Finite Group Databasebeta

← #10614663

Subgroups — #10614663

Order 640.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#525C211yesmaximal, characteristic, complemented
2#66125C211yesmaximal, characteristic, complemented
3#65528C211yesmaximal, characteristic, complemented
4#65527C211yesmaximal, characteristic, complemented
5#65496C211yesmaximal, characteristic, complemented
6#65341C211yesmaximal, characteristic, complemented
7#65338C211yesmaximal, characteristic, complemented
8#65336C211yesmaximal, characteristic, complemented
9#65334C211yesmaximal, characteristic, complemented
10#65331C211yesmaximal, characteristic, complemented
11#65328C211yesmaximal, characteristic, complemented
12#65326C211yesmaximal, characteristic, complemented
13#65324C211yesmaximal, characteristic, complemented
14#65292C211yesmaximal, characteristic, complemented
15#65285C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C2×#709C2211yescharacteristic
19C5×#123—21no—
20C5×C22⋊Q8C2211yescharacteristic
21C5×C22⋊Q8C2211yescharacteristic
22C5×#121C2211yescharacteristic
23C5×#121C2211yescharacteristic
24C5×#120—21no—
25C4×C5×D4—21no—
26C4×C5×D4—21no—
27C5×#117C2211yescharacteristic
28#4099—21no—
29C2×#689C2211yescharacteristic, complemented
30C2×#688C2211yescharacteristic, complemented
31C2×#687C2211yescharacteristic, complemented
32C2×#686C2211yescharacteristic, complemented
33#4076C2211yescharacteristic
34#4076C2211yescharacteristic
35#4075—21no—
36C4×C22⋊D5—21no—
37C4×C22⋊D5—21no—
38#4072C2211yescharacteristic
39#4070C2211yescharacteristic
40#4070C2211yescharacteristic
41C2×#681C2211yescharacteristic
42C2×#681C2211yescharacteristic
43#3967C2211yescharacteristic, complemented
44#3967C2211yescharacteristic, complemented
45#3964C2211yescharacteristic, complemented
46#3964C2211yescharacteristic, complemented
47C22⋊#168C2211yescharacteristic
48C22⋊#168C2211yescharacteristic
49C22⋊#168C2211yescharacteristic
50C22⋊#168C2211yescharacteristic
51#3942C2211yescharacteristic, complemented
52#3942C2211yescharacteristic, complemented
53#3941C2211yescharacteristic, complemented
54#3941C2211yescharacteristic, complemented
55#3940C2211yescharacteristic, complemented
56#3940C2211yescharacteristic, complemented
57#3939C2211yescharacteristic, complemented
58#3939C2211yescharacteristic, complemented
59#3197—51nomaximal, Sylow
60C24×C5D411yescharacteristic
61C5×#41—41no—
62C5×#41—41no—
63C2×C5×Q8C2311yescharacteristic
64C2×C5×D4—21no—
65C2×C5×D4—21no—
66C2×C5×D4—21no—
67C2×C5×D4C2311yescharacteristic
68C22×C4×C5—21no—
69C22×C4×C5—41no—
70C22×C4×C5—41no—
71C22×C4×C5D411yescharacteristic
72C22×C4×C5D411yescharacteristic
73C22×C4×C5D411yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C2×C22⋊D5—21no—
78C2×C22⋊D5—21no—
79C2×C22⋊D5—21no—
80#709—41no—
81#709—41no—
82C2×D20C2311yescharacteristic
83C2×C4×D5—21no—
84C2×Dic10C2311yescharacteristic
85C5×C4⋊C4—21no—
86C5×C4⋊C4—21no—
87C5×C4⋊C4C2311yescharacteristic
88C5×C4⋊C4C2311yescharacteristic
89C5×C22⋊C4—21no—
90C5×C22⋊C4—21no—
91C5×C22⋊C4—21no—
92C5×C22⋊C4—21no—
93C5×C22⋊C4—21no—
94C42×C5—21no—
95C22⋊Dic5—21no—
96C22⋊Dic5—21no—
97C22⋊Dic5—21no—
98#689—41no—
99#688—41no—
100#687—41no—
101#686—41no—
102#685—21no—
103#685—21no—
104C4⋊Dic5C2311yescharacteristic
105C4⋊Dic5C2311yescharacteristic
106#683—21no—
107#683—21no—
108C4×Dic5—21no—
109#681—41no—
110#681—41no—
111C2×#168C2311yescharacteristic
112C2×#168C2311yescharacteristic
113C2×#168C2311yescharacteristic
114C2×#168C2311yescharacteristic
115#525—51no—
116#525D511yescharacteristic, complemented
117#485—51no—
118#485—51no—
119#482—51no—
120#482—51no—
121#470—51no—
122#469—51no—
123#468—51no—
124#468—51no—
125#452—51no—
126#452—51no—
127#449—51no—
128#449—51no—
129#407—51no—
130C23×C5—21no—
131C23×C5—41no—
132C23×C5—41no—
133C23×C5—41no—
134C23×C5C2×D411yescharacteristic
135C23×C5C2×D411yescharacteristic
136C23×C5C2×D411yescharacteristic
137C22×D5—21no—
138C5×Q8—21no—
139C5×Q8—21no—
140C5×D4—21no—
141C5×D4—21no—
142C5×D4—42no—
143C5×D4—42no—
144C5×D4—42no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—42no—
158C2×C4×C5C2×D411yescharacteristic
159C2×C4×C5C2×D411yescharacteristic
160C2×C4×C5C2×D411yescharacteristic
161C2×C4×C5C2411yescharacteristic
162C22⋊D5—42no—
163C22⋊D5—42no—
164C22⋊D5—42no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168D20—21no—
169D20—21no—
170C4×D5—42no—
171Dic10—21no—
172Dic10—21no—
173#168—21no—
174#168—21no—
175#168—21no—
176#168—21no—
177C2×#41—51no—
178C2×#41C2×D511yescharacteristic
179C23×C4C2×D511yescharacteristic
180C2×Q16—51no—
181C2×SD16—51no—
182C2×SD16—51no—
183C2×D8—51no—
184C2×M16—51no—
185C2×M16—51no—
186#123—21no—
187#123—101no—
188C22⋊Q8—51no—
189C22⋊Q8—51no—
190C22⋊Q8C2×D511yescharacteristic
191C22⋊Q8C2×D511yescharacteristic
192#121—51no—
193#121—51no—
194#121C2×D511yescharacteristic
195#121C2×D511yescharacteristic
196#120—21no—
197#120—101no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—101no—
201C4×D4—101no—
202#117—51no—
203#117C2×D511yescharacteristic
204#107—51no—
205#107—51no—
206#106—51no—
207#106—51no—
208#103—51no—
209#103—51no—
210#103—51no—
211#103—51no—
212#102—51no—
213#102—51no—
214#102—51no—
215#102—51no—
216C22⋊C8—51no—
217C22⋊C8—51no—
218C22⋊C8—51no—
219C22⋊C8—51no—
220C22×C5—21no—
221C22×C5—21no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—44no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5—41no—
233C22×C5C22×D411yescharacteristic
234C2×D5—41no—
235C2×D5—41no—
236C2×D5—41no—
237C4×C5—21no—
238C4×C5—21no—
239C4×C5—21no—
240C4×C5—41no—
241C4×C5—41no—
242C4×C5—41no—
243C4×C5H(5,𝔽2)11yescharacteristic
244C4×C5H(5,𝔽2)11yescharacteristic
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248C24C22⋊D511yescharacteristic
249#41—41no—
250#41—41no—
251#41—201no—
252#41—201no—
253C2×Q8—51no—
254C2×Q8C22×D511yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—51no—
259C2×D4—101no—
260C2×D4—101no—
261C2×D4—101no—
262C2×D4C22×D511yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—101no—
267C22×C4C22⋊D511yescharacteristic
268C22×C4C22⋊D511yescharacteristic
269C22×C4C22⋊D511yescharacteristic
270C22×C4C22×D511yescharacteristic
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273Q16—201no—
274SD16—201no—
275SD16—201no—
276D8—201no—
277M16—201no—
278M16—201no—
279C2×C8—51no—
280C2×C8—51no—
281C2×C8—51no—
282C2×C8—51no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—51no—
286C4⋊C4—51no—
287C4⋊C4—101no—
288C4⋊C4—101no—
289C4⋊C4C22×D511yescharacteristic
290C4⋊C4C22×D511yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—101no—
297C22⋊C4—101no—
298C22⋊C4—101no—
299C22⋊C4—101no—
300C22⋊C4—101no—
301C42—21no—
302C42—101no—
303C2×C5—41no—
304C2×C5—41no—
305C2×C5—41no—
306C2×C5—81no—
307C2×C5#53411yescharacteristic
308C2×C5#57511yescharacteristic
309C2×C5#57511yescharacteristic
310D5—81no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—101no—
316C23C2×C22⋊D511yescharacteristic
317C23C2×C22⋊D511yescharacteristic
318C23C2×C22⋊D511yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—101no—
322Q8—101no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—101no—
329D4—101no—
330D4—202no—
331D4—202no—
332D4—202no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—202no—
350C2×C4C2×C22⋊D511yescharacteristic
351C2×C4C2×C22⋊D511yescharacteristic
352C2×C4C2×C22⋊D511yescharacteristic
353C2×C4C23×D511yescharacteristic
354C8—101no—
355C8—101no—
356C8—101no—
357C8—101no—
358C5#319711yescharacteristic, complemented, Sylow
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—201no—
373C22—201no—
374C22—201no—
375C22C22×C22⋊D511yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—201no—
383C4—201no—
384C4—201no—
385C4#414411yescharacteristic
386C4#414411yescharacteristic
387C2—41no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—401no—
392C2#6613811yescharacteristic
393C2#6615611yescharacteristic
394C2#6615611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#525C2
socleC22×C5C22×D4
hypercenter#525D5
hypocenterC5#3197
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C5×#525C2semidirect
2. #66125C2semidirect
3. #65528C2semidirect
4. #65527C2semidirect
5. #65496C2semidirect
6. #65341C2semidirect
7. #65338C2semidirect
8. #65336C2semidirect
9. #65334C2semidirect
10. #65331C2semidirect
11. #65328C2semidirect
12. #65326C2semidirect
13. #65324C2semidirect
14. #65292C2semidirect
15. #65285C2semidirect
29. C2×#689C22semidirect
30. C2×#688C22semidirect
31. C2×#687C22semidirect
32. C2×#686C22semidirect
43. #3967C22semidirect
44. #3967C22semidirect
45. #3964C22semidirect
46. #3964C22semidirect
51. #3942C22semidirect
52. #3942C22semidirect
53. #3941C22semidirect
54. #3941C22semidirect
55. #3940C22semidirect
56. #3940C22semidirect
57. #3939C22semidirect
58. #3939C22semidirect
116. #525D5semidirect
358. C5#3197semidirect

Maximal subgroups

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

subgroupindex
C5×#5252
#661252
#655282
#655272
#654962
#653412
#653382
#653362
#653342
#653312
#653282
#653262
#653242
#652922
#652852
#31975