Finite Group Databasebeta

← #10613306

Subgroups — #10613306

Order 640.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66063C211yesmaximal, characteristic, complemented
2#66053C211yesmaximal, characteristic
3C5×#486C211yesmaximal, characteristic, complemented
4#65369C211yesmaximal, characteristic, complemented
5#65360C211yesmaximal, characteristic, complemented
6#65073C211yesmaximal, characteristic, complemented
7#65068C211yesmaximal, characteristic, complemented
8#65066C211yesmaximal, characteristic, complemented
9#65063C211yesmaximal, characteristic, complemented
10#65055C211yesmaximal, characteristic, complemented
11#65052C211yesmaximal, characteristic, complemented
12#65050C211yesmaximal, characteristic, complemented
13#65048C211yesmaximal, characteristic, complemented
14#64979C211yesmaximal, characteristic, complemented
15#64972C211yesmaximal, characteristic, complemented
16C5×C4⋊Q8C2211yescharacteristic
17C5×#127C2211yescharacteristic
18#4092—21no—
19#4090—21no—
20#4085—21no—
21#4085—21no—
22#4084C2211yescharacteristic
23#4084C2211yescharacteristic
24#4081—21no—
25#4081—21no—
26D4×Dic5C2211yescharacteristic
27D4×Dic5C2211yescharacteristic
28#4042—21no—
29#4038C2211yescharacteristic
30D5×C4⋊C4C2211yescharacteristic
31#4036C2211yescharacteristic
32#4035—21no—
33#4034C2211yescharacteristic
34#4018C2211yescharacteristic
35#4016C2211yescharacteristic
36#4016C2211yescharacteristic
37C5×#102C2211yescharacteristic
38C5×#102C2211yescharacteristic
39C5×#102C2211yescharacteristic
40C5×#102C2211yescharacteristic
41C4×C5×C8C2211yescharacteristic
42#3964C2211yescharacteristic
43#3964C2211yescharacteristic
44#3964C2211yescharacteristic
45#3964C2211yescharacteristic
46#3952C2211yescharacteristic
47#3952C2211yescharacteristic
48#3950C2211yescharacteristic
49#3950C2211yescharacteristic
50#3949C2211yescharacteristic
51#3949C2211yescharacteristic
52#3946C2211yescharacteristic
53#3946C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3939C2211yescharacteristic
57#3939C2211yescharacteristic
58#3935C2211yescharacteristic
59#3241—51nomaximal, Sylow
60C2×C5×Q8—21no—
61C2×C5×D4—21no—
62C2×C5×D4—21no—
63C2×C5×D4C2311yescharacteristic
64C2×C5×D4C2311yescharacteristic
65C2×C22⋊D5—21no—
66C2×C22⋊D5—21no—
67C22×Dic5—21no—
68C22×Dic5—21no—
69C2×C4×D5—21no—
70C2×C4×D5C2311yescharacteristic
71C2×Dic10—21no—
72C2×C5×C8C2311yescharacteristic
73C2×C5×C8C2311yescharacteristic
74C5×C4⋊C4—21no—
75C5×C4⋊C4C2311yescharacteristic
76C5×C4⋊C4C2311yescharacteristic
77C42×C5C2311yescharacteristic
78C22⋊Dic5—21no—
79C22⋊Dic5—21no—
80C22⋊Dic5—21no—
81C22⋊Dic5—21no—
82#685—21no—
83#685D411yescharacteristic
84#685D411yescharacteristic
85C4⋊Dic5—21no—
86C4⋊Dic5C2311yescharacteristic
87C4⋊Dic5C2311yescharacteristic
88C4⋊Dic5C2311yescharacteristic
89C4⋊Dic5C2311yescharacteristic
90#683—21no—
91#683—21no—
92#683—21no—
93#683—21no—
94#683—21no—
95#683—21no—
96#683—21no—
97#683D411yescharacteristic
98#683D411yescharacteristic
99C4×Dic5—21no—
100C4×Dic5C2311yescharacteristic
101C2×#168C2311yescharacteristic
102C2×#168C2311yescharacteristic
103#544—51no—
104#540—51no—
105#501—51no—
106#499—51no—
107#490—51no—
108#486D511yescharacteristic, complemented
109#482—51no—
110#482—51no—
111#481—51no—
112#480—51no—
113#478—51no—
114#478—51no—
115#476—51no—
116#474—51no—
117#433—51no—
118C23×C5—21no—
119C23×C5—21no—
120C22×D5C2×D411yescharacteristic
121C5×Q8—81no—
122C5×D4—21no—
123C5×D4—21no—
124C5×D4—21no—
125C5×D4—21no—
126C5×D4—81no—
127C5×D4—81no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5C2×D411yescharacteristic
131C2×C4×C5C2×D411yescharacteristic
132C2×C4×C5C2411yescharacteristic
133C22⋊D5—81no—
134C22⋊D5—81no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—81no—
140C2×Dic5—81no—
141C2×Dic5C2×D411yescharacteristic
142C2×Dic5C2×D411yescharacteristic
143C2×Dic5C2×D411yescharacteristic
144C4×D5—21no—
145C4×D5—21no—
146C4×D5—81no—
147Dic10—81no—
148C5×C8—21no—
149C5×C8—21no—
150#168—21no—
151#168—21no—
152C4⋊Q8—51no—
153C4⋊Q8—101no—
154C4⋊Q8C2×D511yescharacteristic
155#127C2×D511yescharacteristic
156#125—51no—
157#125—51no—
158#125—51no—
159#125—101no—
160#123—101no—
161#123—101no—
162C22⋊Q8—101no—
163C22⋊Q8—101no—
164#121—51no—
165#121—51no—
166#121—101no—
167#121—101no—
168C4×D4—51no—
169C4×D4—51no—
170#117—51no—
171#117—51no—
172C2×C4⋊C4—51no—
173#107—51no—
174#107—51no—
175#107—51no—
176#107—51no—
177#106—51no—
178#106—51no—
179#106—51no—
180#106—51no—
181C4⋊C8—51no—
182C4⋊C8—51no—
183#102—51no—
184#102—51no—
185#102—51no—
186#102—51no—
187#102C2×D511yescharacteristic
188#102C2×D511yescharacteristic
189#102C2×D511yescharacteristic
190#102C2×D511yescharacteristic
191C22⋊C8—51no—
192C22⋊C8—51no—
193#97—51no—
194C4×C8C2×D511yescharacteristic
195C22×C5—41no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—41no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5C22×D411yescharacteristic
202C2×D5—21no—
203C2×D5—21no—
204C2×D5—21no—
205C4×C5—21no—
206C4×C5—21no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C52−1+411yescharacteristic
210C4×C52−1+411yescharacteristic
211Dic5—21no—
212Dic5—21no—
213Dic5—21no—
214Dic5—41no—
215Dic5—41no—
216Dic5—41no—
217Dic5—41no—
218C2×Q8—21no—
219C2×Q8—101no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—101no—
223C2×D4—101no—
224C2×D4C22×D511yescharacteristic
225C2×D4C22×D511yescharacteristic
226C22×C4—51no—
227C22×C4—101no—
228C22×C4—101no—
229C22×C4—101no—
230C2×C8—51no—
231C2×C8—51no—
232C2×C8C22×D511yescharacteristic
233C2×C8C22×D511yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—51no—
236C4⋊C4—51no—
237C4⋊C4—51no—
238C4⋊C4—51no—
239C4⋊C4—51no—
240C4⋊C4—51no—
241C4⋊C4—101no—
242C4⋊C4—101no—
243C4⋊C4—101no—
244C4⋊C4—101no—
245C4⋊C4—101no—
246C4⋊C4—101no—
247C4⋊C4—101no—
248C4⋊C4—101no—
249C4⋊C4C22×D511yescharacteristic
250C4⋊C4C22×D511yescharacteristic
251C22⋊C4—51no—
252C22⋊C4—51no—
253C22⋊C4—101no—
254C22⋊C4—101no—
255C22⋊C4—101no—
256C22⋊C4—101no—
257C22⋊C4—101no—
258C42—51no—
259C42—101no—
260C42C22×D511yescharacteristic
261C2×C5—81no—
262C2×C5—81no—
263C2×C5#53611yescharacteristic
264C2×C5#57611yescharacteristic
265C2×C5#57711yescharacteristic
266D5—41no—
267C23—21no—
268C23—21no—
269C23—51no—
270Q8—81no—
271Q8—401no—
272D4—21no—
273D4—21no—
274D4—21no—
275D4—21no—
276D4—81no—
277D4—81no—
278D4—401no—
279D4—401no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—51no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—101no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—101no—
290C2×C4—101no—
291C2×C4—401no—
292C2×C4—401no—
293C2×C4—401no—
294C2×C4D4×D511yescharacteristic
295C2×C4D4×D511yescharacteristic
296C2×C4C23×D511yescharacteristic
297C8—21no—
298C8—21no—
299C8—101no—
300C8—101no—
301C5#324111yescharacteristic, complemented, Sylow
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—101no—
309C22—101no—
310C22—101no—
311C22C2×D4×D511yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—101no—
317C4—101no—
318C4—101no—
319C4—201no—
320C4—201no—
321C4—201no—
322C4—201no—
323C4#415011yescharacteristic
324C4#415011yescharacteristic
325C2—81no—
326C2—81no—
327C2—201no—
328C2#6600911yescharacteristic
329C2#6609111yescharacteristic
330C2#6609611yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#486C2
socleC22×C5C22×D4
prime radical#66053C2
hypercenter#486D5
hypocenterC5#3241
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #66063C2semidirect
3. C5×#486C2semidirect
4. #65369C2semidirect
5. #65360C2semidirect
6. #65073C2semidirect
7. #65068C2semidirect
8. #65066C2semidirect
9. #65063C2semidirect
10. #65055C2semidirect
11. #65052C2semidirect
12. #65050C2semidirect
13. #65048C2semidirect
14. #64979C2semidirect
15. #64972C2semidirect
108. #486D5semidirect
301. C5#3241semidirect

Maximal subgroups

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

subgroupindex
#660632
#660532
C5×#4862
#653692
#653602
#650732
#650682
#650662
#650632
#650552
#650522
#650502
#650482
#649792
#649722
#32415