Finite Group Databasebeta

← #10613717

Subgroups — #10613717

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#66032C211yesmaximal, characteristic
3C5×#501C211yesmaximal, characteristic, complemented
4#65377C211yesmaximal, characteristic, complemented
5#65353C211yesmaximal, characteristic, complemented
6#65180C211yesmaximal, characteristic, complemented
7#65179C211yesmaximal, characteristic, complemented
8#65174C211yesmaximal, characteristic, complemented
9#65171C211yesmaximal, characteristic, complemented
10#65160C211yesmaximal, characteristic, complemented
11#65159C211yesmaximal, characteristic, complemented
12#65155C211yesmaximal, characteristic, complemented
13#65152C211yesmaximal, characteristic, complemented
14#65002C211yesmaximal, characteristic, complemented
15#64998C211yesmaximal, characteristic, complemented
16C5×C4⋊Q8C2211yescharacteristic
17C5×#125C2211yescharacteristic
18#4092—21no—
19#4090—21no—
20#4042—21no—
21#4041—21no—
22#4041—21no—
23#4041C2211yescharacteristic
24#4041C2211yescharacteristic
25#4040—21no—
26#4040—21no—
27#4039C2211yescharacteristic
28#4039C2211yescharacteristic
29#4038C2211yescharacteristic
30D5×C4⋊C4C2211yescharacteristic
31#4036C2211yescharacteristic
32#4035—21no—
33#4034C2211yescharacteristic
34#4020C2211yescharacteristic
35#4018C2211yescharacteristic
36#4016C2211yescharacteristic
37C5×#107C2211yescharacteristic
38C5×#107C2211yescharacteristic
39C5×#106C2211yescharacteristic
40C5×#106C2211yescharacteristic
41C5×#97C2211yescharacteristic
42#3953C2211yescharacteristic
43#3953C2211yescharacteristic
44#3953C2211yescharacteristic
45#3953C2211yescharacteristic
46#3952C2211yescharacteristic
47#3952C2211yescharacteristic
48#3946C2211yescharacteristic
49#3946C2211yescharacteristic
50#3941C2211yescharacteristic
51#3941C2211yescharacteristic
52#3941C2211yescharacteristic
53#3941C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3939C2211yescharacteristic
57#3939C2211yescharacteristic
58C4×#168C2211yescharacteristic
59#3241—51nomaximal, Sylow
60C2×C5×Q8—21no—
61C2×D20—21no—
62C2×D20—21no—
63C2×D20—21no—
64C2×D20—21no—
65C2×D20C2311yescharacteristic
66C2×D20C2311yescharacteristic
67C2×C4×D5—21no—
68C2×C4×D5—21no—
69C2×C4×D5—21no—
70C2×C4×D5C2311yescharacteristic
71C2×Dic10—21no—
72C2×C5×C8C2311yescharacteristic
73C2×C5×C8C2311yescharacteristic
74C5×C4⋊C4—21no—
75C5×C4⋊C4—21no—
76C5×C4⋊C4—21no—
77C5×C4⋊C4C2311yescharacteristic
78C5×C4⋊C4C2311yescharacteristic
79C5×C4⋊C4C2311yescharacteristic
80C5×C4⋊C4C2311yescharacteristic
81C42×C5C2311yescharacteristic
82#685—21no—
83#685—21no—
84#685—21no—
85#685—21no—
86#685—21no—
87#685D411yescharacteristic
88#685D411yescharacteristic
89C4⋊Dic5—21no—
90C4⋊Dic5C2311yescharacteristic
91C4⋊Dic5C2311yescharacteristic
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#501D511yescharacteristic, complemented
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—
118C22×D5—21no—
119C22×D5—21no—
120C22×D5C2×D411yescharacteristic
121C5×Q8—81no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5C2×D411yescharacteristic
127C2×C4×C5C2×D411yescharacteristic
128C2×C4×C5C2411yescharacteristic
129C2×Dic5—21no—
130C2×Dic5—21no—
131C2×Dic5C2×D411yescharacteristic
132C2×Dic5C2×D411yescharacteristic
133C2×Dic5C2×D411yescharacteristic
134D20—21no—
135D20—21no—
136D20—21no—
137D20—21no—
138D20—81no—
139D20—81no—
140D20—81no—
141D20—81no—
142C4×D5—21no—
143C4×D5—21no—
144C4×D5—81no—
145C4×D5—81no—
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#127—51no—
156#125—51no—
157#125—51no—
158#125—101no—
159#125C2×D511yescharacteristic
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#107C2×D511yescharacteristic
176#107C2×D511yescharacteristic
177#106—51no—
178#106—51no—
179#106C2×D511yescharacteristic
180#106C2×D511yescharacteristic
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#97C2×D511yescharacteristic
194C4×C8—51no—
195C22×C5C22×D411yescharacteristic
196C2×D5—21no—
197C2×D5—21no—
198C2×D5—21no—
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—41no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C52−1+411yescharacteristic
212C4×C52−1+411yescharacteristic
213Dic5—21no—
214Dic5—21no—
215Dic5—21no—
216Dic5—41no—
217Dic5—41no—
218C2×Q8—21no—
219C2×Q8—101no—
220C2×D4—51no—
221C2×D4—51no—
222C2×D4—101no—
223C2×D4—101no—
224C2×D4—101no—
225C2×D4—101no—
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—21no—
236C4⋊C4—21no—
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⋊C4C22×D511yescharacteristic
248C4⋊C4C22×D511yescharacteristic
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#53611yescharacteristic
262C2×C5#57611yescharacteristic
263C2×C5#57711yescharacteristic
264D5—41no—
265D5—81no—
266D5—81no—
267C23—51no—
268C23—101no—
269C23—101no—
270Q8—81no—
271Q8—401no—
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—51no—
285C2×C4—51no—
286C2×C4—51no—
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—101no—
303C22—101no—
304C22—101no—
305C22—201no—
306C22—201no—
307C22—201no—
308C22—201no—
309C22—201no—
310C22—201no—
311C22C2×D4×D511yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—101no—
319C4—101no—
320C4—101no—
321C4—201no—
322C4—201no—
323C4#414711yescharacteristic
324C4#414711yescharacteristic
325C2—201no—
326C2—401no—
327C2—401no—
328C2#6605811yescharacteristic
329C2#6610811yescharacteristic
330C2#6611211yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#501C2
socleC22×C5C22×D4
prime radical#66032C2
hypercenter#501D5
hypocenterC5#3241
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #66063C2semidirect
3. C5×#501C2semidirect
4. #65377C2semidirect
5. #65353C2semidirect
6. #65180C2semidirect
7. #65179C2semidirect
8. #65174C2semidirect
9. #65171C2semidirect
10. #65160C2semidirect
11. #65159C2semidirect
12. #65155C2semidirect
13. #65152C2semidirect
14. #65002C2semidirect
15. #64998C2semidirect
105. #501D5semidirect
301. C5#3241semidirect

Maximal subgroups

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

subgroupindex
#660632
#660322
C5×#5012
#653772
#653532
#651802
#651792
#651742
#651712
#651602
#651592
#651552
#651522
#650022
#649982
#32415