Finite Group Databasebeta

← #10613062

Subgroups — #10613062

Order 640.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65946C211yesmaximal, characteristic, complemented
2#65841C211yesmaximal, characteristic, complemented
3C5×#481C211yesmaximal, characteristic, complemented
4#65328C211yesmaximal, characteristic, complemented
5#65255C211yesmaximal, characteristic, complemented
6#65159C211yesmaximal, characteristic, complemented
7#65158C211yesmaximal, characteristic, complemented
8#65153C211yesmaximal, characteristic, complemented
9#65151C211yesmaximal, characteristic, complemented
10#65076C211yesmaximal, characteristic, complemented
11#65062C211yesmaximal, characteristic, complemented
12D5×#102C211yesmaximal, characteristic, complemented
13#65054C211yesmaximal, characteristic, complemented
14#65023C211yesmaximal, characteristic, complemented
15#65017C211yesmaximal, characteristic, complemented
16C2×D4×D5C2211yescharacteristic
17C2×#709—21no—
18C2×#709C2211yescharacteristic
19C5×#121C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic, complemented
21#4084—21no—
22#4083—21no—
23#4082C2211yescharacteristic
24C2×#686C2211yescharacteristic
25#4076C2211yescharacteristic, complemented
26#4075—21no—
27C4×C22⋊D5C2211yescharacteristic
28C2×#681C2211yescharacteristic, complemented
29C2×D40C2211yescharacteristic
30C2×#677C2211yescharacteristic, complemented
31C2×C8×D5C2211yescharacteristic
32#4043C2211yescharacteristic
33#4042—21no—
34#4041C2211yescharacteristic
35#4040—21no—
36#4039C2211yescharacteristic
37#4039C2211yescharacteristic, complemented
38D5×C4⋊C4C2211yescharacteristic
39#4034C2211yescharacteristic, complemented
40#4029—21no—
41D5×C22⋊C4—21no—
42C5×#106C2211yescharacteristic, complemented
43C5×#106C2211yescharacteristic, 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#3941C2211yescharacteristic
56#3941C2211yescharacteristic
57#3941C2211yescharacteristic, complemented
58#3940C2211yescharacteristic, complemented
59#3306—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⋊D5—21no—
68C2×C22⋊D5D411yescharacteristic
69D4×D5—22no—
70D4×D5—41no—
71#709—22no—
72#709—41no—
73#709—44no—
74C2×D20—21no—
75C2×D20C2311yescharacteristic
76C2×D20C2311yescharacteristic
77C2×C4×D5—21no—
78C2×C4×D5—21no—
79C2×C4×D5—21no—
80C2×C4×D5—21no—
81C2×C4×D5C2311yescharacteristic
82C2×Dic10—21no—
83C2×Dic10C2311yescharacteristic
84C2×C5×C8C2311yescharacteristic
85C2×C5×C8C2311yescharacteristic
86C5×C4⋊C4—21no—
87C5×C4⋊C4C2311yescharacteristic
88C5×C4⋊C4C2311yescharacteristic
89C5×C4⋊C4C2311yescharacteristic
90C5×C22⋊C4—21no—
91C22⋊Dic5—21no—
92C22⋊Dic5—21no—
93C22⋊Dic5D411yescharacteristic
94#686—41no—
95#685—21no—
96#685—21no—
97#685—21no—
98#685—21no—
99#685—21no—
100#685D411yescharacteristic
101C4⋊Dic5—21no—
102C4⋊Dic5C2311yescharacteristic
103#683—21no—
104#683—21no—
105#683D411yescharacteristic
106C4×Dic5C2311yescharacteristic
107#681—41no—
108C2×#168C2311yescharacteristic
109C2×#168C2311yescharacteristic
110D40—22no—
111#677—41no—
112C8×D5—41no—
113#547—51no—
114#546—51no—
115#489—51no—
116#483—51no—
117#481D511yescharacteristic, complemented
118#476—51no—
119#469—51no—
120#465—51no—
121#463—51no—
122#449—51no—
123#447—51no—
124#442—51no—
125#435—51no—
126#417—51no—
127C2×#102—51no—
128C23×C5—21no—
129C23×C5C2×D411yescharacteristic
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—42no—
133C22×D5—42no—
134C22×D5—81no—
135C22×D5C2×D411yescharacteristic
136C5×D4—21no—
137C5×D4—21no—
138C5×D4—81no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—42no—
145C2×C4×C5C2×D411yescharacteristic
146C2×C4×C5C2411yescharacteristic
147C22⋊D5—22no—
148C22⋊D5—42no—
149C22⋊D5—42no—
150C22⋊D5—42no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5C2×D411yescharacteristic
154C2×Dic5C2×D411yescharacteristic
155C2×Dic5C2×D411yescharacteristic
156D20—21no—
157D20—21no—
158D20—21no—
159D20—42no—
160D20#4112yescomplemented
161C4×D5—21no—
162C4×D5—21no—
163C4×D5—42no—
164C4×D5—42no—
165C4×D5—42no—
166C4×D5—81no—
167Dic10—21no—
168Dic10—21no—
169Dic10—42no—
170C5×C8—21no—
171C5×C8—21no—
172#168—21no—
173#168—21no—
174C2×#41—51no—
175C2×#41—101no—
176C22×D4—51no—
177C2×SD16—51no—
178C2×D8—51no—
179C2×D8—51no—
180C2×M16—51no—
181C22×C8—51no—
182C4⋊Q8—51no—
183#124—51no—
184#123—101no—
185#123—101no—
186#123—101no—
187C22⋊Q8—51no—
188C22⋊Q8—101no—
189#121—51no—
190#121—51no—
191#121—101no—
192#121C2×D511yescharacteristic
193#120—101no—
194C4×D4—51no—
195C4×D4—51no—
196C4×D4—51no—
197C2×C4⋊C4—51no—
198C2×C4⋊C4C2×D511yescharacteristic, complemented
199C2×C22⋊C4—101no—
200#106—51no—
201#106C2×D511yescharacteristic, complemented
202#106C2×D511yescharacteristic, complemented
203C4⋊C8—51no—
204#103—51no—
205#103—51no—
206#103—51no—
207#102—51no—
208#102—51no—
209#102—51no—
210#102—51no—
211#102—51no—
212#102C2×D511yescharacteristic
213#102C2×D511yescharacteristic
214C22⋊C8—51no—
215C22⋊C8C2×D511yescharacteristic, complemented
216#97—51no—
217C22×C5—21no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5C22×D411yescharacteristic
224C2×D5—21no—
225C2×D5—21no—
226C2×D5—22no—
227C2×D5—41no—
228C2×D5—41no—
229C2×D5—41no—
230C2×D5—41no—
231C2×D5—41no—
232C2×D5—84no—
233C2×D5AGL(1,ℤ/8)12yes—
234C4×C5—21no—
235C4×C5—41no—
236C4×C5—41no—
237C4×C5—41no—
238C4×C5C2×#4111yescharacteristic
239C4×C5H(5,𝔽2)11yescharacteristic
240Dic5—21no—
241Dic5—21no—
242Dic5—21no—
243Dic5—41no—
244Dic5—41no—
245C24—101no—
246#41—102no—
247#41—201no—
248#41—204no—
249C2×Q8—51no—
250C2×Q8—101no—
251C2×D4—21no—
252C2×D4—51no—
253C2×D4—51no—
254C2×D4—51no—
255C2×D4—101no—
256C2×D4—101no—
257C2×D4—102no—
258C2×D4—101no—
259C2×D4—101no—
260C2×D4—201no—
261C2×D4C22×D511yescharacteristic
262C22×C4—21no—
263C22×C4—51no—
264C22×C4—101no—
265C22×C4—101no—
266C22×C4—101no—
267C22×C4—101no—
268C22×C4C22×D511yescharacteristic
269SD16—201no—
270D8—102no—
271D8—201no—
272M16—201no—
273C2×C8—51no—
274C2×C8—51no—
275C2×C8—201no—
276C2×C8C22×D511yescharacteristic
277C2×C8C22×D511yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—51no—
280C4⋊C4—51no—
281C4⋊C4—101no—
282C4⋊C4—101no—
283C4⋊C4—101no—
284C4⋊C4C22×D511yescharacteristic
285C4⋊C4C22×D511yescharacteristic
286C4⋊C4C22×D511yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—51no—
289C22⋊C4—51no—
290C22⋊C4—101no—
291C22⋊C4—101no—
292C22⋊C4—101no—
293C22⋊C4—101no—
294C22⋊C4—101no—
295C22⋊C4—101no—
296C22⋊C4—101no—
297C42—51no—
298C2×C5—41no—
299C2×C5—81no—
300C2×C5#54611yescharacteristic
301C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
302C2×C5#57711yescharacteristic
303D5—22no—
304D5—42no—
305D5—81no—
306C23—21no—
307C23—51no—
308C23—101no—
309C23—101no—
310C23—202no—
311C23—202no—
312C23—401no—
313C23D4×D511yescharacteristic
314Q8—101no—
315Q8—101no—
316Q8—202no—
317D4—21no—
318D4—21no—
319D4—52no—
320D4—81no—
321D4—102no—
322D4—101no—
323D4—101no—
324D4—101no—
325D4—202no—
326D4—202no—
327D4—202no—
328D4—202no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—51no—
336C2×C4—51no—
337C2×C4—51no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—101no—
342C2×C4—202no—
343C2×C4—202no—
344C2×C4—202no—
345C2×C4—401no—
346C2×C4D4×D511yescharacteristic
347C2×C4C23×D511yescharacteristic
348C8—21no—
349C8—21no—
350C8—101no—
351C8—101no—
352C5#330611yescharacteristic, complemented, Sylow
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—52no—
360C22—101no—
361C22—101no—
362C22—102no—
363C22—201no—
364C22—201no—
365C22—201no—
366C22—201no—
367C22—201no—
368C22—404no—
369C22C2×D4×D511yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—101no—
375C4—101no—
376C4—101no—
377C4—201no—
378C4—201no—
379C4D5×#4111yescharacteristic
380C4#414911yescharacteristic
381C2—41no—
382C2—81no—
383C2—102no—
384C2—202no—
385C2—401no—
386C2#6598811yescharacteristic
387C2#6609611yescharacteristic
388C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#481C2
socleC22×C5C22×D4
hypercenter#481D5
hypocenterC5#3306
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #65946C2semidirect
2. #65841C2semidirect
3. C5×#481C2semidirect
4. #65328C2semidirect
5. #65255C2semidirect
6. #65159C2semidirect
7. #65158C2semidirect
8. #65153C2semidirect
9. #65151C2semidirect
10. #65076C2semidirect
11. #65062C2semidirect
12. D5×#102C2semidirect
13. #65054C2semidirect
14. #65023C2semidirect
15. #65017C2semidirect
20. C2×C5×C4⋊C4C22semidirect
25. #4076C22semidirect
28. C2×#681C22semidirect
30. C2×#677C22semidirect
37. #4039C22semidirect
39. #4034C22semidirect
42. C5×#106C22semidirect
43. C5×#106C22semidirect
46. C5×C22⋊C8C22semidirect
50. #3948C22semidirect
51. #3947C22semidirect
52. #3946C22semidirect
53. #3942C22semidirect
54. #3942C22semidirect
57. #3941C22semidirect
58. #3940C22semidirect
117. #481D5semidirect
160. D20#41semidirect
198. C2×C4⋊C4C2×D5semidirect
201. #106C2×D5semidirect
202. #106C2×D5semidirect
215. C22⋊C8C2×D5semidirect
352. C5#3306semidirect

Maximal subgroups

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

subgroupindex
#659462
#658412
C5×#4812
#653282
#652552
#651592
#651582
#651532
#651512
#650762
#650622
D5×#1022
#650542
#650232
#650172
#33065