
145C2×C4×C5C2411yescharacteristic
146C2×C4×C5C2413yes—
147C2×C4×C5C2413yes—
148C2×C4×C5C2413yes—
149C2×C4×C5C2411yescharacteristic
150C22⋊D5—26no—
151C22⋊D5#4114yescomplemented
152C2×Dic5C2411yescharacteristic
153C2×Dic5C2411yescharacteristic
154C2×Dic5C2413yes—
155C2×Dic5C2413yes—
156C2×Dic5C2413yes—
157C2×Dic5C2411yescharacteristic
158D20—26no—
159D20—26no—
160C4×D5—26no—
161C4×D5—26no—
162C4×D5—26no—
163C4×D5—22no—
164C4×D5—26no—
165C4×D5—26no—
166Dic10—26no—
167Dic10—26no—
168C2×#41—51no—
169C2×#41—53no—
170C22×Q8—51no—
171C22×Q8C2×D511yescharacteristic
172C4⋊Q8—53no—
173C4⋊Q8—53no—
174#126—56no—
175#126—56no—
176#125—53no—
177#125—56no—
178#125—53no—
179#124—53no—
180#124—53no—
181#123—56no—
182#123—56no—
183C22⋊Q8—53no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8—56no—
187C22⋊Q8—53no—
188#121—53no—
189#121—53no—
190C4×Q8—51no—
191C4×Q8—51no—
192C4×Q8—53no—
193C4×Q8—53no—
194C4×Q8—53no—
195C4×Q8—53no—
196C4×Q8C2×D511yescharacteristic
197C4×Q8C2×D513yes—
198C4×Q8C2×D511yescharacteristic
199C4×Q8C2×D513yes—
200C4×D4—51no—
201C4×D4—53no—
202C4×D4—53no—
203C4×D4—53no—
204C4×D4—53no—
205C4×D4—53no—
206C4×D4—53no—
207C4×D4—53no—
208#117—53no—
209#117—53no—
210#117—53no—
211#117—53no—
212#117—56no—
213C2×C4⋊C4—53no—
214C2×C4⋊C4C2×D513yes—
215C2×C42—53no—
216C2×C42—53no—
217C2×C42C2×D513yes—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5C2×#4112yes—
221C22×C5C2511yescharacteristic
222C2×D5—21no—
223C2×D5—21no—
224C2×D5—23no—
225C2×D5—23no—
226C2×D5—23no—
227C2×D5C2×#4112yes—
228C4×C5—23no—
229C4×C5—23no—
230C4×C5—23no—
231C4×C5—21no—
232C4×C5C2×#4116yes—
233C4×C52−1+412yes—
234Dic5—23no—
235Dic5—23no—
236Dic5—21no—
237Dic5—21no—
238Dic5—23no—
239Dic5C2×#4112yes—
240#41—1012no—
241#41—104no—
242C2×Q8—24no—
243C2×Q8—53no—
244C2×Q8—53no—
245C2×Q8—104no—
246C2×Q8C22×D513yes—
247C2×Q8C22×D511yescharacteristic
248C2×D4—51no—
249C2×D4—53no—
250C2×D4—53no—
251C2×D4—53no—
252C22×C4—53no—
253C22×C4—53no—
254C22×C4—53no—
255C22×C4—53no—
256C22×C4—51no—
257C22×C4—53no—
258C22×C4C22×D513yes—
259C22×C4C22×D513yes—
260C22×C4C22×D511yescharacteristic
261C4⋊C4—53no—
262C4⋊C4—56no—
263C4⋊C4—51no—
264C4⋊C4—53no—
265C4⋊C4—53no—
266C4⋊C4—53no—
267C4⋊C4—53no—
268C4⋊C4—53no—
269C4⋊C4—53no—
270C4⋊C4—56no—
271C4⋊C4—56no—
272C4⋊C4C22×D513yes—
273C4⋊C4C22×D513yes—
274C4⋊C4C22×D516yes—
275C22⋊C4—51no—
276C22⋊C4—53no—
277C22⋊C4—51no—
278C22⋊C4—53no—
279C22⋊C4—56no—
280C22⋊C4—53no—
281C22⋊C4—53no—
282C22⋊C4—56no—
283C22⋊C4—56no—
284C42—53no—
285C42—53no—
286C42—53no—
287C42—53no—
288C42—51no—
289C42—53no—
290C42C22×D513yes—
291C42C22×D513yes—
292C42C22×D513yes—
293C42C22×D513yes—
294C2×C5—22no—
295C2×C5C22×#4111yescharacteristic
296C2×C5C22×#4111yescharacteristic
297C2×C5C2×2−1+411yescharacteristic
298D5—22no—
299D5—43no—
300C23—51no—
301C23—53no—
302C23C23×D511yescharacteristic
303Q8—26no—
304Q8—106no—
305Q8—106no—
306Q8#70914yescomplemented
307D4—54no—
308D4—106no—
309D4—106no—
310D4—106no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—22no—
314C2×C4—51no—
315C2×C4—51no—
316C2×C4—51no—
317C2×C4—53no—
318C2×C4—53no—
319C2×C4—53no—
320C2×C4—106no—
321C2×C4—106no—
322C2×C4—106no—
323C2×C4—106no—
324C2×C4—102no—
325C2×C4—106no—
326C2×C4C23×D513yes—
327C2×C4C23×D511yescharacteristic
328C2×C4C23×D513yes—
329C2×C4C23×D513yes—
330C2×C4C23×D513yes—
331C2×C4C23×D511yescharacteristic
332C5#348211yescharacteristic, complemented, Sylow
333C22—21no—
334C22—21no—
335C22—52no—
336C22—101no—
337C22—101no—
338C22—103no—
339C22—103no—
340C22—103no—
341C22D5×#4112yes—
342C22C24×D511yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—23no—
347C4—52no—
348C4—101no—
349C4—103no—
350C4—101no—
351C4—103no—
352C4—103no—
353C4C2×#70916yes—
354C4#414712yes—
355C2—22no—
356C2—102no—
357C2—203no—
358C2C22×#70911yescharacteristic
359C2C2×#414711yescharacteristic
360C2C2×D5×#4111yescharacteristic
subgroupG/H
centerC2×C4C23×D5
commutator, socle, cosocleC22×C5C25
FrattiniC22C24×D5
FittingC2×C4×C5×Q8C2
hypercenterC2×C4×Q8D5
hypocenterC5#3482
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×Q8C2semidirect
2. #66151C2semidirect
3. Q8×C22⋊D5C2semidirect
4. #65915C2semidirect
5. #65914C2semidirect
6. #65912C2semidirect
7. #65910C2semidirect
8. C4×#713C2semidirect
9. C4×Q8×D5C2semidirect
10. #65903C2semidirect
11. #65902C2semidirect
12. #65847C2semidirect
13. #65841C2semidirect
14. #65813C2semidirect
15. C4×#709C2semidirect
135. C2×C4×Q8D5semidirect
140. C5×Q8#41semidirect
151. C22⋊D5#41semidirect
306. Q8#709semidirect
332. C5#3482semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×Q82
#661512
Q8×C22⋊D52
#659152
#659142
#659122
#659102
C4×#7132
C4×Q8×D52
#659032
#659022
#658472
#658412
#658132
C4×#7092
#34825