Finite Group Databasebeta

← #10618613

Subgroups — #10618613

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
1C5×#548C211yesmaximal, characteristic, complemented
2#66165C211yesmaximal, characteristic, complemented
3#41×Dic5C211yesmaximal, characteristic, complemented
4Q8×C22⋊D5C211yesmaximal, characteristic, complemented
5C2×Q8×Dic5C211yesmaximal, characteristic, complemented
6#66014C213yesmaximal, complemented
7#66005C213yesmaximal, complemented
8#65973C213yesmaximal, complemented
9#65956C213yesmaximal, complemented
10#65945C213yesmaximal, complemented
11#65934C213yesmaximal, complemented
12C4×Q8×D5C211yesmaximal, characteristic, complemented
13#65903C211yesmaximal, characteristic, complemented
14#65875C213yesmaximal, complemented
15C4×#711C213yesmaximal, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C2×#711C2213yes—
20C5×#124C2213yes—
21C5×C22⋊Q8C2213yes—
22C5×#121C2213yes—
23C4×C5×Q8C2211yescharacteristic
24C4×C5×D4C2213yes—
25#4092C2213yes—
26Q8×Dic5C2211yescharacteristic
27Q8×Dic5C2211yescharacteristic
28Q8×Dic5C2213yes—
29Q8×Dic5C2213yes—
30Q8×Dic5C2211yescharacteristic
31#4090C2213yes—
32#4084C2213yes—
33#4082C2213yes—
34#4081C2216yes—
35D4×Dic5C2213yes—
36D4×Dic5C2213yes—
37D4×Dic5C2213yes—
38C4×C22⋊D5C2213yes—
39C4×C22⋊D5C2213yes—
40#4072C2213yes—
41C2×C4⋊Dic5C2213yes—
42#4070C2213yes—
43#4070C2213yes—
44C2×C4×Dic5C2213yes—
45C2×C4×Dic5C2213yes—
46#4038C2213yes—
47D5×C4⋊C4C2213yes—
48#4036C2213yes—
49#4035C2216yes—
50#4034C2213yes—
51#4033C2213yes—
52#4033C2213yes—
53#4029C2216yes—
54#4027C2211yescharacteristic
55#4027C2213yes—
56#4025C2216yes—
57#4025C2216yes—
58#4024C2216yes—
59#4023C2213yes—
60#4023C2213yes—
61#4023C2216yes—
62C42×D5C2213yes—
63#4016C2213yes—
64C4×Dic10C2213yes—
65C4×Dic10C2213yes—
66#3482—51nomaximal, Sylow
67C5×#41—24no—
68C2×C5×Q8—24no—
69C2×C5×Q8C2313yes—
70C2×C5×Q8C2311yescharacteristic
71C2×C5×D4C2313yes—
72C2×C5×D4C2313yes—
73C22×C4×C5C2313yes—
74C22×C4×C5C2313yes—
75C2×C22⋊D5C2313yes—
76C2×C22⋊D5C2311yescharacteristic
77C22×Dic5C2313yes—
78C22×Dic5C2313yes—
79C22×Dic5C2313yes—
80C22×Dic5C2311yescharacteristic
81Q8×D5—24no—
82#711—212no—
83C2×C4×D5C2313yes—
84C2×C4×D5C2313yes—
85C2×C4×D5C2311yescharacteristic
86C2×Dic10C2313yes—
87C2×Dic10C2313yes—
88C5×C4⋊C4C2313yes—
89C5×C4⋊C4C2313yes—
90C5×C22⋊C4C2311yescharacteristic
91C5×C22⋊C4C2313yes—
92C5×C22⋊C4C2316yes—
93C42×C5C2313yes—
94C22⋊Dic5C2316yes—
95C22⋊Dic5C2313yes—
96C22⋊Dic5C2313yes—
97C22⋊Dic5C2316yes—
98#685C2313yes—
99#685C2311yescharacteristic
100C4⋊Dic5C2313yes—
101C4⋊Dic5C2313yes—
102C4⋊Dic5C2313yes—
103C4⋊Dic5C2313yes—
104C4⋊Dic5C2316yes—
105#683C2311yescharacteristic
106#683C2313yes—
107#683C2316yes—
108#683C2316yes—
109#683C2313yes—
110#683C2316yes—
111#683C2313yes—
112C4×Dic5C2313yes—
113C4×Dic5C2313yes—
114C4×Dic5C2313yes—
115C4×Dic5C2313yes—
116C4×Dic5C2313yes—
117C4×Dic5C2313yes—
118C4×Dic5C2313yes—
119C4×Dic5C2313yes—
120C4×Dic5C2311yescharacteristic
121#557—51no—
122#556—53no—
123#552—53no—
124#552—53no—
125D4×Q8—51no—
126#548—51no—
127#548D511yescharacteristic, complemented
128#547—53no—
129#533—53no—
130#529—53no—
131#529—53no—
132C4×#41—51no—
133C4×#41—53no—
134C2×C4×Q8—51no—
135C2×C4×Q8—51no—
136C23×C5C2413yes—
137C23×C5C2411yescharacteristic
138C22×D5C2411yescharacteristic
139C5×Q8—26no—
140C5×Q8#4114yescomplemented
141C5×D4—26no—
142C5×D4—26no—
143C2×C4×C5—26no—
144C2×C4×C5—26no—
145C2×C4×C5C2411yescharacteristic
146C2×C4×C5C2411yescharacteristic
147C2×C4×C5C2413yes—
148C2×C4×C5C2413yes—
149C2×C4×C5C2413yes—
150C22⋊D5—26no—
151C22⋊D5#4114yescomplemented
152C2×Dic5—26no—
153C2×Dic5—26no—
154C2×Dic5—22no—
155C2×Dic5—26no—
156C2×Dic5C2411yescharacteristic
157C2×Dic5C2413yes—
158C2×Dic5C2413yes—
159C2×Dic5C2413yes—
160C2×Dic5C2413yes—
161C2×Dic5C2411yescharacteristic
162C2×Dic5C2411yescharacteristic
163C4×D5—26no—
164C4×D5—26no—
165C4×D5—22no—
166Dic10—26no—
167Dic10—26no—
168C2×#41—53no—
169C2×#41C2×D511yescharacteristic
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#124C2×D513yes—
181#123—56no—
182#123—56no—
183C22⋊Q8—53no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8—56no—
187C22⋊Q8C2×D513yes—
188#121—53no—
189#121C2×D513yes—
190C4×Q8—51no—
191C4×Q8—53no—
192C4×Q8—51no—
193C4×Q8—53no—
194C4×Q8—51no—
195C4×Q8—53no—
196C4×Q8—53no—
197C4×Q8—53no—
198C4×Q8—53no—
199C4×Q8C2×D511yescharacteristic
200C4×D4—53no—
201C4×D4—53no—
202C4×D4—53no—
203C4×D4—53no—
204C4×D4—53no—
205C4×D4—51no—
206C4×D4—53no—
207C4×D4C2×D513yes—
208#117—53no—
209#117—53no—
210#117—53no—
211#117—53no—
212#117—56no—
213C2×C4⋊C4—53no—
214C2×C4⋊C4—53no—
215C2×C42—53no—
216C2×C42—53no—
217C2×C42—53no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5—23no—
221C22×C5—23no—
222C22×C5—23no—
223C22×C5C2×#4112yes—
224C22×C5C2511yescharacteristic
225C2×D5—21no—
226C2×D5—21no—
227C2×D5C2×#4112yes—
228C4×C5—23no—
229C4×C5—21no—
230C4×C5—23no—
231C4×C5—21no—
232C4×C5C2×#4116yes—
233Dic5—23no—
234Dic5—23no—
235Dic5—23no—
236Dic5—21no—
237Dic5—23no—
238Dic5C2×#4112yes—
239Dic52−1+412yes—
240#41—24no—
241#41—1012no—
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×D4C22×D513yes—
251C2×D4C22×D513yes—
252C22×C4—51no—
253C22×C4—53no—
254C22×C4—53no—
255C22×C4—53no—
256C22×C4—51no—
257C22×C4—53no—
258C22×C4—53no—
259C22×C4C22×D513yes—
260C22×C4C22×D513yes—
261C4⋊C4—53no—
262C4⋊C4—53no—
263C4⋊C4—56no—
264C4⋊C4—56no—
265C4⋊C4—51no—
266C4⋊C4—53no—
267C4⋊C4—53no—
268C4⋊C4—53no—
269C4⋊C4—53no—
270C4⋊C4—53no—
271C4⋊C4—56no—
272C4⋊C4—56no—
273C4⋊C4C22×D513yes—
274C4⋊C4C22×D513yes—
275C22⋊C4—53no—
276C22⋊C4—56no—
277C22⋊C4—53no—
278C22⋊C4—56no—
279C22⋊C4—51no—
280C22⋊C4—53no—
281C22⋊C4C22×D511yescharacteristic
282C22⋊C4C22×D513yes—
283C22⋊C4C22×D516yes—
284C42—51no—
285C42—53no—
286C42—53no—
287C42—53no—
288C42—53no—
289C42—53no—
290C42—53no—
291C42—53no—
292C42—53no—
293C42C22×D513yes—
294C2×C5—22no—
295C2×C5—43no—
296C2×C5C22×#4111yescharacteristic
297C2×C5C22×#4111yescharacteristic
298C2×C5C2×2−1+411yescharacteristic
299D5—22no—
300C23—51no—
301C23C23×D513yes—
302C23C23×D511yescharacteristic
303Q8—26no—
304Q8—106no—
305Q8—106no—
306Q8#71114yescomplemented
307D4—26no—
308D4—26no—
309D4—54no—
310D4—106no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—51no—
314C2×C4—51no—
315C2×C4—51no—
316C2×C4—53no—
317C2×C4—53no—
318C2×C4—53no—
319C2×C4—53no—
320C2×C4—102no—
321C2×C4—106no—
322C2×C4—106no—
323C2×C4—102no—
324C2×C4—106no—
325C2×C4—106no—
326C2×C4—106no—
327C2×C4C23×D511yescharacteristic
328C2×C4C23×D511yescharacteristic
329C2×C4C23×D513yes—
330C2×C4C23×D513yes—
331C2×C4C23×D513yes—
332C5#348211yescharacteristic, complemented, Sylow
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—52no—
339C22—101no—
340C22—101no—
341C22D5×#4112yes—
342C22C24×D511yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—52no—
348C4—52no—
349C4—103no—
350C4—103no—
351C4—103no—
352C4—101no—
353C4—103no—
354C4C2×#71116yes—
355C2—22no—
356C2—43no—
357C2—102no—
358C2C22×#71111yescharacteristic
359C2C2×D5×#4111yescharacteristic
360C2D5×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#548C2
hypercenter#548D5
hypocenterC5#3482
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C5×#548C2semidirect
2. #66165C2semidirect
3. #41×Dic5C2semidirect
4. Q8×C22⋊D5C2semidirect
5. C2×Q8×Dic5C2semidirect
6. #66014C2semidirect
7. #66005C2semidirect
8. #65973C2semidirect
9. #65956C2semidirect
10. #65945C2semidirect
11. #65934C2semidirect
12. C4×Q8×D5C2semidirect
13. #65903C2semidirect
14. #65875C2semidirect
15. C4×#711C2semidirect
127. #548D5semidirect
140. C5×Q8#41semidirect
151. C22⋊D5#41semidirect
306. Q8#711semidirect
332. C5#3482semidirect

Maximal subgroups

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

subgroupindex
C5×#5482
#661652
#41×Dic52
Q8×C22⋊D52
C2×Q8×Dic52
#660142
#660052
#659732
#659562
#659452
#659342
C4×Q8×D52
#659032
#658752
C4×#7112
#34825