Finite Group Databasebeta

← #10613020

Subgroups — #10613020

Order 640.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65946C211yesmaximal, characteristic
2#65845C211yesmaximal, characteristic, complemented
3C5×#480C211yesmaximal, characteristic, complemented
4#65331C211yesmaximal, characteristic, complemented
5#65255C211yesmaximal, characteristic, complemented
6#65179C211yesmaximal, characteristic, complemented
7#65178C211yesmaximal, characteristic, complemented
8#65176C211yesmaximal, characteristic, complemented
9#65173C211yesmaximal, characteristic, complemented
10#65065C211yesmaximal, characteristic, complemented
11D5×#102C211yesmaximal, characteristic, complemented
12#65056C211yesmaximal, characteristic, complemented
13#65054C211yesmaximal, characteristic, complemented
14#65022C211yesmaximal, characteristic, complemented
15#65017C211yesmaximal, characteristic, complemented
16C2×D4×D5C2211yescharacteristic
17C2×#709C2211yescharacteristic
18C5×#121C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic, complemented
20#4084—21no—
21#4083—21no—
22#4082C2211yescharacteristic
23C2×#687C2211yescharacteristic
24C4×C22⋊D5—21no—
25C4×C22⋊D5C2211yescharacteristic
26#4070—21no—
27#4070C2211yescharacteristic, complemented
28C2×#681C2211yescharacteristic, complemented
29C2×Dic20C2211yescharacteristic, complemented
30C2×#677C2211yescharacteristic
31C2×C8×D5C2211yescharacteristic
32#4043C2211yescharacteristic
33#4043C2211yescharacteristic
34#4042—21no—
35#4039C2211yescharacteristic
36D5×C4⋊C4C2211yescharacteristic
37#4035—21no—
38#4034C2211yescharacteristic, complemented
39#4033C2211yescharacteristic, complemented
40#4029—21no—
41D5×C22⋊C4—21no—
42C5×#107C2211yescharacteristic, complemented
43C5×#107C2211yescharacteristic, complemented
44C5×#102C2211yescharacteristic
45C5×#102C2211yescharacteristic
46C5×C22⋊C8C2211yescharacteristic, complemented
47#3964C2211yescharacteristic
48#3953C2211yescharacteristic
49#3952C2211yescharacteristic
50#3948C2211yescharacteristic, complemented
51#3947C2211yescharacteristic, complemented
52#3946C2211yescharacteristic, complemented
53#3942C2211yescharacteristic, complemented
54#3942C2211yescharacteristic, complemented
55#3942C2211yescharacteristic, complemented
56#3941C2211yescharacteristic
57#3941C2211yescharacteristic
58#3939C2211yescharacteristic, complemented
59#3314—51nomaximal, Sylow
60C23×D5—21no—
61C2×C5×D4—21no—
62C2×C5×D4C2311yescharacteristic
63C22×C4×C5—21no—
64C22×C4×C5C2311yescharacteristic
65C2×C22⋊D5—21no—
66C2×C22⋊D5—21no—
67C2×C22⋊D5D411yescharacteristic
68D4×D5—22no—
69D4×D5—41no—
70#709—22no—
71#709—41no—
72C2×D20C2311yescharacteristic
73C2×C4×D5—21no—
74C2×C4×D5—21no—
75C2×C4×D5—21no—
76C2×C4×D5C2311yescharacteristic
77C2×Dic10—21no—
78C2×Dic10C2311yescharacteristic
79C2×Dic10C2311yescharacteristic
80C2×C5×C8C2311yescharacteristic
81C2×C5×C8C2311yescharacteristic
82C5×C4⋊C4—21no—
83C5×C4⋊C4C2311yescharacteristic
84C5×C4⋊C4C2311yescharacteristic
85C5×C4⋊C4C2311yescharacteristic
86C5×C22⋊C4—21no—
87C22⋊Dic5—21no—
88C22⋊Dic5—21no—
89C22⋊Dic5—21no—
90C22⋊Dic5D411yescharacteristic
91#687—41no—
92#685—21no—
93#685—21no—
94#685—21no—
95#685D411yescharacteristic
96C4⋊Dic5—21no—
97C4⋊Dic5—21no—
98C4⋊Dic5C2311yescharacteristic
99#683—21no—
100#683—21no—
101#683—21no—
102#683—21no—
103#683D411yescharacteristic
104C4×Dic5—21no—
105C4×Dic5C2311yescharacteristic
106#681—41no—
107C2×#168C2311yescharacteristic
108C2×#168C2311yescharacteristic
109Dic20—41no—
110#677—22no—
111C8×D5—41no—
112#554—51no—
113#546—51no—
114#489—51no—
115#484—51no—
116#480D511yescharacteristic, complemented
117#474—51no—
118#468—51no—
119#467—51no—
120#464—51no—
121#452—51no—
122#450—51no—
123#440—51no—
124#435—51no—
125#417—51no—
126C2×#102—51no—
127C23×C5—21no—
128C23×C5C2×D411yescharacteristic
129C22×D5—21no—
130C22×D5—42no—
131C22×D5—42no—
132C22×D5—81no—
133C22×D5C2×D411yescharacteristic
134C5×D4—21no—
135C5×D4—21no—
136C5×D4—81no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—81no—
143C2×C4×C5C2×D411yescharacteristic
144C2×C4×C5C2411yescharacteristic
145C22⋊D5—22no—
146C22⋊D5—42no—
147C22⋊D5—42no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5C2×D411yescharacteristic
152C2×Dic5C2×D411yescharacteristic
153C2×Dic5C2×D411yescharacteristic
154D20—21no—
155D20#4112yes—
156C4×D5—21no—
157C4×D5—21no—
158C4×D5—42no—
159C4×D5—81no—
160C4×D5—81no—
161Dic10—21no—
162Dic10—21no—
163Dic10—21no—
164Dic10—21no—
165Dic10—81no—
166C5×C8—21no—
167C5×C8—21no—
168#168—21no—
169#168—21no—
170C2×#41—51no—
171C22×D4—51no—
172C2×Q16—51no—
173C2×SD16—51no—
174C2×SD16—51no—
175C2×M16—51no—
176C22×C8—51no—
177C4⋊Q8—51no—
178#125—101no—
179#124—51no—
180#123—101no—
181C22⋊Q8—51no—
182C22⋊Q8—51no—
183C22⋊Q8—51no—
184C22⋊Q8—101no—
185C22⋊Q8—101no—
186#121—101no—
187#121C2×D511yescharacteristic
188#120—101no—
189C4×Q8—51no—
190C4×D4—51no—
191C4×D4—51no—
192C4×D4—101no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4C2×D511yescharacteristic, complemented
195C2×C22⋊C4—101no—
196#107—51no—
197#107C2×D511yescharacteristic, complemented
198#107C2×D511yescharacteristic, complemented
199C4⋊C8—51no—
200#103—51no—
201#103—51no—
202#103—51no—
203#103—51no—
204#102—51no—
205#102—51no—
206#102—51no—
207#102—51no—
208#102C2×D511yescharacteristic
209#102C2×D511yescharacteristic
210C22⋊C8—51no—
211C22⋊C8C2×D511yescharacteristic, complemented
212#97—51no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5C22×D411yescharacteristic
220C2×D5—21no—
221C2×D5—21no—
222C2×D5—22no—
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—84no—
226C2×D5#13712yes—
227C4×C5—21no—
228C4×C5—41no—
229C4×C5—41no—
230C4×C5—41no—
231C4×C5C2×#4111yescharacteristic
232C4×C5H(5,𝔽2)11yescharacteristic
233Dic5—21no—
234Dic5—21no—
235Dic5—21no—
236Dic5—41no—
237Dic5—41no—
238Dic5—41no—
239C24—101no—
240#41—102no—
241#41—201no—
242C2×Q8—51no—
243C2×Q8—51no—
244C2×Q8—101no—
245C2×D4—21no—
246C2×D4—51no—
247C2×D4—51no—
248C2×D4—101no—
249C2×D4—102no—
250C2×D4—101no—
251C2×D4—201no—
252C2×D4C22×D511yescharacteristic
253C22×C4—21no—
254C22×C4—51no—
255C22×C4—101no—
256C22×C4—101no—
257C22×C4—101no—
258C22×C4C22×D511yescharacteristic
259Q16—201no—
260SD16—102no—
261SD16—201no—
262M16—201no—
263C2×C8—51no—
264C2×C8—51no—
265C2×C8—201no—
266C2×C8C22×D511yescharacteristic
267C2×C8C22×D511yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—51no—
270C4⋊C4—51no—
271C4⋊C4—101no—
272C4⋊C4—101no—
273C4⋊C4—101no—
274C4⋊C4—101no—
275C4⋊C4—101no—
276C4⋊C4—101no—
277C4⋊C4C22×D511yescharacteristic
278C4⋊C4C22×D511yescharacteristic
279C4⋊C4C22×D511yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—51no—
282C22⋊C4—51no—
283C22⋊C4—101no—
284C22⋊C4—101no—
285C22⋊C4—101no—
286C22⋊C4—101no—
287C22⋊C4—101no—
288C22⋊C4—101no—
289C42—51no—
290C42—101no—
291C2×C5—41no—
292C2×C5—81no—
293C2×C5#54611yescharacteristic
294C2×C5C2×#13711yescharacteristic
295C2×C5#57611yescharacteristic
296D5—22no—
297D5—42no—
298C23—21no—
299C23—51no—
300C23—101no—
301C23—202no—
302C23—202no—
303C23—401no—
304C23D4×D511yescharacteristic
305Q8—101no—
306Q8—101no—
307Q8—101no—
308Q8—101no—
309Q8—401no—
310D4—21no—
311D4—21no—
312D4—52no—
313D4—81no—
314D4—102no—
315D4—101no—
316D4—202no—
317D4—202no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—51no—
324C2×C4—51no—
325C2×C4—51no—
326C2×C4—81no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—202no—
333C2×C4—401no—
334C2×C4—401no—
335C2×C4D4×D511yescharacteristic
336C2×C4C23×D511yescharacteristic
337C8—21no—
338C8—21no—
339C8—101no—
340C8—101no—
341C5#331411yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—52no—
349C22—101no—
350C22—101no—
351C22—102no—
352C22—201no—
353C22—201no—
354C22—404no—
355C22C2×D4×D511yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—101no—
361C4—101no—
362C4—101no—
363C4—201no—
364C4—201no—
365C4—201no—
366C4D5×#4111yescharacteristic
367C4#414911yescharacteristic
368C2—41no—
369C2—81no—
370C2—102no—
371C2—202no—
372C2#6598811yescharacteristic
373C2#6609111yescharacteristic
374C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#480C2
socleC22×C5C22×D4
prime radical#65946C2
hypercenter#480D5
hypocenterC5#3314
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #65845C2semidirect
3. C5×#480C2semidirect
4. #65331C2semidirect
5. #65255C2semidirect
6. #65179C2semidirect
7. #65178C2semidirect
8. #65176C2semidirect
9. #65173C2semidirect
10. #65065C2semidirect
11. D5×#102C2semidirect
12. #65056C2semidirect
13. #65054C2semidirect
14. #65022C2semidirect
15. #65017C2semidirect
19. C2×C5×C4⋊C4C22semidirect
27. #4070C22semidirect
28. C2×#681C22semidirect
29. C2×Dic20C22semidirect
38. #4034C22semidirect
39. #4033C22semidirect
42. C5×#107C22semidirect
43. C5×#107C22semidirect
46. C5×C22⋊C8C22semidirect
50. #3948C22semidirect
51. #3947C22semidirect
52. #3946C22semidirect
53. #3942C22semidirect
54. #3942C22semidirect
55. #3942C22semidirect
58. #3939C22semidirect
116. #480D5semidirect
194. C2×C4⋊C4C2×D5semidirect
197. #107C2×D5semidirect
198. #107C2×D5semidirect
211. C22⋊C8C2×D5semidirect
341. C5#3314semidirect

Maximal subgroups

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

subgroupindex
#659462
#658452
C5×#4802
#653312
#652552
#651792
#651782
#651762
#651732
#650652
D5×#1022
#650562
#650542
#650222
#650172
#33145