Finite Group Databasebeta

← #10618621

Subgroups — #10618621

Order 640.

1010 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×#548C211yesmaximal, characteristic, complemented
2#66165C211yesmaximal, characteristic, complemented
3#41×Dic5C211yesmaximal, characteristic, complemented
4Q8×C22⋊D5C211yesmaximal, characteristic, complemented
5#66146C211yesmaximal, characteristic, complemented
6#66016C213yesmaximal, complemented
7#66005C213yesmaximal, complemented
8#65975C213yesmaximal, complemented
9#65956C213yesmaximal, complemented
10#65950C213yesmaximal, complemented
11#65937C213yesmaximal, complemented
12#65906C211yesmaximal, characteristic, complemented
13Q8×Dic10C211yesmaximal, characteristic, complemented
14#65876C213yesmaximal, complemented
15#65872C213yesmaximal, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C5×#124C2213yes—
20C5×C22⋊Q8C2213yes—
21C5×#121C2213yes—
22C4×C5×Q8C2211yescharacteristic
23C4×C5×D4C2213yes—
24#4092C2213yes—
25Q8×Dic5C2211yescharacteristic
26Q8×Dic5C2211yescharacteristic
27Q8×Dic5C2213yes—
28Q8×Dic5C2213yes—
29Q8×Dic5C2211yescharacteristic
30#4090C2213yes—
31#4085C2213yes—
32#4084C2213yes—
33#4082C2216yes—
34#4082C2213yes—
35#4081C2213yes—
36D4×Dic5C2213yes—
37D4×Dic5C2213yes—
38C4×C22⋊D5C2213yes—
39C4×C22⋊D5C2213yes—
40#4072C2213yes—
41#4072C2213yes—
42#4072C2213yes—
43#4070C2213yes—
44#4070C2213yes—
45C2×C4×Dic5C2213yes—
46#4044C2213yes—
47#4038C2213yes—
48#4035C2213yes—
49#4034C2213yes—
50#4034C2213yes—
51#4033C2216yes—
52#4033C2213yes—
53#4032C2213yes—
54#4031C2211yescharacteristic
55#4031C2216yes—
56#4025C2213yes—
57#4025C2216yes—
58#4025C2216yes—
59#4024C2213yes—
60#4024C2216yes—
61#4023C2216yes—
62#4018C2213yes—
63#4015C2213yes—
64C4×Dic10C2213yes—
65C4×Dic10C2213yes—
66#3518—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⋊D5C2311yescharacteristic
76C2×C22⋊D5C2313yes—
77C22×Dic5C2313yes—
78C22×Dic5C2313yes—
79Q8×D5—24no—
80C2×C4×D5C2313yes—
81C2×Dic10C2313yes—
82C2×Dic10C2311yescharacteristic
83C2×Dic10C2316yes—
84C5×C4⋊C4C2313yes—
85C5×C4⋊C4C2313yes—
86C5×C22⋊C4C2311yescharacteristic
87C5×C22⋊C4C2313yes—
88C5×C22⋊C4C2316yes—
89C42×C5C2313yes—
90C22⋊Dic5C2316yes—
91C22⋊Dic5C2313yes—
92C22⋊Dic5C2316yes—
93C22⋊Dic5C2313yes—
94C22⋊Dic5C2313yes—
95C22⋊Dic5C2311yescharacteristic
96#685C2311yescharacteristic
97#685C2313yes—
98#685C2313yes—
99#685C2311yescharacteristic
100C4⋊Dic5C2313yes—
101C4⋊Dic5C2313yes—
102C4⋊Dic5C2313yes—
103C4⋊Dic5C2313yes—
104C4⋊Dic5C2313yes—
105C4⋊Dic5C2313yes—
106#683C2313yes—
107#683C2313yes—
108#683C2313yes—
109#683C2313yes—
110#683C2313yes—
111#683C2316yes—
112#683C2313yes—
113C4×Dic5C2316yes—
114C4×Dic5C2313yes—
115C4×Dic5C2313yes—
116C4×Dic5C2313yes—
117C4×Dic5C2316yes—
118C4×Dic5C2313yes—
119C4×Dic5C2311yescharacteristic
120Q82—51no—
121#556—53no—
122#556—53no—
123#556—53no—
124#555—53no—
125D4×Q8—51no—
126#548—51no—
127#548D511yescharacteristic, complemented
128#542—53no—
129#542—53no—
130#533—53no—
131#529—53no—
132#519—51no—
133#519—51no—
134C4×#41—51no—
135C23×C5C2411yescharacteristic
136C23×C5C2413yes—
137C22×D5C2411yescharacteristic
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—22no—
150C22⋊D5—43no—
151C2×Dic5—26no—
152C2×Dic5—43no—
153C2×Dic5C2413yes—
154C2×Dic5C2411yescharacteristic
155C2×Dic5C2411yescharacteristic
156C2×Dic5C2413yes—
157C2×Dic5C2413yes—
158C2×Dic5C2413yes—
159C2×Dic5C2411yescharacteristic
160C4×D5—26no—
161Dic10—26no—
162Dic10—22no—
163Dic10—46no—
164C2×#41C2×D511yescharacteristic
165C22×Q8—51no—
166C22×Q8C2×D511yescharacteristic
167C4⋊Q8—53no—
168C4⋊Q8—53no—
169C4⋊Q8—53no—
170C4⋊Q8—53no—
171#126—53no—
172#126—53no—
173#126—56no—
174#126—56no—
175#125—53no—
176#124—53no—
177#124—56no—
178#124—51no—
179#124—56no—
180#124C2×D513yes—
181#123—53no—
182#123—53no—
183C22⋊Q8—53no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8—53no—
187C22⋊Q8—56no—
188C22⋊Q8C2×D513yes—
189#121—53no—
190#121—53no—
191#121C2×D513yes—
192C4×Q8—51no—
193C4×Q8—51no—
194C4×Q8—53no—
195C4×Q8—51no—
196C4×Q8—53no—
197C4×Q8—53no—
198C4×Q8—56no—
199C4×Q8—53no—
200C4×Q8—53no—
201C4×Q8C2×D511yescharacteristic
202C4×D4—53no—
203C4×D4—53no—
204C4×D4—53no—
205C4×D4—53no—
206C4×D4C2×D513yes—
207#117—53no—
208#117—53no—
209#117—53no—
210#117—53no—
211#117—56no—
212#117—53no—
213C2×C42—53no—
214C22×C5—21no—
215C22×C5—23no—
216C22×C5—23no—
217C22×C5—21no—
218C22×C5—23no—
219C22×C5—21no—
220C22×C5C2511yescharacteristic
221C2×D5—21no—
222C2×D5—21no—
223C2×D5—21no—
224C4×C5—21no—
225C4×C5—23no—
226C4×C5—21no—
227C4×C5—23no—
228C4×C5C2×#4116yes—
229Dic5—23no—
230Dic5—21no—
231Dic5—23no—
232Dic5—23no—
233Dic5—23no—
234Dic5—21no—
235Dic52−1+412yes—
236#41—24no—
237C2×Q8—24no—
238C2×Q8—51no—
239C2×Q8—53no—
240C2×Q8—56no—
241C2×Q8—104no—
242C2×Q8C22×D513yes—
243C2×Q8C22×D511yescharacteristic
244C2×D4—51no—
245C2×D4—53no—
246C2×D4C22×D513yes—
247C2×D4C22×D513yes—
248C22×C4—53no—
249C22×C4—53no—
250C22×C4—53no—
251C22×C4C22×D513yes—
252C22×C4C22×D513yes—
253C4⋊C4—53no—
254C4⋊C4—53no—
255C4⋊C4—53no—
256C4⋊C4—53no—
257C4⋊C4—53no—
258C4⋊C4—53no—
259C4⋊C4—53no—
260C4⋊C4—53no—
261C4⋊C4—53no—
262C4⋊C4—53no—
263C4⋊C4—56no—
264C4⋊C4—53no—
265C4⋊C4—53no—
266C4⋊C4C22×D513yes—
267C4⋊C4C22×D513yes—
268C22⋊C4—53no—
269C22⋊C4—51no—
270C22⋊C4—56no—
271C22⋊C4—56no—
272C22⋊C4—53no—
273C22⋊C4—53no—
274C22⋊C4—53no—
275C22⋊C4—51no—
276C22⋊C4—51no—
277C22⋊C4—53no—
278C22⋊C4C22×D516yes—
279C22⋊C4C22×D511yescharacteristic
280C22⋊C4C22×D513yes—
281C42—51no—
282C42—53no—
283C42—53no—
284C42—53no—
285C42—56no—
286C42—56no—
287C42—53no—
288C42C22×D513yes—
289C2×C5—41no—
290C2×C5—43no—
291C2×C5C22×#4111yescharacteristic
292C2×C5C2×2−1+411yescharacteristic
293C2×C5#58511yescharacteristic
294D5—41no—
295C23—51no—
296C23C23×D511yescharacteristic
297C23C23×D513yes—
298Q8—26no—
299Q8—102no—
300Q8—106no—
301Q8—206no—
302Q8#71114yes—
303D4—26no—
304D4—43no—
305D4—102no—
306D4—203no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—51no—
310C2×C4—53no—
311C2×C4—51no—
312C2×C4—51no—
313C2×C4—53no—
314C2×C4—53no—
315C2×C4—53no—
316C2×C4—106no—
317C2×C4—106no—
318C2×C4—203no—
319C2×C4C23×D511yescharacteristic
320C2×C4C23×D513yes—
321C2×C4C23×D513yes—
322C2×C4C23×D511yescharacteristic
323C2×C4C23×D513yes—
324C5#351811yescharacteristic, complemented, Sylow
325C22—21no—
326C22—23no—
327C22—23no—
328C22—21no—
329C22—23no—
330C22—21no—
331C22—101no—
332C22—101no—
333C22—101no—
334C22C24×D511yescharacteristic
335C4—21no—
336C4—23no—
337C4—23no—
338C4—21no—
339C4—52no—
340C4—103no—
341C4—101no—
342C4—103no—
343C4—103no—
344C4—103no—
345C4—101no—
346C4C2×#71116yes—
347C2—41no—
348C2—43no—
349C2—201no—
350C2C22×#71111yescharacteristic
351C2#6628311yescharacteristic
352C2D5×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#548C2
hypercenter#548D5
hypocenterC5#3518
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. #66146C2semidirect
6. #66016C2semidirect
7. #66005C2semidirect
8. #65975C2semidirect
9. #65956C2semidirect
10. #65950C2semidirect
11. #65937C2semidirect
12. #65906C2semidirect
13. Q8×Dic10C2semidirect
14. #65876C2semidirect
15. #65872C2semidirect
127. #548D5semidirect
324. C5#3518semidirect

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
#661462
#660162
#660052
#659752
#659562
#659502
#659372
#659062
Q8×Dic102
#658762
#658722
#35185