Finite Group Databasebeta

← #10612137

Subgroups — #10612137

Order 640.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×D5×SD16C212yesmaximal, complemented
2#66051C211yesmaximal, characteristic
3Q8×D20C211yesmaximal, characteristic, complemented
4C5×#460C211yesmaximal, characteristic, complemented
5#65459C212yesmaximal, complemented
6#65365C211yesmaximal, characteristic, complemented
7C4×#688C211yesmaximal, characteristic, complemented
8#65130C211yesmaximal, characteristic, complemented
9#65123C211yesmaximal, characteristic, complemented
10#65067C212yesmaximal, complemented
11#65065C212yesmaximal, complemented
12C2×Q8×D5C2212yescomplemented
13C2×D4×D5—21no—
14C2×D4×D5C2212yes—
15C2×C5×SD16C2212yescomplemented
16C5×#127C2211yescharacteristic
17C4×C5×Q8C2211yescharacteristic, complemented
18C2×#688C2211yescharacteristic
19#4084—21no—
20#4084C2212yes—
21#4083—22no—
22C2×#687C2212yescomplemented
23C2×#687C2212yescomplemented
24D5×SD16—28no—
25C2×#677C2212yes—
26C2×C8×D5C2212yescomplemented
27#4043—22no—
28#4043C2212yescomplemented
29C4×D20—21no—
30C4×D20C2211yescharacteristic
31C4×D20C2211yescharacteristic
32#4015—21no—
33#4015C2211yescharacteristic, complemented
34C5×C4⋊C8C2211yescharacteristic, complemented
35C5×#102C2212yescomplemented
36#3967C2211yescharacteristic, complemented
37#3952C2212yescomplemented
38#3948C2212yescomplemented
39#3941C2211yescharacteristic
40#3940C2211yescharacteristic, complemented
41C4×#168C2211yescharacteristic, complemented
42D4×SD16—51nomaximal, Sylow
43C23×D5—22no—
44C2×C5×Q8C2311yescharacteristic
45C2×C5×D4—21no—
46C2×C5×D4—21no—
47C2×C5×D4C2312yes—
48C2×C22⋊D5—22no—
49C2×C22⋊D5—22no—
50Q8×D5—24no—
51Q8×D5—42no—
52D4×D5—24no—
53D4×D5—42no—
54D4×D5—44no—
55C2×D20D411yescharacteristic
56C2×D20C2311yescharacteristic
57C2×C4×D5—22no—
58C2×C4×D5—21no—
59C2×C4×D5C2312yes—
60C2×Dic10—22no—
61C2×Dic10C2312yes—
62C5×SD16—24no—
63C2×C5×C8C2312yes—
64C5×C4⋊C4—21no—
65C5×C4⋊C4C2311yescharacteristic
66C42×C5—21no—
67C42×C5C2311yescharacteristic
68C22⋊Dic5—22no—
69#688D414yescomplemented
70#687—24no—
71#687—42no—
72#685—22no—
73#685—21no—
74#685D412yes—
75C4⋊Dic5—21no—
76C4⋊Dic5—21no—
77C4⋊Dic5—22no—
78C4⋊Dic5D411yescharacteristic
79C4⋊Dic5C2311yescharacteristic
80C2×#168C2311yescharacteristic
81C2×#168C2311yescharacteristic
82#677—24no—
83C8×D5—24no—
84C22×SD16—52no—
85D4×Q8—51no—
86D42—51no—
87#492—51no—
88#465—52no—
89#461—51no—
90#460D511yescharacteristic, complemented
91#450—52no—
92#448—52no—
93C4×SD16—51no—
94C8×D4—51no—
95C23×C5—22no—
96C22×D5—21no—
97C22×D5—44no—
98C22×D5—44no—
99C22×D5—82no—
100C22×D5C2×D412yes—
101C5×Q8—21no—
102C5×Q8C2×D412yescomplemented
103C5×D4—22no—
104C5×D4—22no—
105C5×D4—42no—
106C5×D4—81no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5C2×D411yescharacteristic
110C2×C4×C5C2×D411yescharacteristic
111C2×C4×C5C2411yescharacteristic
112C22⋊D5—44no—
113C22⋊D5—44no—
114C2×Dic5—22no—
115C2×Dic5—21no—
116C2×Dic5C2×D412yes—
117D20—21no—
118D20SD1614yescomplemented
119D20C2×D412yes—
120C4×D5—22no—
121C4×D5—22no—
122C4×D5—44no—
123C4×D5—42no—
124Dic10—22no—
125Dic10—22no—
126Dic10—44no—
127C5×C8—22no—
128#168—21no—
129#168C2×D412yescomplemented
130C22×Q8—52no—
131C22×D4—52no—
132C22×D4—101no—
133C2×SD16—51no—
134C2×SD16—52no—
135C2×SD16—52no—
136C2×SD16—52no—
137C2×SD16—108no—
138C2×SD16C2×D512yescomplemented
139C22×C8—52no—
140C4⋊Q8—51no—
141C4⋊Q8—101no—
142#127C2×D511yescharacteristic
143C22⋊Q8—52no—
144C22⋊Q8—102no—
145#121—52no—
146#121—101no—
147#120—102no—
148C4×Q8C2×D511yescharacteristic, complemented
149C4×D4—51no—
150C4×D4—51no—
151C4×D4—101no—
152#106—51no—
153C4⋊C8C2×D511yescharacteristic, complemented
154#103—51no—
155#103—52no—
156#102—51no—
157#102C2×D512yescomplemented
158C22⋊C8—52no—
159C4×C8—51no—
160C22×C5—42no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5C22×D411yescharacteristic
164C2×D5—22no—
165C2×D5—22no—
166C2×D5—22no—
167C2×D5—41no—
168C2×D5—41no—
169C2×D5—88no—
170C2×D5C2×SD1614yescomplemented
171C4×C5—22no—
172C4×C5—21no—
173C4×C5—41no—
174C4×C5C2×SD1612yescomplemented
175C4×C5C22×D411yescharacteristic
176C4×C5H(5,𝔽2)11yescharacteristic
177Dic5—22no—
178Dic5—42no—
179Dic5—41no—
180C24—102no—
181C2×Q8—52no—
182C2×Q8—102no—
183C2×Q8—104no—
184C2×Q8—202no—
185C2×Q8C22×D511yescharacteristic
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—51no—
189C2×D4—51no—
190C2×D4—102no—
191C2×D4—104no—
192C2×D4—102no—
193C2×D4—202no—
194C2×D4—204no—
195C2×D4C22×D512yes—
196C22×C4—52no—
197C22×C4—102no—
198C22×C4—101no—
199SD16—24no—
200SD16—54no—
201SD16—104no—
202SD16—104no—
203SD16—202no—
204C2×C8—51no—
205C2×C8—51no—
206C2×C8—104no—
207C2×C8C22×D512yes—
208C4⋊C4—21no—
209C4⋊C4—51no—
210C4⋊C4—51no—
211C4⋊C4—102no—
212C4⋊C4—101no—
213C4⋊C4—101no—
214C4⋊C4C22×D511yescharacteristic
215C22⋊C4—52no—
216C22⋊C4—102no—
217C22⋊C4—102no—
218C22⋊C4—101no—
219C42—21no—
220C42C22×D511yescharacteristic
221C2×C5—82no—
222C2×C5D4211yescharacteristic
223C2×C5C22×SD1611yescharacteristic
224C2×C5#57711yescharacteristic
225D5—24no—
226D5—42no—
227C23—22no—
228C23—52no—
229C23—101no—
230C23—204no—
231C23—204no—
232C23—402no—
233Q8—21no—
234Q8—102no—
235Q8—102no—
236Q8—204no—
237Q8D4×D512yescomplemented
238D4—22no—
239D4—22no—
240D4—42no—
241D4—52no—
242D4—54no—
243D4—81no—
244D4—101no—
245D4—204no—
246D4—204no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—52no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4—101no—
253C2×C4—102no—
254C2×C4—204no—
255C2×C4—202no—
256C2×C4D4×D511yescharacteristic
257C2×C4D4×D511yescharacteristic
258C2×C4C23×D511yescharacteristic
259C8—22no—
260C8—52no—
261C8—101no—
262C5D4×SD1611yescharacteristic, complemented, Sylow
263C22—42no—
264C22—42no—
265C22—42no—
266C22—54no—
267C22—102no—
268C22—102no—
269C22—102no—
270C22—201no—
271C22—201no—
272C22—408no—
273C22C2×D4×D511yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—102no—
278C4—201no—
279C4—202no—
280C4D5×SD1612yescomplemented
281C4C2×D4×D511yescharacteristic
282C4#414911yescharacteristic
283C2—82no—
284C2—104no—
285C2—202no—
286C2#6594311yescharacteristic
287C2C2×D5×SD1611yescharacteristic
288C2#6610911yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#460C2
socleC22×C5C22×D4
prime radical#66051C2
hypercenter#460D5
hypocenterC5D4×SD16
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. C2×D5×SD16C2semidirect
3. Q8×D20C2semidirect
4. C5×#460C2semidirect
5. #65459C2semidirect
6. #65365C2semidirect
7. C4×#688C2semidirect
8. #65130C2semidirect
9. #65123C2semidirect
10. #65067C2semidirect
11. #65065C2semidirect
12. C2×Q8×D5C22semidirect
15. C2×C5×SD16C22semidirect
17. C4×C5×Q8C22semidirect
22. C2×#687C22semidirect
23. C2×#687C22semidirect
26. C2×C8×D5C22semidirect
28. #4043C22semidirect
33. #4015C22semidirect
34. C5×C4⋊C8C22semidirect
35. C5×#102C22semidirect
36. #3967C22semidirect
37. #3952C22semidirect
38. #3948C22semidirect
40. #3940C22semidirect
41. C4×#168C22semidirect
69. #688D4semidirect
90. #460D5semidirect
102. C5×Q8C2×D4semidirect
118. D20SD16semidirect
129. #168C2×D4semidirect
138. C2×SD16C2×D5semidirect
148. C4×Q8C2×D5semidirect
153. C4⋊C8C2×D5semidirect
157. #102C2×D5semidirect
170. C2×D5C2×SD16semidirect
174. C4×C5C2×SD16semidirect
237. Q8D4×D5semidirect
262. C5D4×SD16semidirect
280. C4D5×SD16semidirect

Maximal subgroups

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

subgroupindex
C2×D5×SD162
#660512
Q8×D202
C5×#4602
#654592
#653652
C4×#6882
#651302
#651232
#650672
#650652
D4×SD165