Finite Group Databasebeta

← #10615366

Subgroups — #10615366

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
1C5×#554C211yesmaximal, characteristic, complemented
2#65884C211yesmaximal, characteristic
3#65378C211yesmaximal, characteristic, complemented
4#65376C211yesmaximal, characteristic, complemented
5#65340C211yesmaximal, characteristic, complemented
6#65338C211yesmaximal, characteristic, complemented
7#65336C211yesmaximal, characteristic, complemented
8#65335C211yesmaximal, characteristic, complemented
9#65317C211yesmaximal, characteristic, complemented
10#65315C211yesmaximal, characteristic, complemented
11#65300C211yesmaximal, characteristic, complemented
12#65259C211yesmaximal, characteristic, complemented
13#65256C211yesmaximal, characteristic, complemented
14#65255C211yesmaximal, characteristic, complemented
15C2×#3941C211yesmaximal, characteristic, complemented
16C2×#709C2211yescharacteristic
17C22×D20C2211yescharacteristic
18C5×C4⋊Q8C2211yescharacteristic, complemented
19C5×#125—21no—
20C5×C22⋊Q8—21no—
21C5×C22⋊Q8—21no—
22C5×C22⋊Q8C2211yescharacteristic
23C5×C22⋊Q8C2211yescharacteristic, complemented
24C4×C5×Q8C2211yescharacteristic, complemented
25C4×C5×D4—21no—
26C4×C5×D4C2211yescharacteristic
27C2×C5×C4⋊C4C2211yescharacteristic
28C2×C5×C4⋊C4C2211yescharacteristic, complemented
29C2×#689C2211yescharacteristic, complemented
30C2×#688C2211yescharacteristic
31C2×#688C2211yescharacteristic
32#4083—21no—
33#4076C2211yescharacteristic
34#4070C2211yescharacteristic
35C2×#681C2211yescharacteristic, complemented
36C22×#168C2211yescharacteristic
37#4040—21no—
38#4030—21no—
39D5×C22⋊C4—21no—
40#4021C2211yescharacteristic
41C4×D20C2211yescharacteristic
42#3967C2211yescharacteristic, complemented
43C22⋊#168C2211yescharacteristic
44C22⋊#168C2211yescharacteristic, complemented
45#3942C2211yescharacteristic, complemented
46#3942C2211yescharacteristic, complemented
47#3942C2211yescharacteristic, complemented
48#3941C2211yescharacteristic
49#3941C2211yescharacteristic
50#3941C2211yescharacteristic
51#3941C2211yescharacteristic
52#3941C2211yescharacteristic
53#3941C2211yescharacteristic
54#3939C2211yescharacteristic, complemented
55#3939C2211yescharacteristic, complemented
56#3939C2211yescharacteristic, complemented
57C4⋊#168C2211yescharacteristic, complemented
58#3935C2211yescharacteristic, complemented
59#3314—51nomaximal, Sylow
60C23×D5—21no—
61C2×C5×Q8—21no—
62C2×C5×Q8C2311yescharacteristic
63C2×C5×D4D411yescharacteristic
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C2×C22⋊D5—21no—
69C2×C22⋊D5—21no—
70#709—22no—
71#709—41no—
72C2×D20—22no—
73C2×D20—21no—
74C2×D20—41no—
75C2×D20C2311yescharacteristic
76C2×D20C2311yescharacteristic
77C2×C4×D5—21no—
78C2×C4×D5—21no—
79C2×Dic10C2311yescharacteristic
80C5×C4⋊C4—21no—
81C5×C4⋊C4—21no—
82C5×C4⋊C4—21no—
83C5×C4⋊C4—21no—
84C5×C4⋊C4—21no—
85C5×C4⋊C4—21no—
86C5×C4⋊C4D411yescharacteristic
87C5×C4⋊C4C2311yescharacteristic
88C5×C4⋊C4C2311yescharacteristic
89C5×C4⋊C4C2311yescharacteristic
90C5×C22⋊C4—21no—
91C5×C22⋊C4—21no—
92C5×C22⋊C4D411yescharacteristic
93C5×C22⋊C4D411yescharacteristic
94C42×C5—21no—
95C42×C5C2311yescharacteristic
96C22⋊Dic5—21no—
97#689—41no—
98#688—22no—
99#688—41no—
100#685—21no—
101#685—21no—
102#685—21no—
103#685—21no—
104C4⋊Dic5C2311yescharacteristic
105#683—21no—
106#681—41no—
107C2×#168—41no—
108C2×#168C2311yescharacteristic
109C2×#168C2311yescharacteristic
110C2×#168C2311yescharacteristic
111C2×#168C2311yescharacteristic
112#554D511yescharacteristic, complemented
113#546—51no—
114#489—51no—
115#484—51no—
116#480—51no—
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×C5C2×D411yescharacteristic
128C23×C5C2×D411yescharacteristic
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—42no—
132C22×D5—42no—
133C22×D5—81no—
134C5×Q8—21no—
135C5×Q8—21no—
136C5×Q8—81no—
137C5×D4—22no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—81no—
147C2×C4×C5—81no—
148C2×C4×C5C2×D411yescharacteristic
149C2×C4×C5C2×D411yescharacteristic
150C2×C4×C5C2×D411yescharacteristic
151C2×C4×C5C2×D411yescharacteristic
152C2×C4×C5C2411yescharacteristic
153C22⋊D5—42no—
154C22⋊D5—81no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157D20—21no—
158D20—21no—
159D20—21no—
160D20—42no—
161D20#4112yes—
162C4×D5—42no—
163C4×D5—81no—
164Dic10—21no—
165Dic10—21no—
166#168—21no—
167#168—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⋊Q8C2×D511yescharacteristic, complemented
178#125—21no—
179#124—51no—
180#123—101no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—51no—
184C22⋊Q8C2×D511yescharacteristic
185C22⋊Q8C2×D511yescharacteristic, complemented
186#121—51no—
187#121—101no—
188#120—101no—
189C4×Q8C2×D511yescharacteristic, complemented
190C4×D4—21no—
191C4×D4—51no—
192C4×D4C2×D511yescharacteristic
193C2×C4⋊C4C2×D511yescharacteristic
194C2×C4⋊C4C2×D511yescharacteristic, complemented
195C2×C22⋊C4—101no—
196#107—51no—
197#107—51no—
198#107—51no—
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#102—51no—
209#102—51no—
210C22⋊C8—51no—
211C22⋊C8—51no—
212#97—51no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C5#13712yes—
219C22×C5C22×D411yescharacteristic
220C2×D5—22no—
221C2×D5—41no—
222C2×D5—41no—
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—84no—
227C4×C5—21no—
228C4×C5—21no—
229C4×C5—21no—
230C4×C5—21no—
231C4×C5—41no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5—41no—
235C4×C5C2×#4111yescharacteristic
236C4×C5H(5,𝔽2)11yescharacteristic
237Dic5—41no—
238Dic5—41no—
239C24—101no—
240#41—102no—
241#41—201no—
242C2×Q8—21no—
243C2×Q8—51no—
244C2×Q8C22×D511yescharacteristic
245C2×D4—51no—
246C2×D4—51no—
247C2×D4—102no—
248C2×D4—101no—
249C2×D4—101no—
250C2×D4—101no—
251C2×D4—201no—
252C2×D4C22⋊D511yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—101no—
256C22×C4—101no—
257C22×C4C22×D511yescharacteristic
258C22×C4C22×D511yescharacteristic
259Q16—201no—
260SD16—102no—
261SD16—201no—
262M16—201no—
263C2×C8—51no—
264C2×C8—51no—
265C2×C8—51no—
266C2×C8—51no—
267C2×C8—201no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—51no—
275C4⋊C4—101no—
276C4⋊C4C22⋊D511yescharacteristic
277C4⋊C4C22×D511yescharacteristic
278C4⋊C4C22×D511yescharacteristic
279C4⋊C4C22×D511yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—101no—
283C22⋊C4—101no—
284C22⋊C4—101no—
285C22⋊C4—101no—
286C22⋊C4—101no—
287C22⋊C4C22⋊D511yescharacteristic
288C22⋊C4C22⋊D511yescharacteristic
289C42—21no—
290C42C22×D511yescharacteristic
291C2×C5—22no—
292C2×C5—41no—
293C2×C5#54611yescharacteristic
294C2×C5C2×#13711yescharacteristic
295C2×C5#57611yescharacteristic
296D5—42no—
297D5—81no—
298C23—101no—
299C23—101no—
300C23—202no—
301C23—202no—
302C23—401no—
303C23C2×C22⋊D511yescharacteristic
304C23C2×C22⋊D511yescharacteristic
305Q8—21no—
306Q8—21no—
307Q8—81no—
308Q8—101no—
309Q8—101no—
310D4—22no—
311D4—52no—
312D4—101no—
313D4—101no—
314D4—101no—
315D4—202no—
316D4—202no—
317D4—401no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—202no—
331C2×C4—401no—
332C2×C4C2×C22⋊D511yescharacteristic
333C2×C4C2×C22⋊D511yescharacteristic
334C2×C4C2×C22⋊D511yescharacteristic
335C2×C4C2×C22⋊D511yescharacteristic
336C2×C4C23×D511yescharacteristic
337C8—101no—
338C8—101no—
339C8—101no—
340C8—101no—
341C5#331411yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—102no—
348C22—201no—
349C22—201no—
350C22—201no—
351C22—201no—
352C22—201no—
353C22—404no—
354C22#408812yes—
355C22C22×C22⋊D511yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—201no—
365C4—201no—
366C4D5×#4111yescharacteristic
367C4#414911yescharacteristic
368C2—22no—
369C2—41no—
370C2—202no—
371C2—401no—
372C2C2×#408811yescharacteristic
373C2#6616311yescharacteristic
374C2#6616911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#554C2
socleC22×C5C22×D4
prime radical#65884C2
hypercenter#554D5
hypocenterC5#3314
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C5×#554C2semidirect
3. #65378C2semidirect
4. #65376C2semidirect
5. #65340C2semidirect
6. #65338C2semidirect
7. #65336C2semidirect
8. #65335C2semidirect
9. #65317C2semidirect
10. #65315C2semidirect
11. #65300C2semidirect
12. #65259C2semidirect
13. #65256C2semidirect
14. #65255C2semidirect
15. C2×#3941C2semidirect
18. C5×C4⋊Q8C22semidirect
23. C5×C22⋊Q8C22semidirect
24. C4×C5×Q8C22semidirect
28. C2×C5×C4⋊C4C22semidirect
29. C2×#689C22semidirect
35. C2×#681C22semidirect
42. #3967C22semidirect
44. C22⋊#168C22semidirect
45. #3942C22semidirect
46. #3942C22semidirect
47. #3942C22semidirect
54. #3939C22semidirect
55. #3939C22semidirect
56. #3939C22semidirect
57. C4⋊#168C22semidirect
58. #3935C22semidirect
112. #554D5semidirect
177. C4⋊Q8C2×D5semidirect
185. C22⋊Q8C2×D5semidirect
189. C4×Q8C2×D5semidirect
194. C2×C4⋊C4C2×D5semidirect
341. C5#3314semidirect

Maximal subgroups

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

subgroupindex
C5×#5542
#658842
#653782
#653762
#653402
#653382
#653362
#653352
#653172
#653152
#653002
#652592
#652562
#652552
C2×#39412
#33145