Finite Group Databasebeta

← #10611871

Subgroups — #10611871

Order 640.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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×#4082C211yesmaximal, characteristic
2#65956C211yesmaximal, characteristic, complemented
3C5×#450C211yesmaximal, characteristic
4#65507C211yesmaximal, characteristic
5#65456C212yesmaximal, complemented
6#65455C212yesmaximal, complemented
7#65339C211yesmaximal, characteristic, complemented
8#65056C212yesmaximal, complemented
9#65054C212yesmaximal, complemented
10#65028C211yesmaximal, characteristic, complemented
11#65008C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×SD16C2212yes—
15C5×C22⋊Q8C2211yescharacteristic
16C2×C22⋊Dic5—21no—
17C2×C22⋊Dic5—21no—
18Q8×Dic5C2211yescharacteristic, complemented
19#4090C2211yescharacteristic, complemented
20C2×#689C2212yescomplemented
21#4082—21no—
22#4082—21no—
23#4082C2212yes—
24#4082C2212yes—
25C2×#687C2212yes—
26#4070C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28C2×Dic20C2212yescomplemented
29#4035—21no—
30#4034C2211yescharacteristic, complemented
31#4033—21no—
32#4033C2211yescharacteristic, complemented
33#4024—21no—
34#4023—21no—
35C5×#102C2212yes—
36C5×C22⋊C8C2211yescharacteristic
37#3964C2212yes—
38C22⋊#168C2211yescharacteristic
39#3948C2212yescomplemented
40#3947C2212yescomplemented
41#3946C2212yescomplemented
42#3942C2212yescomplemented
43#3215—51nomaximal, Sylow
44C24×C5—21no—
45C2×C5×Q8C2311yescharacteristic
46C2×C5×D4—21no—
47C2×C5×D4—42no—
48C2×C5×D4C2312yes—
49C22×C4×C5C2311yescharacteristic
50C22×Dic5—21no—
51C22×Dic5D411yescharacteristic
52C22×Dic5D411yescharacteristic
53C2×Dic10—21no—
54C2×Dic10—21no—
55C2×Dic10—42no—
56C2×Dic10C2312yes—
57C2×Dic10C2311yescharacteristic
58C5×SD16—42no—
59C2×C5×C8C2312yes—
60C5×C4⋊C4—21no—
61C5×C4⋊C4C2311yescharacteristic
62C5×C22⋊C4—21no—
63C22⋊Dic5—21no—
64C22⋊Dic5—22no—
65C22⋊Dic5—22no—
66C22⋊Dic5—22no—
67C22⋊Dic5—22no—
68#689—42no—
69#687—42no—
70C4⋊Dic5—21no—
71C4⋊Dic5C2311yescharacteristic
72#683—21no—
73#683—21no—
74#683—21no—
75#683—21no—
76#683—21no—
77C4×Dic5—21no—
78C4×Dic5—21no—
79C4×Dic5D411yescharacteristic
80C4×Dic5D411yescharacteristic
81C4×Dic5C2311yescharacteristic
82C4×Dic5C2311yescharacteristic
83C2×#168C2312yes—
84Dic20—42no—
85#533—51no—
86C2×#124—51no—
87#497—52no—
88#489—52no—
89#464—52no—
90#464—52no—
91#451—51no—
92#451—51no—
93#450—51no—
94#450D511yescharacteristic
95#432—51no—
96C23×C5—22no—
97C23×C5—42no—
98C23×C5—42no—
99C23×C5—42no—
100C23×C5C2×D411yescharacteristic
101C5×Q8—22no—
102C5×D4—22no—
103C5×D4—22no—
104C5×D4—42no—
105C2×C4×C5—22no—
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5C2×D411yescharacteristic
109C2×C4×C5C2411yescharacteristic
110C2×Dic5—22no—
111C2×Dic5—21no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—41no—
115C2×Dic5—42no—
116C2×Dic5C2×D411yescharacteristic
117C2×Dic5C2×D411yescharacteristic
118C2×Dic5C2×D411yescharacteristic
119C2×Dic5C2×D411yescharacteristic
120Dic10—22no—
121Dic10—22no—
122Dic10—22no—
123Dic10—42no—
124Dic10—42no—
125C5×C8—22no—
126#168—22no—
127C22×Q8—51no—
128C22×D4C2×D511yescharacteristic
129C2×Q16—52no—
130C2×Q16—52no—
131C2×SD16—52no—
132C2×SD16C2×D512yes—
133C4⋊Q8—51no—
134C4⋊Q8—51no—
135#125—101no—
136#124—52no—
137#124—52no—
138#124—101no—
139#124—101no—
140C22⋊Q8—51no—
141C22⋊Q8—101no—
142C22⋊Q8C2×D511yescharacteristic
143C4×Q8—51no—
144C4×Q8—51no—
145C4×Q8—101no—
146#117—101no—
147C2×C22⋊C4—101no—
148C2×C22⋊C4—101no—
149C2×C42—51no—
150C4⋊C8—52no—
151#103—52no—
152#103—52no—
153#102—52no—
154#102C2×D512yes—
155C22⋊C8—51no—
156C22⋊C8C2×D511yescharacteristic
157#97—52no—
158C22×C5—21no—
159C22×C5—21no—
160C22×C5—42no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5—84no—
164C22×C5#13712yes—
165C22×C5C22×D411yescharacteristic
166C4×C5—21no—
167C4×C5—41no—
168C4×C5—41no—
169C4×C5H(5,𝔽2)12yes—
170Dic5—21no—
171Dic5—21no—
172Dic5—21no—
173Dic5—21no—
174Dic5—42no—
175Dic5—41no—
176Dic5—41no—
177C24—21no—
178C2×Q8—52no—
179C2×Q8—51no—
180C2×Q8—101no—
181C2×Q8—101no—
182C2×Q8—202no—
183C2×Q8C22×D511yescharacteristic
184C2×D4—21no—
185C2×D4—42no—
186C2×D4C22×D512yes—
187C22×C4—51no—
188C22×C4—51no—
189C22×C4—101no—
190C22×C4C22×D511yescharacteristic
191Q16—202no—
192Q16—202no—
193SD16—42no—
194SD16—202no—
195C2×C8—52no—
196C2×C8C22×D512yes—
197C4⋊C4—21no—
198C4⋊C4—51no—
199C4⋊C4—101no—
200C4⋊C4—101no—
201C4⋊C4—101no—
202C4⋊C4—101no—
203C4⋊C4—101no—
204C4⋊C4—101no—
205C4⋊C4C22×D511yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—102no—
208C22⋊C4—102no—
209C22⋊C4—102no—
210C22⋊C4—102no—
211C22⋊C4—101no—
212C42—51no—
213C42—51no—
214C42—51no—
215C42—51no—
216C42—101no—
217C42—101no—
218C2×C5—22no—
219C2×C5—82no—
220C2×C5#53511yescharacteristic
221C2×C5C2×#13711yescharacteristic
222C2×C5#57511yescharacteristic
223C23—22no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23D4×D511yescharacteristic
228Q8—22no—
229Q8—102no—
230Q8—102no—
231Q8—102no—
232Q8—202no—
233Q8—202no—
234D4—22no—
235D4—22no—
236D4—42no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—51no—
241C2×C4—51no—
242C2×C4—51no—
243C2×C4—51no—
244C2×C4—102no—
245C2×C4—101no—
246C2×C4—101no—
247C2×C4—102no—
248C2×C4—201no—
249C2×C4—202no—
250C2×C4D4×D511yescharacteristic
251C2×C4C23×D511yescharacteristic
252C8—22no—
253C8—102no—
254C5#321511yescharacteristic, complemented, Sylow
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—84no—
261C22#406112yes—
262C22C2×D4×D511yescharacteristic
263C4—21no—
264C4—41no—
265C4—41no—
266C4—101no—
267C4—101no—
268C4—101no—
269C4—101no—
270C4—201no—
271C4—202no—
272C4—201no—
273C4#414412yes—
274C2—22no—
275C2—82no—
276C2#6592611yescharacteristic
277C2C2×#406111yescharacteristic
278C2#6610711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#450C2
socleC22×C5C22×D4
prime radicalC22×C5×D4C22
hypercenter#450D5
hypocenterC5#3215
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #65956C2semidirect
5. #65456C2semidirect
6. #65455C2semidirect
7. #65339C2semidirect
8. #65056C2semidirect
9. #65054C2semidirect
10. #65028C2semidirect
11. #65008C2semidirect
18. Q8×Dic5C22semidirect
19. #4090C22semidirect
20. C2×#689C22semidirect
28. C2×Dic20C22semidirect
30. #4034C22semidirect
32. #4033C22semidirect
39. #3948C22semidirect
40. #3947C22semidirect
41. #3946C22semidirect
42. #3942C22semidirect
254. C5#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
#659562
C5×#4502
#655072
#654562
#654552
#653392
#650562
#650542
#650282
#650082
#32155