Finite Group Databasebeta

← #10618623

Subgroups — #10618623

Order 640.

1104 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#66168C211yesmaximal, characteristic, complemented
3#66161C211yesmaximal, characteristic, complemented
4#66151C211yesmaximal, characteristic, complemented
5C2×Q8×Dic5C211yesmaximal, characteristic, complemented
6#66016C213yesmaximal, complemented
7#66015C213yesmaximal, complemented
8#65977C213yesmaximal, complemented
9#65969C213yesmaximal, complemented
10#65945C213yesmaximal, complemented
11#65930C213yesmaximal, complemented
12#65908C211yesmaximal, characteristic, complemented
13Q8×Dic10C211yesmaximal, characteristic, complemented
14#65889C213yesmaximal, complemented
15#65872C213yesmaximal, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C2×#713C2211yescharacteristic
19C2×#711C2216yes—
20C5×#124C2213yes—
21C5×C22⋊Q8C2213yes—
22C5×#121C2213yes—
23C4×C5×Q8C2211yescharacteristic
24C4×C5×D4C2213yes—
25#4093C2213yes—
26#4092C2213yes—
27Q8×Dic5C2211yescharacteristic
28Q8×Dic5C2211yescharacteristic
29Q8×Dic5C2213yes—
30Q8×Dic5C2213yes—
31Q8×Dic5C2211yescharacteristic
32#4086C2213yes—
33#4085C2213yes—
34#4084C2213yes—
35#4081C2213yes—
36D4×Dic5C2213yes—
37D4×Dic5C2213yes—
38#4076C2213yes—
39#4076C2213yes—
40C4×C22⋊D5C2213yes—
41C4×C22⋊D5C2213yes—
42#4072C2213yes—
43C2×C4⋊Dic5C2213yes—
44C2×C4⋊Dic5C2213yes—
45C2×C4×Dic5C2213yes—
46#4044C2213yes—
47#4043C2216yes—
48#4039C2213yes—
49D5×C4⋊C4C2213yes—
50#4035C2213yes—
51#4034C2213yes—
52#4034C2213yes—
53#4032C2213yes—
54#4032C2216yes—
55#4031C2213yes—
56#4031C2216yes—
57#4029C2216yes—
58#4027C2216yes—
59#4025C2213yes—
60#4024C2211yescharacteristic
61#4024C2216yes—
62C4×D20C2213yes—
63#4018C2213yes—
64#4015C2213yes—
65C4×Dic10C2213yes—
66#3522—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⋊D5C2313yes—
77C2×C22⋊D5C2316yes—
78C22×Dic5C2313yes—
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2316yes—
81#713—24no—
82#711—412no—
83C2×D20C2313yes—
84C2×C4×D5C2313yes—
85C2×C4×D5C2316yes—
86C2×Dic10C2311yescharacteristic
87C2×Dic10C2316yes—
88C5×C4⋊C4C2313yes—
89C5×C4⋊C4C2313yes—
90C5×C22⋊C4C2311yescharacteristic
91C5×C22⋊C4C2313yes—
92C5×C22⋊C4C2316yes—
93C42×C5C2313yes—
94C22⋊Dic5C2313yes—
95C22⋊Dic5C2311yescharacteristic
96C22⋊Dic5C2313yes—
97C22⋊Dic5C2313yes—
98#685C2313yes—
99#685C2313yes—
100#685C2316yes—
101#685C2313yes—
102#685C2313yes—
103C4⋊Dic5C2313yes—
104C4⋊Dic5C2313yes—
105C4⋊Dic5C2313yes—
106C4⋊Dic5C2313yes—
107C4⋊Dic5C2316yes—
108C4⋊Dic5C2311yescharacteristic
109C4⋊Dic5C2313yes—
110#683C2313yes—
111#683C2313yes—
112#683C2311yescharacteristic
113#683C2313yes—
114#683C2313yes—
115#683C2311yescharacteristic
116C4×Dic5C2313yes—
117C4×Dic5C2313yes—
118C4×Dic5C2313yes—
119C4×Dic5C2313yes—
120C4×Dic5C2313yes—
121#559—53no—
122Q82—51no—
123#554—53no—
124#550—51no—
125#548—51no—
126#548D511yescharacteristic, complemented
127#547—53no—
128#547—53no—
129#542—53no—
130#542—53no—
131#532—53no—
132#529—53no—
133#520—51no—
134#520—51no—
135C2×C4×Q8—51no—
136C23×C5C2413yes—
137C23×C5C2411yescharacteristic
138C22×D5C2413yes—
139C5×Q8—26no—
140C5×Q8#4114yescomplemented
141C5×D4—26no—
142C5×D4—26no—
143C2×C4×C5—26no—
144C2×C4×C5—26no—
145C2×C4×C5C2413yes—
146C2×C4×C5C2413yes—
147C2×C4×C5C2411yescharacteristic
148C2×C4×C5C2411yescharacteristic
149C2×C4×C5C2413yes—
150C22⋊D5—26no—
151C22⋊D5—26no—
152C22⋊D5—46no—
153C2×Dic5—26no—
154C2×Dic5—22no—
155C2×Dic5—46no—
156C2×Dic5C2411yescharacteristic
157C2×Dic5C2411yescharacteristic
158C2×Dic5C2413yes—
159C2×Dic5C2413yes—
160C2×Dic5C2413yes—
161C2×Dic5C2411yescharacteristic
162C2×Dic5C2411yescharacteristic
163D20—26no—
164C4×D5—26no—
165C4×D5—46no—
166Dic10—22no—
167Dic10—46no—
168C2×#41—51no—
169C2×#41—56no—
170C2×#41C2×D511yescharacteristic
171C22×Q8C2×D511yescharacteristic
172C4⋊Q8—53no—
173C4⋊Q8—53no—
174C4⋊Q8—53no—
175#127—53no—
176#126—53no—
177#126—53no—
178#125—53no—
179#124—53no—
180#124—53no—
181#124—56no—
182#124C2×D513yes—
183#123—53no—
184#123—53no—
185#123—56no—
186#123—56no—
187C22⋊Q8—53no—
188C22⋊Q8—56no—
189C22⋊Q8—51no—
190C22⋊Q8—56no—
191C22⋊Q8C2×D513yes—
192#121—53no—
193#121—53no—
194#121—53no—
195#121—53no—
196#121C2×D513yes—
197C4×Q8—51no—
198C4×Q8—53no—
199C4×Q8—51no—
200C4×Q8—53no—
201C4×Q8—51no—
202C4×Q8—53no—
203C4×Q8C2×D511yescharacteristic
204C4×D4—53no—
205C4×D4—53no—
206C4×D4—53no—
207C4×D4—53no—
208C4×D4—53no—
209C4×D4—56no—
210C4×D4—53no—
211C4×D4C2×D513yes—
212#117—53no—
213#117—53no—
214C2×C4⋊C4—53no—
215C2×C4⋊C4—53no—
216C2×C4⋊C4—53no—
217C2×C42—53no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5—23no—
221C22×C5—23no—
222C22×C5—23no—
223C22×C5H(5,𝔽2)12yes—
224C22×C5C2511yescharacteristic
225C2×D5—23no—
226C2×D5—23no—
227C2×D5—23no—
228C4×C5—21no—
229C4×C5—23no—
230C4×C5—23no—
231C4×C5—21no—
232C4×C5C2×#4116yes—
233Dic5—21no—
234Dic5—23no—
235Dic5—23no—
236Dic5—23no—
237Dic5—21no—
238Dic5—21no—
239Dic52−1+412yes—
240#41—24no—
241#41—104no—
242#41—2012no—
243C2×Q8—24no—
244C2×Q8—51no—
245C2×Q8—56no—
246C2×Q8C22×D513yes—
247C2×Q8C22×D511yescharacteristic
248C2×D4—53no—
249C2×D4—53no—
250C2×D4—56no—
251C2×D4—53no—
252C2×D4C22×D513yes—
253C2×D4C22×D513yes—
254C22×C4—51no—
255C22×C4—53no—
256C22×C4—56no—
257C22×C4—53no—
258C22×C4—56no—
259C22×C4C22×D513yes—
260C22×C4C22×D513yes—
261C4⋊C4—53no—
262C4⋊C4—53no—
263C4⋊C4—53no—
264C4⋊C4—53no—
265C4⋊C4—53no—
266C4⋊C4—56no—
267C4⋊C4—51no—
268C4⋊C4—51no—
269C4⋊C4—51no—
270C4⋊C4—53no—
271C4⋊C4—53no—
272C4⋊C4—53no—
273C4⋊C4—53no—
274C4⋊C4C22×D513yes—
275C4⋊C4C22×D513yes—
276C22⋊C4—53no—
277C22⋊C4—53no—
278C22⋊C4—51no—
279C22⋊C4—53no—
280C22⋊C4—53no—
281C22⋊C4—53no—
282C22⋊C4—53no—
283C22⋊C4—56no—
284C22⋊C4—53no—
285C22⋊C4C22×D516yes—
286C22⋊C4C22×D511yescharacteristic
287C22⋊C4C22×D513yes—
288C42—53no—
289C42—53no—
290C42—53no—
291C42—53no—
292C42—53no—
293C42C22×D513yes—
294C2×C5—22no—
295C2×C5—43no—
296C2×C5C22×#4111yescharacteristic
297C2×C5C2×H(5,𝔽2)11yescharacteristic
298C2×C5C2×2−1+411yescharacteristic
299D5—43no—
300C23—53no—
301C23C23×D513yes—
302C23C23×D511yescharacteristic
303Q8—26no—
304Q8—102no—
305Q8—206no—
306Q8#71114yescomplemented
307D4—26no—
308D4—26no—
309D4—106no—
310D4—106no—
311D4—106no—
312D4—206no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—51no—
316C2×C4—51no—
317C2×C4—51no—
318C2×C4—53no—
319C2×C4—53no—
320C2×C4—53no—
321C2×C4—51no—
322C2×C4—102no—
323C2×C4—106no—
324C2×C4—106no—
325C2×C4—206no—
326C2×C4—206no—
327C2×C4C23×D513yes—
328C2×C4C23×D511yescharacteristic
329C2×C4C23×D513yes—
330C2×C4C23×D511yescharacteristic
331C2×C4C23×D513yes—
332C5#352211yescharacteristic, complemented, Sylow
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—103no—
339C22—103no—
340C22—103no—
341C22#414912yes—
342C22C24×D511yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—52no—
348C4—101no—
349C4—103no—
350C4—103no—
351C4—103no—
352C4—101no—
353C4—101no—
354C4C2×#71116yes—
355C2—22no—
356C2—43no—
357C2—203no—
358C2C22×#71111yescharacteristic
359C2C2×#414911yescharacteristic
360C2D5×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#548C2
hypercenter#548D5
hypocenterC5#3522
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. #66168C2semidirect
3. #66161C2semidirect
4. #66151C2semidirect
5. C2×Q8×Dic5C2semidirect
6. #66016C2semidirect
7. #66015C2semidirect
8. #65977C2semidirect
9. #65969C2semidirect
10. #65945C2semidirect
11. #65930C2semidirect
12. #65908C2semidirect
13. Q8×Dic10C2semidirect
14. #65889C2semidirect
15. #65872C2semidirect
126. #548D5semidirect
140. C5×Q8#41semidirect
306. Q8#711semidirect
332. C5#3522semidirect

Maximal subgroups

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

subgroupindex
C5×#5482
#661682
#661612
#661512
C2×Q8×Dic52
#660162
#660152
#659772
#659692
#659452
#659302
#659082
Q8×Dic102
#658892
#658722
#35225