Finite Group Databasebeta

← #10613720

Subgroups — #10613720

Order 640.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66067C211yesmaximal, characteristic
2#66023C211yesmaximal, characteristic, complemented
3C5×#501C211yesmaximal, characteristic, complemented
4#65368C211yesmaximal, characteristic, complemented
5#65357C211yesmaximal, characteristic, complemented
6#65180C211yesmaximal, characteristic, complemented
7#65177C211yesmaximal, characteristic, complemented
8#65174C211yesmaximal, characteristic
9#65166C211yesmaximal, characteristic, complemented
10#65159C211yesmaximal, characteristic, complemented
11#65152C211yesmaximal, characteristic
12#65151C211yesmaximal, characteristic, complemented
13#65143C211yesmaximal, characteristic, complemented
14#65003C211yesmaximal, characteristic, complemented
15#64998C211yesmaximal, characteristic, complemented
16C5×C4⋊Q8C2211yescharacteristic
17C5×#125C2211yescharacteristic
18#4093—21no—
19#4092—21no—
20#4044—21no—
21#4043—21no—
22#4043C2211yescharacteristic
23#4042—21no—
24#4041C2211yescharacteristic
25#4040—21no—
26#4039C2211yescharacteristic
27#4038C2211yescharacteristic
28D5×C4⋊C4C2211yescharacteristic
29#4036C2211yescharacteristic
30#4035—21no—
31#4034—21no—
32#4034C2211yescharacteristic
33#4033C2211yescharacteristic
34#4021C2211yescharacteristic
35#4018C2211yescharacteristic
36#4015C2211yescharacteristic
37C5×#107C2211yescharacteristic
38C5×#107C2211yescharacteristic
39C5×#106C2211yescharacteristic
40C5×#106C2211yescharacteristic
41C5×#97C2211yescharacteristic
42#3953C2211yescharacteristic
43#3953C2211yescharacteristic
44#3952C2211yescharacteristic
45#3952C2211yescharacteristic
46#3948C2211yescharacteristic
47#3948C2211yescharacteristic
48#3946C2211yescharacteristic
49#3946C2211yescharacteristic
50#3942C2211yescharacteristic
51#3942C2211yescharacteristic
52#3941C2211yescharacteristic
53#3941C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3940C2211yescharacteristic
57#3940C2211yescharacteristic
58C4×#168C2211yescharacteristic
59#3244—51nomaximal, Sylow
60C2×C5×Q8—21no—
61C2×D20—21no—
62C2×D20C2311yescharacteristic
63C2×C4×D5—21no—
64C2×C4×D5—21no—
65C2×C4×D5C2311yescharacteristic
66C2×Dic10—21no—
67C2×Dic10—21no—
68C2×Dic10C2311yescharacteristic
69C2×C5×C8C2311yescharacteristic
70C2×C5×C8C2311yescharacteristic
71C5×C4⋊C4—21no—
72C5×C4⋊C4—21no—
73C5×C4⋊C4—21no—
74C5×C4⋊C4C2311yescharacteristic
75C5×C4⋊C4C2311yescharacteristic
76C5×C4⋊C4C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C42×C5C2311yescharacteristic
79#685—21no—
80#685—21no—
81#685—21no—
82#685—21no—
83#685—21no—
84#685—21no—
85#685D411yescharacteristic
86#685D411yescharacteristic
87C4⋊Dic5—21no—
88C4⋊Dic5—21no—
89C4⋊Dic5—21no—
90C4⋊Dic5C2311yescharacteristic
91C4⋊Dic5C2311yescharacteristic
92#683—21no—
93#683—21no—
94#683—21no—
95#683—21no—
96#683D411yescharacteristic
97#683D411yescharacteristic
98C4×Dic5—21no—
99C4×Dic5—21no—
100C4×Dic5C2311yescharacteristic
101C2×#168C2311yescharacteristic
102C2×#168C2311yescharacteristic
103#544—51no—
104#542—51no—
105#501D511yescharacteristic, complemented
106#498—51no—
107#489—51no—
108#488—51no—
109#485—51no—
110#483—51no—
111#482—51no—
112#481—51no—
113#479—51no—
114#478—51no—
115#476—51no—
116#475—51no—
117#433—51no—
118C22×D5—21no—
119C22×D5C2×D411yescharacteristic
120C5×Q8—81no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5C2×D411yescharacteristic
126C2×C4×C5C2×D411yescharacteristic
127C2×C4×C5C2411yescharacteristic
128C2×Dic5—21no—
129C2×Dic5—21no—
130C2×Dic5—21no—
131C2×Dic5C2×D411yescharacteristic
132C2×Dic5C2×D411yescharacteristic
133C2×Dic5C2×D411yescharacteristic
134D20—21no—
135D20—21no—
136D20—81no—
137C4×D5—21no—
138C4×D5—21no—
139C4×D5—81no—
140C4×D5—81no—
141Dic10—21no—
142Dic10—21no—
143Dic10—81no—
144Dic10—81no—
145C5×C8—21no—
146C5×C8—21no—
147#168—21no—
148#168—21no—
149C4⋊Q8—51no—
150C4⋊Q8—51no—
151C4⋊Q8—101no—
152C4⋊Q8C2×D511yescharacteristic
153#126—101no—
154#125—51no—
155#125—101no—
156#125C2×D511yescharacteristic
157#124—51no—
158#124—101no—
159#123—101no—
160C22⋊Q8—51no—
161C22⋊Q8—101no—
162C22⋊Q8—101no—
163C22⋊Q8—101no—
164#121—51no—
165C4×Q8—51no—
166C4×D4—51no—
167#117—51no—
168#117—51no—
169C2×C4⋊C4—51no—
170#107C2×D511yescharacteristic
171#107C2×D511yescharacteristic
172#106—51no—
173#106—51no—
174#106—51no—
175#106—51no—
176#106C2×D511yescharacteristic
177#106C2×D511yescharacteristic
178C4⋊C8—51no—
179C4⋊C8—51no—
180#103—51no—
181#103—51no—
182#103—51no—
183#103—51no—
184#102—51no—
185#102—51no—
186#102—51no—
187#102—51no—
188C22⋊C8—51no—
189C22⋊C8—51no—
190#97C2×D511yescharacteristic
191C4×C8—51no—
192C22×C5C22×D411yescharacteristic
193C2×D5—21no—
194C2×D5—21no—
195C2×D5—21no—
196C2×D5—41no—
197C2×D5—41no—
198C2×D5—41no—
199C4×C5—21no—
200C4×C5—21no—
201C4×C5—41no—
202C4×C5—41no—
203C4×C5—41no—
204C4×C5—41no—
205C4×C52−1+411yescharacteristic
206C4×C52−1+411yescharacteristic
207Dic5—21no—
208Dic5—21no—
209Dic5—21no—
210Dic5—41no—
211Dic5—41no—
212Dic5—41no—
213C2×Q8—21no—
214C2×Q8—51no—
215C2×Q8—101no—
216C2×Q8—101no—
217C2×D4—51no—
218C2×D4—101no—
219C22×C4—51no—
220C22×C4—101no—
221C22×C4—101no—
222C2×C8—51no—
223C2×C8—51no—
224C2×C8C22×D511yescharacteristic
225C2×C8C22×D511yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—51no—
230C4⋊C4—51no—
231C4⋊C4—51no—
232C4⋊C4—51no—
233C4⋊C4—101no—
234C4⋊C4—101no—
235C4⋊C4—101no—
236C4⋊C4—101no—
237C4⋊C4—101no—
238C4⋊C4—101no—
239C4⋊C4—101no—
240C4⋊C4C22×D511yescharacteristic
241C4⋊C4C22×D511yescharacteristic
242C4⋊C4C22×D511yescharacteristic
243C4⋊C4C22×D511yescharacteristic
244C22⋊C4—51no—
245C22⋊C4—51no—
246C22⋊C4—101no—
247C22⋊C4—101no—
248C22⋊C4—101no—
249C22⋊C4—101no—
250C22⋊C4—101no—
251C22⋊C4—101no—
252C42—51no—
253C42—101no—
254C42—101no—
255C42C22×D511yescharacteristic
256C2×C5#53611yescharacteristic
257C2×C5#57711yescharacteristic
258C2×C5#57711yescharacteristic
259D5—41no—
260D5—81no—
261C23—51no—
262C23—101no—
263Q8—81no—
264Q8—101no—
265Q8—101no—
266Q8—401no—
267Q8—401no—
268D4—101no—
269D4—101no—
270D4—401no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—51no—
276C2×C4—51no—
277C2×C4—51no—
278C2×C4—101no—
279C2×C4—101no—
280C2×C4—101no—
281C2×C4—101no—
282C2×C4—101no—
283C2×C4—401no—
284C2×C4—401no—
285C2×C4D4×D511yescharacteristic
286C2×C4D4×D511yescharacteristic
287C2×C4C23×D511yescharacteristic
288C8—21no—
289C8—21no—
290C8—101no—
291C8—101no—
292C5#324411yescharacteristic, complemented, Sylow
293C22—101no—
294C22—101no—
295C22—101no—
296C22—201no—
297C22—201no—
298C22—201no—
299C22C2×D4×D511yescharacteristic
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—101no—
307C4—101no—
308C4—101no—
309C4—201no—
310C4—201no—
311C4—201no—
312C4#414711yescharacteristic
313C4#414711yescharacteristic
314C2—201no—
315C2—401no—
316C2#6605811yescharacteristic
317C2#6610911yescharacteristic
318C2#6611211yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#501C2
socleC22×C5C22×D4
prime radical#4041C22
hypercenter#501D5
hypocenterC5#3244
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
2. #66023C2semidirect
3. C5×#501C2semidirect
4. #65368C2semidirect
5. #65357C2semidirect
6. #65180C2semidirect
7. #65177C2semidirect
9. #65166C2semidirect
10. #65159C2semidirect
12. #65151C2semidirect
13. #65143C2semidirect
14. #65003C2semidirect
15. #64998C2semidirect
105. #501D5semidirect
292. C5#3244semidirect

Maximal subgroups

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

subgroupindex
#660672
#660232
C5×#5012
#653682
#653572
#651802
#651772
#651742
#651662
#651592
#651522
#651512
#651432
#650032
#649982
#32445