Finite Group Databasebeta

← #10615208

Subgroups — #10615208

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
1C5×#544C211yesmaximal, characteristic, complemented
2#65869C211yesmaximal, characteristic
3#65377C211yesmaximal, characteristic, complemented
4#65376C211yesmaximal, characteristic, complemented
5#65373C211yesmaximal, characteristic, complemented
6#65369C211yesmaximal, characteristic, complemented
7#65356C211yesmaximal, characteristic, complemented
8#65355C211yesmaximal, characteristic, complemented
9#65355C211yesmaximal, characteristic, complemented
10#65353C211yesmaximal, characteristic, complemented
11#65292C211yesmaximal, characteristic, complemented
12#65292C211yesmaximal, characteristic, complemented
13#65289C211yesmaximal, characteristic, complemented
14#65257C211yesmaximal, characteristic, complemented
15#65256C211yesmaximal, characteristic, complemented
16C5×C4⋊Q8—21no—
17C5×C4⋊Q8C2211yescharacteristic
18C5×C4⋊Q8C2211yescharacteristic
19C5×#125—21no—
20C5×#125C2211yescharacteristic
21C5×#125C2211yescharacteristic
22C5×C22⋊Q8—21no—
23C5×C22⋊Q8—21no—
24C5×#117C2211yescharacteristic
25C5×#117C2211yescharacteristic
26C2×C5×C4⋊C4C2211yescharacteristic
27#4076C2211yescharacteristic
28#4076C2211yescharacteristic
29#4040—21no—
30#4040—21no—
31#4030—21no—
32#4030—21no—
33#4020C2211yescharacteristic
34C4×D20C2211yescharacteristic
35C4×D20C2211yescharacteristic
36#4016C2211yescharacteristic
37C22⋊#168C2211yescharacteristic
38C22⋊#168C2211yescharacteristic
39#3941C2211yescharacteristic
40#3941C2211yescharacteristic
41#3941C2211yescharacteristic
42#3941C2211yescharacteristic
43#3941C2211yescharacteristic
44#3941C2211yescharacteristic
45#3941C2211yescharacteristic
46#3941C2211yescharacteristic
47#3940C2211yescharacteristic
48#3940C2211yescharacteristic
49#3940C2211yescharacteristic
50#3940C2211yescharacteristic
51#3939C2211yescharacteristic
52#3939C2211yescharacteristic
53#3939C2211yescharacteristic
54#3939C2211yescharacteristic
55C4⋊#168C2211yescharacteristic
56C4⋊#168C2211yescharacteristic
57#3935C2211yescharacteristic
58C4×#168C2211yescharacteristic
59#3241—51nomaximal, Sylow
60C2×C5×Q8—21no—
61C2×C5×Q8—21no—
62C22×C4×C5—21no—
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—21no—
65C2×C22⋊D5—21no—
66C2×D20—21no—
67C2×D20—21no—
68C2×D20C2311yescharacteristic
69C2×D20C2311yescharacteristic
70C2×C4×D5—21no—
71C2×C4×D5—21no—
72C5×C4⋊C4—21no—
73C5×C4⋊C4—21no—
74C5×C4⋊C4—21no—
75C5×C4⋊C4—21no—
76C5×C4⋊C4—21no—
77C5×C4⋊C4—21no—
78C5×C4⋊C4—21no—
79C5×C4⋊C4D411yescharacteristic
80C5×C4⋊C4D411yescharacteristic
81C5×C4⋊C4C2311yescharacteristic
82C5×C4⋊C4C2311yescharacteristic
83C5×C4⋊C4C2311yescharacteristic
84C5×C4⋊C4C2311yescharacteristic
85C5×C22⋊C4—21no—
86C5×C22⋊C4D411yescharacteristic
87C5×C22⋊C4D411yescharacteristic
88C42×C5—21no—
89C42×C5C2311yescharacteristic
90C42×C5C2311yescharacteristic
91#685—21no—
92#685—21no—
93#685—21no—
94#685—21no—
95C4⋊Dic5C2311yescharacteristic
96C4⋊Dic5C2311yescharacteristic
97#683—21no—
98#683—21no—
99C2×#168C2311yescharacteristic
100C2×#168C2311yescharacteristic
101C2×#168C2311yescharacteristic
102C2×#168C2311yescharacteristic
103#544D511yescharacteristic, complemented
104#540—51no—
105#501—51no—
106#499—51no—
107#490—51no—
108#486—51no—
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×C5C2×D411yescharacteristic
119C22×D5—21no—
120C22×D5—21no—
121C5×Q8—81no—
122C5×Q8—81no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—81no—
130C2×C4×C5C2×D411yescharacteristic
131C2×C4×C5C2×D411yescharacteristic
132C2×C4×C5C2×D411yescharacteristic
133C2×C4×C5C2×D411yescharacteristic
134C2×C4×C5C2×D411yescharacteristic
135C2×C4×C5C2411yescharacteristic
136C22⋊D5—81no—
137C22⋊D5—81no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140D20—21no—
141D20—21no—
142D20—21no—
143D20—21no—
144D20—81no—
145D20—81no—
146C4×D5—81no—
147C4×D5—81no—
148#168—21no—
149#168—21no—
150#168—21no—
151#168—21no—
152C4⋊Q8—21no—
153C4⋊Q8C2×D511yescharacteristic
154C4⋊Q8C2×D511yescharacteristic
155#127—51no—
156#125—21no—
157#125—51no—
158#125C2×D511yescharacteristic
159#125C2×D511yescharacteristic
160#123—101no—
161#123—101no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164#121—51no—
165#121—51no—
166#121—101no—
167#121—101no—
168C4×D4—51no—
169C4×D4—51no—
170#117C2×D511yescharacteristic
171#117C2×D511yescharacteristic
172C2×C4⋊C4C2×D511yescharacteristic
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#102—51no—
188#102—51no—
189#102—51no—
190#102—51no—
191C22⋊C8—51no—
192C22⋊C8—51no—
193#97—51no—
194C4×C8—51no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5C22×D411yescharacteristic
199C2×D5—41no—
200C2×D5—41no—
201C2×D5—41no—
202C2×D5—41no—
203C2×D5—41no—
204C2×D5—41no—
205C4×C5—21no—
206C4×C5—21no—
207C4×C5—21no—
208C4×C5—21no—
209C4×C5—21no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C52−1+411yescharacteristic
215C4×C52−1+411yescharacteristic
216Dic5—41no—
217Dic5—41no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×D4—51no—
221C2×D4—51no—
222C2×D4—101no—
223C2×D4—101no—
224C2×D4—101no—
225C2×D4—101no—
226C22×C4—21no—
227C22×C4—101no—
228C22×C4—101no—
229C22×C4C22×D511yescharacteristic
230C2×C8—51no—
231C2×C8—51no—
232C2×C8—51no—
233C2×C8—51no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—51no—
242C4⋊C4—51no—
243C4⋊C4—101no—
244C4⋊C4—101no—
245C4⋊C4C22⋊D511yescharacteristic
246C4⋊C4C22⋊D511yescharacteristic
247C4⋊C4C22×D511yescharacteristic
248C4⋊C4C22×D511yescharacteristic
249C4⋊C4C22×D511yescharacteristic
250C4⋊C4C22×D511yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—101no—
253C22⋊C4—101no—
254C22⋊C4—101no—
255C22⋊C4—101no—
256C22⋊C4C22⋊D511yescharacteristic
257C22⋊C4C22⋊D511yescharacteristic
258C42—21no—
259C42C22×D511yescharacteristic
260C42C22×D511yescharacteristic
261C2×C5—41no—
262C2×C5#53611yescharacteristic
263C2×C5#57611yescharacteristic
264C2×C5#57711yescharacteristic
265D5—81no—
266D5—81no—
267C23—101no—
268C23—101no—
269C23C2×C22⋊D511yescharacteristic
270Q8—81no—
271Q8—81no—
272D4—101no—
273D4—101no—
274D4—101no—
275D4—101no—
276D4—401no—
277D4—401no—
278D4—401no—
279D4—401no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—401no—
290C2×C4—401no—
291C2×C4C2×C22⋊D511yescharacteristic
292C2×C4C2×C22⋊D511yescharacteristic
293C2×C4C2×C22⋊D511yescharacteristic
294C2×C4C2×C22⋊D511yescharacteristic
295C2×C4C2×C22⋊D511yescharacteristic
296C2×C4C23×D511yescharacteristic
297C8—101no—
298C8—101no—
299C8—101no—
300C8—101no—
301C5#324111yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—21no—
305C22—201no—
306C22—201no—
307C22—201no—
308C22—201no—
309C22—201no—
310C22—201no—
311C22C22×C22⋊D511yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—201no—
322C4—201no—
323C4#414711yescharacteristic
324C4#414711yescharacteristic
325C2—41no—
326C2—401no—
327C2—401no—
328C2#6615011yescharacteristic
329C2#6616911yescharacteristic
330C2#6617111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#544C2
socleC22×C5C22×D4
prime radical#65869C2
hypercenter#544D5
hypocenterC5#3241
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
1. C5×#544C2semidirect
3. #65377C2semidirect
4. #65376C2semidirect
5. #65373C2semidirect
6. #65369C2semidirect
7. #65356C2semidirect
8. #65355C2semidirect
9. #65355C2semidirect
10. #65353C2semidirect
11. #65292C2semidirect
12. #65292C2semidirect
13. #65289C2semidirect
14. #65257C2semidirect
15. #65256C2semidirect
103. #544D5semidirect
301. C5#3241semidirect

Maximal subgroups

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

subgroupindex
C5×#5442
#658692
#653772
#653762
#653732
#653692
#653562
#653552
#653532
#652922
#652892
#652572
#652562
#32415