Finite Group Databasebeta

← #10617807

Subgroups — #10617807

Order 640.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#520C211yesmaximal, characteristic, complemented
2#66165C211yesmaximal, characteristic, complemented
3#66165C211yesmaximal, characteristic, complemented
4Q8×D20C211yesmaximal, characteristic, complemented
5#65906C211yesmaximal, characteristic, complemented
6C4×Q8×D5C211yesmaximal, characteristic, complemented
7#65903C211yesmaximal, characteristic, complemented
8#65895C213yesmaximal, complemented
9#65892C213yesmaximal, complemented
10#65891C213yesmaximal, complemented
11#65876C213yesmaximal, complemented
12#65862C213yesmaximal, complemented
13#65852C213yesmaximal, complemented
14#65847C213yesmaximal, complemented
15#65845C213yesmaximal, complemented
16C2×C5×#41C2211yescharacteristic
17C2×Q8×D5C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C4×C5×Q8C2211yescharacteristic
20C4×C5×Q8C2211yescharacteristic
21C4×C5×D4C2213yes—
22C4×C5×D4C2213yes—
23C5×#117C2213yes—
24C2×C5×C4⋊C4C2213yes—
25Q8×Dic5C2211yescharacteristic
26Q8×Dic5C2211yescharacteristic
27#4084C2213yes—
28#4084C2213yes—
29#4082C2213yes—
30#4082C2213yes—
31#4075C2216yes—
32C4×C22⋊D5C2213yes—
33C4×C22⋊D5C2216yes—
34C4×C22⋊D5C2213yes—
35#4072C2216yes—
36#4070C2213yes—
37#4070C2216yes—
38#4070C2213yes—
39#4044C2213yes—
40#4043C2213yes—
41#4043C2213yes—
42#4042C2213yes—
43#4040C2211yescharacteristic
44#4038C2213yes—
45D5×C4⋊C4C2213yes—
46#4036C2213yes—
47#4035C2216yes—
48#4035C2213yes—
49#4034C2213yes—
50#4033C2213yes—
51#4033C2213yes—
52#4031C2213yes—
53#4030C2213yes—
54#4029C2213yes—
55#4025C2216yes—
56#4025C2216yes—
57#4025C2213yes—
58C4×D20C2213yes—
59#4018C2213yes—
60C42×D5C2213yes—
61#4016C2213yes—
62#4015C2213yes—
63C4×Dic10C2213yes—
64C4×Dic10C2213yes—
65C4×Dic10C2213yes—
66#3516—51nomaximal, Sylow
67C5×#41—24no—
68C2×C5×Q8C2311yescharacteristic
69C2×C5×D4C2313yes—
70C22×C4×C5C2313yes—
71C22×C4×C5C2316yes—
72C2×C22⋊D5C2313yes—
73C2×C22⋊D5C2313yes—
74Q8×D5—24no—
75Q8×D5—24no—
76C2×D20C2311yescharacteristic
77C2×C4×D5C2313yes—
78C2×C4×D5C2313yes—
79C2×C4×D5C2311yescharacteristic
80C2×C4×D5C2313yes—
81C2×Dic10C2313yes—
82C2×Dic10C2313yes—
83C2×Dic10C2313yes—
84C5×C4⋊C4C2313yes—
85C5×C4⋊C4C2313yes—
86C5×C4⋊C4C2313yes—
87C5×C4⋊C4C2311yescharacteristic
88C5×C22⋊C4C2313yes—
89C5×C22⋊C4C2313yes—
90C42×C5C2313yes—
91C42×C5C2313yes—
92C22⋊Dic5C2316yes—
93C22⋊Dic5C2316yes—
94C22⋊Dic5C2313yes—
95C22⋊Dic5C2313yes—
96#685C2311yescharacteristic
97#685C2313yes—
98#685C2313yes—
99#685C2313yes—
100#685C2311yescharacteristic
101#685C2311yescharacteristic
102C4⋊Dic5C2313yes—
103C4⋊Dic5C2313yes—
104C4⋊Dic5C2313yes—
105C4⋊Dic5C2313yes—
106C4⋊Dic5C2316yes—
107#683C2311yescharacteristic
108#683C2313yes—
109#683C2313yes—
110#683C2316yes—
111#683C2316yes—
112#683C2313yes—
113#683C2313yes—
114#683C2313yes—
115C4×Dic5C2313yes—
116C4×Dic5C2313yes—
117C4×Dic5C2313yes—
118C4×Dic5C2316yes—
119C4×Dic5C2313yes—
120#557—51no—
121#556—53no—
122#554—53no—
123#553—53no—
124#552—53no—
125D4×Q8—51no—
126#548—51no—
127#548—51no—
128#542—53no—
129#541—53no—
130#529—53no—
131#529—53no—
132#520D511yescharacteristic, complemented
133#519—51no—
134C2×C4×Q8—51no—
135C23×C5C2413yes—
136C22×D5C2411yescharacteristic
137C22×D5C2411yescharacteristic
138C5×Q8#4114yes—
139C5×D4—26no—
140C2×C4×C5—26no—
141C2×C4×C5—46no—
142C2×C4×C5C2411yescharacteristic
143C2×C4×C5C2413yes—
144C2×C4×C5C2411yescharacteristic
145C2×C4×C5C2411yescharacteristic
146C2×C4×C5C2413yes—
147C2×C4×C5C2413yes—
148C22⋊D5—26no—
149C22⋊D5—43no—
150C2×Dic5C2411yescharacteristic
151C2×Dic5C2413yes—
152C2×Dic5C2411yescharacteristic
153C2×Dic5C2413yes—
154C2×Dic5C2413yes—
155C2×Dic5C2413yes—
156D20—22no—
157C4×D5—22no—
158C4×D5—26no—
159C4×D5—26no—
160C4×D5—43no—
161Dic10—26no—
162Dic10—26no—
163Dic10—43no—
164C2×#41C2×D511yescharacteristic
165C22×Q8—51no—
166C22×Q8—51no—
167C4⋊Q8—53no—
168C4⋊Q8—53no—
169#126—53no—
170#126—56no—
171#126—53no—
172#126—56no—
173#125—53no—
174#125—53no—
175#125—56no—
176#125—53no—
177#124—53no—
178#124—53no—
179#124—53no—
180#123—56no—
181#123—51no—
182#123—53no—
183C22⋊Q8—53no—
184C22⋊Q8—53no—
185C22⋊Q8—56no—
186C22⋊Q8—53no—
187C22⋊Q8—53no—
188C22⋊Q8—53no—
189#121—53no—
190#121—53no—
191#121—53no—
192C4×Q8—51no—
193C4×Q8—51no—
194C4×Q8—53no—
195C4×Q8—53no—
196C4×Q8—53no—
197C4×Q8—53no—
198C4×Q8—53no—
199C4×Q8C2×D511yescharacteristic
200C4×Q8C2×D511yescharacteristic
201C4×D4—53no—
202C4×D4—53no—
203C4×D4—56no—
204C4×D4—53no—
205C4×D4C2×D513yes—
206C4×D4C2×D513yes—
207#117—56no—
208#117—53no—
209#117—53no—
210#117C2×D513yes—
211C2×C4⋊C4—53no—
212C2×C4⋊C4C2×D513yes—
213C2×C42—53no—
214C22×C5—23no—
215C22×C5—23no—
216C22×C5—23no—
217C22×C5C2511yescharacteristic
218C2×D5—21no—
219C2×D5—21no—
220C2×D5—21no—
221C2×D5—21no—
222C2×D5—21no—
223C2×D52−1+412yes—
224C4×C5—21no—
225C4×C5—21no—
226C4×C5—21no—
227C4×C5—23no—
228C4×C5—23no—
229C4×C5C2×#4116yes—
230Dic5—23no—
231Dic5—23no—
232Dic5—21no—
233Dic5—23no—
234Dic5—21no—
235Dic5—23no—
236#41—24no—
237C2×Q8—53no—
238C2×Q8—53no—
239C2×Q8—53no—
240C2×Q8—104no—
241C2×Q8—104no—
242C2×Q8C22×D511yescharacteristic
243C2×D4—53no—
244C2×D4—53no—
245C2×D4—51no—
246C2×D4C22×D513yes—
247C22×C4—51no—
248C22×C4—53no—
249C22×C4—53no—
250C22×C4—53no—
251C22×C4C22×D513yes—
252C22×C4C22×D516yes—
253C4⋊C4—53no—
254C4⋊C4—56no—
255C4⋊C4—53no—
256C4⋊C4—53no—
257C4⋊C4—53no—
258C4⋊C4—53no—
259C4⋊C4—56no—
260C4⋊C4—53no—
261C4⋊C4—53no—
262C4⋊C4—56no—
263C4⋊C4—53no—
264C4⋊C4—51no—
265C4⋊C4—53no—
266C4⋊C4C22×D513yes—
267C4⋊C4C22×D513yes—
268C4⋊C4C22×D513yes—
269C4⋊C4C22×D511yescharacteristic
270C22⋊C4—53no—
271C22⋊C4—56no—
272C22⋊C4—56no—
273C22⋊C4—53no—
274C22⋊C4—53no—
275C22⋊C4—51no—
276C22⋊C4—51no—
277C22⋊C4—53no—
278C22⋊C4—51no—
279C22⋊C4—53no—
280C22⋊C4C22×D513yes—
281C22⋊C4C22×D513yes—
282C42—53no—
283C42—53no—
284C42—53no—
285C42—56no—
286C42—53no—
287C42C22×D513yes—
288C42C22×D513yes—
289C2×C5—43no—
290C2×C5C22×#4111yescharacteristic
291C2×C5C2×2−1+411yescharacteristic
292C2×C5#58511yescharacteristic
293D5—22no—
294D5—41no—
295C23—51no—
296C23—51no—
297C23C23×D513yes—
298Q8—106no—
299Q8—106no—
300Q8—203no—
301Q8#70914yes—
302D4—26no—
303D4—102no—
304D4—106no—
305D4—203no—
306C2×C4—26no—
307C2×C4—46no—
308C2×C4—51no—
309C2×C4—51no—
310C2×C4—53no—
311C2×C4—53no—
312C2×C4—53no—
313C2×C4—53no—
314C2×C4—102no—
315C2×C4—106no—
316C2×C4—106no—
317C2×C4—203no—
318C2×C4C23×D511yescharacteristic
319C2×C4C23×D513yes—
320C2×C4C23×D511yescharacteristic
321C2×C4C23×D513yes—
322C2×C4C23×D511yescharacteristic
323C2×C4C23×D513yes—
324C5#351611yescharacteristic, complemented, Sylow
325C22—23no—
326C22—23no—
327C22—23no—
328C22—52no—
329C22—101no—
330C22—101no—
331C22—101no—
332C22—101no—
333C22—101no—
334C22C24×D511yescharacteristic
335C4—21no—
336C4—23no—
337C4—23no—
338C4—21no—
339C4—21no—
340C4—103no—
341C4—103no—
342C4—103no—
343C4—101no—
344C4—101no—
345C4—103no—
346C4C2×#70916yes—
347C2—43no—
348C2—102no—
349C2—201no—
350C2C22×#70911yescharacteristic
351C2#6628511yescharacteristic
352C2D5×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#520C2
hypercenter#520D5
hypocenterC5#3516
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. C5×#520C2semidirect
2. #66165C2semidirect
3. #66165C2semidirect
4. Q8×D20C2semidirect
5. #65906C2semidirect
6. C4×Q8×D5C2semidirect
7. #65903C2semidirect
8. #65895C2semidirect
9. #65892C2semidirect
10. #65891C2semidirect
11. #65876C2semidirect
12. #65862C2semidirect
13. #65852C2semidirect
14. #65847C2semidirect
15. #65845C2semidirect
132. #520D5semidirect
324. C5#3516semidirect

Maximal subgroups

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

subgroupindex
C5×#5202
#661652
Q8×D202
#659062
C4×Q8×D52
#659032
#658952
#658922
#658912
#658762
#658622
#658522
#658472
#658452
#35165