Finite Group Databasebeta

← #10612121

Subgroups — #10612121

Order 640.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#127C211yesmaximal, characteristic, complemented
2#65914C211yesmaximal, characteristic, complemented
3C5×#460C211yesmaximal, characteristic, complemented
4#65465C212yesmaximal, complemented
5#65458C212yesmaximal, complemented
6#65361C211yesmaximal, characteristic, complemented
7#65316C211yesmaximal, characteristic, complemented
8#65132C211yesmaximal, characteristic, complemented
9#65124C211yesmaximal, characteristic, complemented
10#65062C212yesmaximal, complemented
11#65053C212yesmaximal, complemented
12C2×D4×D5—21no—
13C2×D4×D5—21no—
14C2×D4×D5C2212yes—
15C2×C5×SD16C2212yescomplemented
16C5×#127C2211yescharacteristic
17C4×C5×Q8C2211yescharacteristic, complemented
18#4093C2211yescharacteristic, complemented
19#4092C2211yescharacteristic
20C2×#688C2212yescomplemented
21#4086—21no—
22#4086—21no—
23#4086C2212yes—
24C2×#686C2212yescomplemented
25C2×D40C2212yescomplemented
26#4044—21no—
27#4041C2211yescharacteristic
28#4040—21no—
29#4036C2211yescharacteristic, complemented
30#4020C2211yescharacteristic
31C4×D20—21no—
32C4×D20C2211yescharacteristic, complemented
33#4018—21no—
34C42×D5C2211yescharacteristic
35C5×C4⋊C8C2211yescharacteristic, complemented
36C5×#102C2212yescomplemented
37#3964C2212yescomplemented
38#3953C2212yescomplemented
39#3952C2212yes—
40#3947C2212yescomplemented
41#3941C2212yescomplemented
42C4⋊#168C2211yescharacteristic, complemented
43#3217—51nomaximal, Sylow
44C23×D5—22no—
45C2×C5×Q8C2311yescharacteristic
46C2×C5×D4—21no—
47C2×C5×D4—21no—
48C2×C5×D4C2312yes—
49C2×C22⋊D5—22no—
50C2×C22⋊D5—22no—
51C2×C22⋊D5—22no—
52D4×D5—42no—
53D4×D5—42no—
54D4×D5—44no—
55D4×D5—44no—
56C2×D20—21no—
57C2×D20—21no—
58C2×D20—21no—
59C2×D20C2312yes—
60C2×D20C2311yescharacteristic
61C2×C4×D5—21no—
62C2×C4×D5D411yescharacteristic
63C2×C4×D5D411yescharacteristic
64C2×C4×D5C2311yescharacteristic
65C5×SD16—42no—
66C2×C5×C8C2312yes—
67C5×C4⋊C4—21no—
68C5×C4⋊C4C2311yescharacteristic
69C42×C5—21no—
70C42×C5C2311yescharacteristic
71#688—42no—
72#686—42no—
73#685—21no—
74#685—21no—
75#685—21no—
76#685—21no—
77#685—21no—
78C4⋊Dic5—21no—
79C4⋊Dic5C2311yescharacteristic
80#683—21no—
81#683—21no—
82C4×Dic5D411yescharacteristic
83C4×Dic5D411yescharacteristic
84C4×Dic5C2311yescharacteristic
85C2×#168C2312yes—
86D40—42no—
87C2×#127—51no—
88#529—51no—
89#496—52no—
90#490—52no—
91#460—51no—
92#460D511yescharacteristic, complemented
93#459—51no—
94#459—51no—
95#450—52no—
96#447—52no—
97#432—51no—
98C23×C5—22no—
99C22×D5—21no—
100C22×D5—22no—
101C22×D5—44no—
102C22×D5—44no—
103C22×D5—44no—
104C22×D5C2×D411yescharacteristic
105C5×Q8—21no—
106C5×Q8—21no—
107C5×D4—22no—
108C5×D4—22no—
109C5×D4—42no—
110C5×D4—42no—
111C2×C4×C5—21no—
112C2×C4×C5—21no—
113C2×C4×C5C2×D411yescharacteristic
114C2×C4×C5C2×D411yescharacteristic
115C2×C4×C5C2411yescharacteristic
116C22⋊D5—44no—
117C22⋊D5—44no—
118C22⋊D5—44no—
119C2×Dic5—21no—
120C2×Dic5C2×D411yescharacteristic
121C2×Dic5C2×D411yescharacteristic
122C2×Dic5C2×D411yescharacteristic
123D20—21no—
124D20—22no—
125D20—21no—
126D20—22no—
127D20—42no—
128D20—42no—
129D20—42no—
130C4×D5—22no—
131C4×D5—22no—
132C4×D5—21no—
133C4×D5—21no—
134C4×D5—42no—
135C5×C8—22no—
136#168—22no—
137C22×D4—52no—
138C22×D4—101no—
139C22×D4—101no—
140C2×SD16—52no—
141C2×SD16C2×D512yescomplemented
142C2×D8—52no—
143C2×D8—52no—
144#127—52no—
145#127—51no—
146#127—101no—
147#127—101no—
148#127C2×D511yescharacteristic
149#126—101no—
150#125—51no—
151#124—51no—
152#123—101no—
153C22⋊Q8—51no—
154#121—51no—
155C4×Q8C2×D511yescharacteristic, complemented
156C4×D4—51no—
157C4×D4—101no—
158#117—101no—
159C2×C42—51no—
160C4⋊C8—51no—
161C4⋊C8C2×D511yescharacteristic, complemented
162#102—52no—
163#102—52no—
164#102—52no—
165#102C2×D512yescomplemented
166C22⋊C8—52no—
167#97—52no—
168C22×C5—42no—
169C22×C5—42no—
170C22×C5—42no—
171C22×C5C22×D411yescharacteristic
172C2×D5—21no—
173C2×D5—21no—
174C2×D5—42no—
175C2×D5—41no—
176C2×D5—41no—
177C2×D5—42no—
178C2×D5—42no—
179C2×D5—41no—
180C2×D5—88no—
181C2×D5AGL(1,ℤ/8)12yes—
182C4×C5—21no—
183C4×C5—41no—
184C4×C5—41no—
185C4×C5AGL(1,ℤ/8)12yes—
186C4×C5H(5,𝔽2)11yescharacteristic
187C4×C5H(5,𝔽2)11yescharacteristic
188Dic5—21no—
189Dic5—21no—
190Dic5—21no—
191Dic5—41no—
192C24—102no—
193C2×Q8C22×D511yescharacteristic
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—52no—
197C2×D4—51no—
198C2×D4—101no—
199C2×D4—102no—
200C2×D4—102no—
201C2×D4—102no—
202C2×D4—101no—
203C2×D4—101no—
204C2×D4—202no—
205C2×D4—202no—
206C2×D4—204no—
207C2×D4—204no—
208C2×D4C22×D512yes—
209C22×C4—51no—
210C22×C4—51no—
211C22×C4—51no—
212C22×C4—101no—
213SD16—42no—
214SD16—202no—
215D8—202no—
216D8—202no—
217C2×C8—52no—
218C2×C8C22×D512yes—
219C4⋊C4—21no—
220C4⋊C4—51no—
221C4⋊C4—101no—
222C4⋊C4—101no—
223C4⋊C4—101no—
224C4⋊C4C22×D511yescharacteristic
225C22⋊C4—101no—
226C22⋊C4—101no—
227C22⋊C4—101no—
228C22⋊C4—101no—
229C22⋊C4—101no—
230C42—21no—
231C42—51no—
232C42—51no—
233C42—51no—
234C42C22×D511yescharacteristic
235C2×C5—82no—
236C2×C5#53511yescharacteristic
237C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
238C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
239D5—22no—
240D5—81no—
241D5—82no—
242C23—22no—
243C23—51no—
244C23—101no—
245C23—102no—
246C23—204no—
247C23—204no—
248C23—204no—
249Q8—21no—
250Q8—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—42no—
255D4—101no—
256D4—102no—
257D4—101no—
258D4—102no—
259D4—202no—
260D4—202no—
261D4—204no—
262D4—204no—
263D4—204no—
264D4—202no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—51no—
268C2×C4—51no—
269C2×C4—51no—
270C2×C4—102no—
271C2×C4—101no—
272C2×C4—101no—
273C2×C4—102no—
274C2×C4—101no—
275C2×C4—202no—
276C2×C4D4×D511yescharacteristic
277C2×C4D4×D511yescharacteristic
278C2×C4C23×D511yescharacteristic
279C8—22no—
280C8—102no—
281C5#321711yescharacteristic, complemented, Sylow
282C22—42no—
283C22—42no—
284C22—42no—
285C22—52no—
286C22—101no—
287C22—101no—
288C22—201no—
289C22—202no—
290C22—202no—
291C22—201no—
292C22—202no—
293C22—201no—
294C22—408no—
295C22C2×D4×D511yescharacteristic
296C4—21no—
297C4—41no—
298C4—41no—
299C4—101no—
300C4—101no—
301C4—101no—
302C4—201no—
303C4#406012yes—
304C4#414411yescharacteristic
305C4#414911yescharacteristic
306C2—82no—
307C2—102no—
308C2—402no—
309C2—401no—
310C2#6594111yescharacteristic
311C2C2×#406011yescharacteristic
312C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#460C2
socleC22×C5C22×D4
hypercenter#460D5
hypocenterC5#3217
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. D5×#127C2semidirect
2. #65914C2semidirect
3. C5×#460C2semidirect
4. #65465C2semidirect
5. #65458C2semidirect
6. #65361C2semidirect
7. #65316C2semidirect
8. #65132C2semidirect
9. #65124C2semidirect
10. #65062C2semidirect
11. #65053C2semidirect
15. C2×C5×SD16C22semidirect
17. C4×C5×Q8C22semidirect
18. #4093C22semidirect
20. C2×#688C22semidirect
24. C2×#686C22semidirect
25. C2×D40C22semidirect
29. #4036C22semidirect
32. C4×D20C22semidirect
35. C5×C4⋊C8C22semidirect
36. C5×#102C22semidirect
37. #3964C22semidirect
38. #3953C22semidirect
40. #3947C22semidirect
41. #3941C22semidirect
42. C4⋊#168C22semidirect
92. #460D5semidirect
141. C2×SD16C2×D5semidirect
155. C4×Q8C2×D5semidirect
161. C4⋊C8C2×D5semidirect
165. #102C2×D5semidirect
281. C5#3217semidirect

Maximal subgroups

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

subgroupindex
D5×#1272
#659142
C5×#4602
#654652
#654582
#653612
#653162
#651322
#651242
#650622
#650532
#32175