Finite Group Databasebeta

← #10618624

Subgroups — #10618624

Order 640.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 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#41×Dic5C211yesmaximal, characteristic, complemented
4#66151C211yesmaximal, characteristic, complemented
5#66146C211yesmaximal, characteristic, complemented
6#66016C213yesmaximal, complemented
7#66015C213yesmaximal, complemented
8#65973C213yesmaximal, complemented
9#65958C213yesmaximal, complemented
10#65953C213yesmaximal, complemented
11#65948C213yesmaximal, complemented
12#65908C211yesmaximal, characteristic, complemented
13#65904C211yesmaximal, characteristic, complemented
14#65897C213yesmaximal, complemented
15#65872C213yesmaximal, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C2×#713C2211yescharacteristic
19C5×#124C2213yes—
20C5×C22⋊Q8C2213yes—
21C5×#121C2213yes—
22C4×C5×Q8C2211yescharacteristic
23C4×C5×D4C2213yes—
24#4093C2213yes—
25#4092C2213yes—
26Q8×Dic5C2211yescharacteristic
27Q8×Dic5C2211yescharacteristic
28Q8×Dic5C2213yes—
29Q8×Dic5C2213yes—
30Q8×Dic5C2211yescharacteristic
31#4086C2213yes—
32#4085C2213yes—
33#4084C2216yes—
34#4084C2213yes—
35#4081C2213yes—
36D4×Dic5C2213yes—
37D4×Dic5C2213yes—
38#4076C2213yes—
39#4076C2213yes—
40C4×C22⋊D5C2213yes—
41C4×C22⋊D5C2213yes—
42#4072C2213yes—
43#4072C2213yes—
44#4072C2213yes—
45C2×C4×Dic5C2213yes—
46#4044C2213yes—
47#4039C2213yes—
48#4038C2216yes—
49D5×C4⋊C4C2213yes—
50#4036C2213yes—
51#4036C2213yes—
52#4035C2213yes—
53#4032C2213yes—
54#4032C2216yes—
55#4031C2213yes—
56#4031C2216yes—
57#4029C2216yes—
58#4027C2216yes—
59#4025C2211yescharacteristic
60#4025C2216yes—
61#4024C2213yes—
62C4×D20C2213yes—
63#4018C2213yes—
64#4015C2213yes—
65C4×Dic10C2213yes—
66#3524—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⋊D5C2316yes—
76C2×C22⋊D5C2313yes—
77C2×C22⋊D5C2313yes—
78C22×Dic5C2313yes—
79C22×Dic5C2313yes—
80#713—24no—
81C2×D20C2313yes—
82C2×C4×D5C2313yes—
83C2×C4×D5C2316yes—
84C2×Dic10C2313yes—
85C5×C4⋊C4C2313yes—
86C5×C4⋊C4C2313yes—
87C5×C22⋊C4C2311yescharacteristic
88C5×C22⋊C4C2313yes—
89C5×C22⋊C4C2316yes—
90C42×C5C2313yes—
91C22⋊Dic5C2313yes—
92C22⋊Dic5C2316yes—
93C22⋊Dic5C2311yescharacteristic
94C22⋊Dic5C2311yescharacteristic
95C22⋊Dic5C2313yes—
96#685C2313yes—
97#685C2313yes—
98#685C2313yes—
99#685C2316yes—
100#685C2313yes—
101C4⋊Dic5C2311yescharacteristic
102C4⋊Dic5C2316yes—
103C4⋊Dic5C2313yes—
104C4⋊Dic5C2313yes—
105C4⋊Dic5C2313yes—
106C4⋊Dic5C2313yes—
107C4⋊Dic5C2313yes—
108#683C2313yes—
109#683C2313yes—
110#683C2311yescharacteristic
111#683C2313yes—
112#683C2313yes—
113#683C2311yescharacteristic
114C4×Dic5C2313yes—
115C4×Dic5C2313yes—
116C4×Dic5C2313yes—
117C4×Dic5C2316yes—
118C4×Dic5C2311yescharacteristic
119C4×Dic5C2313yes—
120#559—53no—
121#557—51no—
122#555—53no—
123#553—53no—
124#552—53no—
125#550—51no—
126#548—51no—
127#548D511yescharacteristic, complemented
128#542—53no—
129#540—53no—
130#529—53no—
131#529—53no—
132#520—51no—
133#519—51no—
134C4×#41—51no—
135C23×C5C2411yescharacteristic
136C23×C5C2413yes—
137C22×D5C2413yes—
138C5×Q8—26no—
139C5×Q8#4114yes—
140C5×D4—26no—
141C5×D4—43no—
142C2×C4×C5—26no—
143C2×C4×C5—26no—
144C2×C4×C5C2411yescharacteristic
145C2×C4×C5C2413yes—
146C2×C4×C5C2413yes—
147C2×C4×C5C2411yescharacteristic
148C2×C4×C5C2413yes—
149C22⋊D5—43no—
150C22⋊D5—46no—
151C22⋊D5—43no—
152C2×Dic5—43no—
153C2×Dic5—43no—
154C2×Dic5C2411yescharacteristic
155C2×Dic5C2411yescharacteristic
156C2×Dic5C2413yes—
157C2×Dic5C2411yescharacteristic
158C2×Dic5C2413yes—
159C2×Dic5C2413yes—
160C2×Dic5C2411yescharacteristic
161D20—26no—
162C4×D5—26no—
163C4×D5—46no—
164Dic10—43no—
165C2×#41—51no—
166C2×#41C2×D511yescharacteristic
167C22×Q8C2×D511yescharacteristic
168C4⋊Q8—53no—
169#127—53no—
170#126—53no—
171#126—51no—
172#126—56no—
173#125—53no—
174#125—53no—
175#125—53no—
176#124—53no—
177#124—53no—
178#124—56no—
179#124C2×D513yes—
180#123—53no—
181#123—56no—
182#123—53no—
183#123—56no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8C2×D513yes—
187#121—53no—
188#121—56no—
189#121—53no—
190#121—53no—
191#121—53no—
192#121C2×D513yes—
193C4×Q8—51no—
194C4×Q8—51no—
195C4×Q8—53no—
196C4×Q8—51no—
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—56no—
206C4×D4—53no—
207C4×D4C2×D513yes—
208#117—53no—
209#117—53no—
210#117—53no—
211#117—56no—
212#117—53no—
213C2×C4⋊C4—53no—
214C2×C42—53no—
215C22×C5—21no—
216C22×C5—23no—
217C22×C5—23no—
218C22×C5—23no—
219C22×C5—21no—
220C22×C5—21no—
221C22×C5C2511yescharacteristic
222C2×D5—23no—
223C2×D5—23no—
224C2×D5—23no—
225C4×C5—21no—
226C4×C5—23no—
227C4×C5—21no—
228C4×C5—23no—
229C4×C5C2×#4116yes—
230Dic5—21no—
231Dic5—23no—
232Dic5—23no—
233Dic5—23no—
234Dic5—21no—
235Dic5—21no—
236Dic5—21no—
237#41—24no—
238#41—104no—
239C2×Q8—24no—
240C2×Q8—53no—
241C2×Q8C22×D513yes—
242C2×Q8C22×D511yescharacteristic
243C2×D4—53no—
244C2×D4—56no—
245C2×D4—53no—
246C2×D4—53no—
247C2×D4C22×D513yes—
248C2×D4C22×D513yes—
249C22×C4—53no—
250C22×C4—53no—
251C22×C4—53no—
252C22×C4—56no—
253C22×C4C22×D513yes—
254C22×C4C22×D513yes—
255C4⋊C4—53no—
256C4⋊C4—53no—
257C4⋊C4—51no—
258C4⋊C4—53no—
259C4⋊C4—53no—
260C4⋊C4—53no—
261C4⋊C4—53no—
262C4⋊C4—56no—
263C4⋊C4—51no—
264C4⋊C4—53no—
265C4⋊C4—53no—
266C4⋊C4—51no—
267C4⋊C4—53no—
268C4⋊C4C22×D513yes—
269C4⋊C4C22×D513yes—
270C22⋊C4—51no—
271C22⋊C4—53no—
272C22⋊C4—56no—
273C22⋊C4—51no—
274C22⋊C4—53no—
275C22⋊C4—53no—
276C22⋊C4—53no—
277C22⋊C4—53no—
278C22⋊C4—56no—
279C22⋊C4—53no—
280C22⋊C4C22×D516yes—
281C22⋊C4C22×D511yescharacteristic
282C22⋊C4C22×D513yes—
283C42—51no—
284C42—53no—
285C42—53no—
286C42—53no—
287C42—56no—
288C42—53no—
289C42C22×D513yes—
290C2×C5—41no—
291C2×C5—43no—
292C2×C5C22×#4111yescharacteristic
293C2×C5#58511yescharacteristic
294C2×C5#58511yescharacteristic
295D5—43no—
296C23—53no—
297C23C23×D511yescharacteristic
298C23C23×D513yes—
299Q8—26no—
300Q8—203no—
301Q8#71114yes—
302D4—26no—
303D4—43no—
304D4—106no—
305D4—203no—
306D4—203no—
307D4—206no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—51no—
311C2×C4—53no—
312C2×C4—51no—
313C2×C4—53no—
314C2×C4—51no—
315C2×C4—53no—
316C2×C4—51no—
317C2×C4—106no—
318C2×C4—206no—
319C2×C4—203no—
320C2×C4—203no—
321C2×C4C23×D511yescharacteristic
322C2×C4C23×D513yes—
323C2×C4C23×D511yescharacteristic
324C2×C4C23×D513yes—
325C2×C4C23×D513yes—
326C5#352411yescharacteristic, complemented, Sylow
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—23no—
332C22—21no—
333C22—103no—
334C22—103no—
335C22—103no—
336C22C24×D511yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—101no—
342C4—103no—
343C4—101no—
344C4—101no—
345C4—101no—
346C4—103no—
347C4—103no—
348C4C2×#71116yes—
349C2—41no—
350C2—43no—
351C2—203no—
352C2C22×#71111yescharacteristic
353C2#6628311yescharacteristic
354C2#6628711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#548C2
hypercenter#548D5
hypocenterC5#3524
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. #41×Dic5C2semidirect
4. #66151C2semidirect
5. #66146C2semidirect
6. #66016C2semidirect
7. #66015C2semidirect
8. #65973C2semidirect
9. #65958C2semidirect
10. #65953C2semidirect
11. #65948C2semidirect
12. #65908C2semidirect
13. #65904C2semidirect
14. #65897C2semidirect
15. #65872C2semidirect
127. #548D5semidirect
326. C5#3524semidirect

Maximal subgroups

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

subgroupindex
C5×#5482
#661682
#41×Dic52
#661512
#661462
#660162
#660152
#659732
#659582
#659532
#659482
#659082
#659042
#658972
#658722
#35245