Finite Group Databasebeta

← #10613211

Subgroups — #10613211

Order 640.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65973C211yesmaximal, characteristic
2#65847C211yesmaximal, characteristic
3C5×#484C211yesmaximal, characteristic, complemented
4#65341C211yesmaximal, characteristic, complemented
5#65253C211yesmaximal, characteristic, complemented
6#65173C211yesmaximal, characteristic, complemented
7#65170C211yesmaximal, characteristic, complemented
8#65167C211yesmaximal, characteristic, complemented
9#65166C211yesmaximal, characteristic, complemented
10#65105C211yesmaximal, characteristic, complemented
11#65103C211yesmaximal, characteristic, complemented
12#65088C211yesmaximal, characteristic, complemented
13#65079C211yesmaximal, characteristic, complemented
14#65018C211yesmaximal, characteristic
15#65012C211yesmaximal, characteristic, complemented
16C5×C22⋊Q8C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18#4092C2211yescharacteristic
19Q8×Dic5C2211yescharacteristic
20C2×#689C2211yescharacteristic
21#4075—21no—
22C4×C22⋊D5C2211yescharacteristic
23#4072—21no—
24#4072C2211yescharacteristic
25#4070C2211yescharacteristic
26C2×#681C2211yescharacteristic
27C2×#676C2211yescharacteristic
28#4044—21no—
29#4043C2211yescharacteristic
30#4038—21no—
31#4038C2211yescharacteristic
32D5×C4⋊C4C2211yescharacteristic
33#4036C2211yescharacteristic
34#4036C2211yescharacteristic
35#4035—21no—
36#4035—21no—
37#4033C2211yescharacteristic
38#4029—21no—
39#4025—21no—
40C5×#107C2211yescharacteristic
41C5×#107C2211yescharacteristic
42C5×#103C2211yescharacteristic
43C5×#103C2211yescharacteristic
44C5×C22⋊C8C2211yescharacteristic
45#3967C2211yescharacteristic
46#3952C2211yescharacteristic
47#3950C2211yescharacteristic
48#3949C2211yescharacteristic
49#3948C2211yescharacteristic
50#3948C2211yescharacteristic
51#3946C2211yescharacteristic
52C8×Dic5C2211yescharacteristic
53#3942C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3940C2211yescharacteristic
57#3939C2211yescharacteristic
58#3939C2211yescharacteristic
59#3352—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C22×C4×C5—21no—
62C22×C4×C5C2311yescharacteristic
63C2×C22⋊D5D411yescharacteristic
64C2×C4×D5—21no—
65C2×C4×D5C2311yescharacteristic
66C2×Dic10C2311yescharacteristic
67C2×C5×C8C2311yescharacteristic
68C2×C5×C8C2311yescharacteristic
69C5×C4⋊C4—21no—
70C5×C4⋊C4—21no—
71C5×C4⋊C4C2311yescharacteristic
72C5×C4⋊C4C2311yescharacteristic
73C5×C4⋊C4C2311yescharacteristic
74C5×C22⋊C4—21no—
75C22⋊Dic5—21no—
76C22⋊Dic5—21no—
77C22⋊Dic5D411yescharacteristic
78#689—22no—
79#685—21no—
80#685—21no—
81#685—21no—
82#685D411yescharacteristic
83C4⋊Dic5—21no—
84C4⋊Dic5—21no—
85C4⋊Dic5—21no—
86C4⋊Dic5—21no—
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#683D411yescharacteristic
97C4×Dic5—21no—
98C4×Dic5—21no—
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5C2311yescharacteristic
102#681—22no—
103C2×#168C2311yescharacteristic
104C2×#168C2311yescharacteristic
105#676—22no—
106#552—51no—
107#552—51no—
108#499—51no—
109#485—51no—
110#485—51no—
111#484—51no—
112#484D511yescharacteristic, complemented
113#479—51no—
114#479—51no—
115#470—51no—
116#470—51no—
117C4×Q16—51no—
118#436—51no—
119#428—51no—
120#428—51no—
121C23×C5C2×D411yescharacteristic
122C22×D5C2×D411yescharacteristic
123C5×Q8—21no—
124C5×Q8—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—42no—
132C2×C4×C5C2×D411yescharacteristic
133C2×C4×C5C2411yescharacteristic
134C22⋊D5—41no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5C2×D411yescharacteristic
140C2×Dic5C2×D411yescharacteristic
141C2×Dic5C2×D411yescharacteristic
142C4×D5—21no—
143C4×D5—21no—
144C4×D5—42no—
145Dic10—21no—
146Dic10—21no—
147C5×C8—21no—
148C5×C8—21no—
149#168—21no—
150#168#4112yes—
151C2×Q16—51no—
152C2×M16—51no—
153C2×M16—51no—
154#126—101no—
155#126—101no—
156#125—51no—
157#125—51no—
158#125—101no—
159#125—101no—
160#123—101no—
161#123—101no—
162C22⋊Q8—51no—
163C22⋊Q8—51no—
164C22⋊Q8—51no—
165C22⋊Q8C2×D511yescharacteristic
166C4×Q8—51no—
167C4×Q8—51no—
168C4×D4—51no—
169#117—51no—
170#117—51no—
171#117—101no—
172#117—101no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4C2×D511yescharacteristic
175#107—51no—
176#107—51no—
177#107—51no—
178#107C2×D511yescharacteristic
179#107C2×D511yescharacteristic
180#106—51no—
181#106—51no—
182#106—51no—
183#106—51no—
184C4⋊C8—51no—
185#103—51no—
186#103—51no—
187#103—51no—
188#103—51no—
189#103C2×D511yescharacteristic
190#103C2×D511yescharacteristic
191C22⋊C8—51no—
192C22⋊C8C2×D511yescharacteristic
193C4×C8—51no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5C22×D411yescharacteristic
198C2×D5—21no—
199C2×D5—21no—
200C2×D5—21no—
201C4×C5—21no—
202C4×C5—41no—
203C4×C5—41no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5C2×#4111yescharacteristic
207C4×C52−1+411yescharacteristic
208Dic5—21no—
209Dic5—21no—
210Dic5—21no—
211Dic5—41no—
212Dic5—41no—
213Dic5—41no—
214Dic5—41no—
215C2×Q8—51no—
216C2×Q8C22×D511yescharacteristic
217C2×D4—51no—
218C22×C4—21no—
219C22×C4—51no—
220C22×C4—101no—
221C22×C4C22×D511yescharacteristic
222Q16—102no—
223M16—102no—
224M16—102no—
225C2×C8—51no—
226C2×C8—51no—
227C2×C8C22×D511yescharacteristic
228C2×C8C22×D511yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—51no—
232C4⋊C4—51no—
233C4⋊C4—51no—
234C4⋊C4—51no—
235C4⋊C4—101no—
236C4⋊C4—101no—
237C4⋊C4—101no—
238C4⋊C4—101no—
239C4⋊C4—101no—
240C4⋊C4—101no—
241C4⋊C4—101no—
242C4⋊C4—101no—
243C4⋊C4—101no—
244C4⋊C4—101no—
245C4⋊C4C22×D511yescharacteristic
246C4⋊C4C22×D511yescharacteristic
247C4⋊C4C22×D511yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—51no—
250C22⋊C4—51no—
251C22⋊C4—101no—
252C22⋊C4—101no—
253C22⋊C4—101no—
254C22⋊C4—101no—
255C22⋊C4—101no—
256C42—51no—
257C42—101no—
258C42—101no—
259C42—101no—
260C42—101no—
261C2×C5—41no—
262C2×C5#54711yescharacteristic
263C2×C5#57511yescharacteristic
264C2×C5#57811yescharacteristic
265D5—41no—
266C23—51no—
267C23D4×D511yescharacteristic
268Q8—21no—
269Q8—21no—
270Q8—101no—
271Q8—101no—
272D4—201no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—51no—
281C2×C4—51no—
282C2×C4—51no—
283C2×C4—101no—
284C2×C4—101no—
285C2×C4—101no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—202no—
290C2×C4D4×D511yescharacteristic
291C2×C4C23×D511yescharacteristic
292C8—21no—
293C8—21no—
294C8—52no—
295C8—101no—
296C5#335211yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—21no—
300C22—101no—
301C22—101no—
302C22—101no—
303C22C2×D4×D511yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—101no—
310C4—101no—
311C4—101no—
312C4—201no—
313C4—201no—
314C4—201no—
315C4—201no—
316C4D5×#4111yescharacteristic
317C4#415011yescharacteristic
318C2—41no—
319C2—201no—
320C2#6598911yescharacteristic
321C2#6610011yescharacteristic
322C2#6610711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#484C2
socleC22×C5C22×D4
prime radicalC2×C22⋊D5D4
hypercenter#484D5
hypocenterC5#3352
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. C5×#484C2semidirect
4. #65341C2semidirect
5. #65253C2semidirect
6. #65173C2semidirect
7. #65170C2semidirect
8. #65167C2semidirect
9. #65166C2semidirect
10. #65105C2semidirect
11. #65103C2semidirect
12. #65088C2semidirect
13. #65079C2semidirect
15. #65012C2semidirect
112. #484D5semidirect
296. C5#3352semidirect

Maximal subgroups

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

subgroupindex
#659732
#658472
C5×#4842
#653412
#652532
#651732
#651702
#651672
#651662
#651052
#651032
#650882
#650792
#650182
#650122
#33525