Finite Group Databasebeta

← #10612378

Subgroups — #10612378

Order 640.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65977C211yesmaximal, characteristic, complemented
2#65937C211yesmaximal, characteristic, complemented
3C5×#465C211yesmaximal, characteristic, complemented
4#65455C211yesmaximal, characteristic, complemented
5#65454C211yesmaximal, characteristic, complemented
6#65403C211yesmaximal, characteristic, complemented
7#65391C211yesmaximal, characteristic, complemented
8#65341C211yesmaximal, characteristic, complemented
9#65328C211yesmaximal, characteristic
10#65159C211yesmaximal, characteristic, complemented
11#65143C211yesmaximal, characteristic, complemented
12#65087C211yesmaximal, characteristic, complemented
13#65080C211yesmaximal, characteristic, complemented
14#65076C211yesmaximal, characteristic, complemented
15#65050C211yesmaximal, characteristic, complemented
16C2×C5×SD16C2211yescharacteristic
17C22×C5×C8C2211yescharacteristic
18C5×C22⋊Q8C2211yescharacteristic
19C5×#121C2211yescharacteristic
20#4093C2211yescharacteristic
21Q8×Dic5C2211yescharacteristic
22C2×#689C2211yescharacteristic
23#4082—21no—
24#4082C2211yescharacteristic
25#4081—21no—
26D4×Dic5C2211yescharacteristic
27C2×#686C2211yescharacteristic
28#4076C2211yescharacteristic
29#4072C2211yescharacteristic
30#4070C2211yescharacteristic
31C2×#681C2211yescharacteristic
32C2×#677C2211yescharacteristic
33#4044—21no—
34#4043—21no—
35#4039C2211yescharacteristic
36#4034C2211yescharacteristic
37#4034C2211yescharacteristic
38#4033C2211yescharacteristic
39#4031—21no—
40#4029—21no—
41#4025—21no—
42#4024—21no—
43C5×#106C2211yescharacteristic
44C5×#103C2211yescharacteristic
45C5×#102C2211yescharacteristic
46#3967C2211yescharacteristic
47#3964C2211yescharacteristic
48#3953C2211yescharacteristic
49#3949C2211yescharacteristic
50#3948C2211yescharacteristic
51#3946C2211yescharacteristic
52#3946C2211yescharacteristic
53#3946C2211yescharacteristic
54#3946C2211yescharacteristic
55#3942C2211yescharacteristic
56#3941C2211yescharacteristic
57#3940C2211yescharacteristic
58#3940C2211yescharacteristic
59#3263—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C2×C5×D4—21no—
62C2×C5×D4C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—21no—
65C22×Dic5—21no—
66C2×D20C2311yescharacteristic
67C2×C4×D5—21no—
68C2×Dic10—21no—
69C2×Dic10—21no—
70C2×Dic10C2311yescharacteristic
71C5×SD16—41no—
72C2×C5×C8—41no—
73C2×C5×C8C2311yescharacteristic
74C2×C5×C8C2311yescharacteristic
75C5×C4⋊C4—21no—
76C5×C4⋊C4C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C5×C22⋊C4—21no—
79C5×C22⋊C4—21no—
80C22⋊Dic5—21no—
81C22⋊Dic5—21no—
82C22⋊Dic5—21no—
83C22⋊Dic5—21no—
84C22⋊Dic5D411yescharacteristic
85C22⋊Dic5D411yescharacteristic
86#689—41no—
87#686—41no—
88#685—21no—
89#685—21no—
90#685—21no—
91#685—21no—
92C4⋊Dic5—21no—
93C4⋊Dic5D411yescharacteristic
94C4⋊Dic5D411yescharacteristic
95C4⋊Dic5C2311yescharacteristic
96C4⋊Dic5C2311yescharacteristic
97#683—21no—
98#683—21no—
99#683—21no—
100#683—21no—
101C4×Dic5—21no—
102C4×Dic5—21no—
103C4×Dic5C2311yescharacteristic
104C4×Dic5C2311yescharacteristic
105#681—41no—
106C2×#168C2311yescharacteristic
107C2×#168C2311yescharacteristic
108#677—41no—
109#542—51no—
110#542—51no—
111#476—51no—
112#476—51no—
113#475—51no—
114#475—51no—
115#470—51no—
116#469—51no—
117#465—51no—
118#465D511yescharacteristic, complemented
119#464—51no—
120#464—51no—
121#463—51no—
122#463—51no—
123#424—51no—
124C23×C5—21no—
125C23×C5C2×D411yescharacteristic
126C22×D5—21no—
127C5×Q8—21no—
128C5×Q8—21no—
129C5×D4—21no—
130C5×D4—21no—
131C5×D4—81no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5C2×D411yescharacteristic
138C2×C4×C5C2411yescharacteristic
139C22⋊D5—81no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—81no—
144C2×Dic5C2×D411yescharacteristic
145C2×Dic5C2×D411yescharacteristic
146C2×Dic5C2×D411yescharacteristic
147C2×Dic5C2×D411yescharacteristic
148D20—21no—
149D20—21no—
150C4×D5—81no—
151Dic10—21no—
152Dic10—21no—
153Dic10—81no—
154Dic10—81no—
155C5×C8—21no—
156C5×C8—21no—
157#168—21no—
158#168—21no—
159C2×Q16—51no—
160C2×SD16—51no—
161C2×SD16C2×D511yescharacteristic
162C2×D8—51no—
163C2×M16—51no—
164C22×C8C2×D511yescharacteristic
165C4⋊Q8—51no—
166C4⋊Q8—51no—
167#126—101no—
168#126—101no—
169#124—51no—
170#124—51no—
171#124—101no—
172#124—101no—
173#123—101no—
174#123—101no—
175C22⋊Q8—51no—
176C22⋊Q8—101no—
177C22⋊Q8—101no—
178C22⋊Q8C2×D511yescharacteristic
179#121—51no—
180#121C2×D511yescharacteristic
181C4×Q8—51no—
182C4×Q8—51no—
183C4×D4—51no—
184C4×D4—51no—
185#117—51no—
186#106—51no—
187#106—51no—
188#106—51no—
189#106C2×D511yescharacteristic
190C4⋊C8—51no—
191C4⋊C8—51no—
192C4⋊C8—51no—
193C4⋊C8—51no—
194#103—51no—
195#103—51no—
196#103—51no—
197#103C2×D511yescharacteristic
198#102—51no—
199#102—51no—
200#102—51no—
201#102C2×D511yescharacteristic
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5C22×D411yescharacteristic
209C2×D5—41no—
210C2×D5—41no—
211C2×D5—41no—
212C4×C5—21no—
213C4×C5—41no—
214C4×C5—41no—
215C4×C5—41no—
216C4×C5H(5,𝔽2)11yescharacteristic
217C4×C52−1+411yescharacteristic
218Dic5—21no—
219Dic5—21no—
220Dic5—21no—
221Dic5—21no—
222Dic5—41no—
223Dic5—41no—
224Dic5—41no—
225C2×Q8—51no—
226C2×Q8—101no—
227C2×Q8—101no—
228C2×Q8C22×D511yescharacteristic
229C2×D4—21no—
230C2×D4—51no—
231C2×D4—101no—
232C2×D4C22×D511yescharacteristic
233C22×C4—101no—
234C22×C4—101no—
235C22×C4C22×D511yescharacteristic
236Q16—201no—
237SD16—41no—
238SD16—201no—
239D8—201no—
240M16—201no—
241C2×C8—41no—
242C2×C8—51no—
243C2×C8—51no—
244C2×C8C22×D511yescharacteristic
245C2×C8C22×D511yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—51no—
248C4⋊C4—51no—
249C4⋊C4—51no—
250C4⋊C4—51no—
251C4⋊C4—101no—
252C4⋊C4—101no—
253C4⋊C4—101no—
254C4⋊C4—101no—
255C4⋊C4—101no—
256C4⋊C4C22×D511yescharacteristic
257C4⋊C4C22×D511yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—51no—
261C22⋊C4—51no—
262C22⋊C4—101no—
263C22⋊C4—101no—
264C22⋊C4—101no—
265C22⋊C4—101no—
266C22⋊C4—101no—
267C22⋊C4—101no—
268C22⋊C4—101no—
269C22⋊C4—101no—
270C42—51no—
271C42—51no—
272C42—101no—
273C42—101no—
274C2×C5—41no—
275C2×C5—81no—
276C2×C5#53711yescharacteristic
277C2×C5#57711yescharacteristic
278C2×C5#57711yescharacteristic
279D5—81no—
280C23—21no—
281C23—101no—
282C23D4×D511yescharacteristic
283Q8—21no—
284Q8—21no—
285Q8—101no—
286Q8—101no—
287Q8—401no—
288Q8—401no—
289D4—21no—
290D4—21no—
291D4—81no—
292D4—101no—
293D4—101no—
294D4—401no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—51no—
301C2×C4—51no—
302C2×C4—51no—
303C2×C4—51no—
304C2×C4—101no—
305C2×C4—101no—
306C2×C4—101no—
307C2×C4—401no—
308C2×C4—401no—
309C2×C4D4×D511yescharacteristic
310C2×C4C23×D511yescharacteristic
311C8—21no—
312C8—21no—
313C8—101no—
314C8—101no—
315C5#326311yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—201no—
323C22—201no—
324C22—201no—
325C22C2×D4×D511yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—101no—
331C4—101no—
332C4—101no—
333C4—101no—
334C4—201no—
335C4—201no—
336C4—201no—
337C4#414411yescharacteristic
338C4#415011yescharacteristic
339C2—41no—
340C2—81no—
341C2—401no—
342C2#6593111yescharacteristic
343C2#6609611yescharacteristic
344C2#6610411yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#465C2
socleC22×C5C22×D4
prime radical#65328C2
hypercenter#465D5
hypocenterC5#3263
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #65977C2semidirect
2. #65937C2semidirect
3. C5×#465C2semidirect
4. #65455C2semidirect
5. #65454C2semidirect
6. #65403C2semidirect
7. #65391C2semidirect
8. #65341C2semidirect
10. #65159C2semidirect
11. #65143C2semidirect
12. #65087C2semidirect
13. #65080C2semidirect
14. #65076C2semidirect
15. #65050C2semidirect
118. #465D5semidirect
315. C5#3263semidirect

Maximal subgroups

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

subgroupindex
#659772
#659372
C5×#4652
#654552
#654542
#654032
#653912
#653412
#653282
#651592
#651432
#650872
#650802
#650762
#650502
#32635