Finite Group Databasebeta

← #10613605

Subgroups — #10613605

Order 640.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#66062C211yesmaximal, characteristic, complemented
2D5×#124C211yesmaximal, characteristic
3C5×#497C211yesmaximal, characteristic, complemented
4#65475C212yesmaximal, complemented
5#65474C212yesmaximal, complemented
6#65458C212yesmaximal
7#65455C212yesmaximal, complemented
8#65375C211yesmaximal, characteristic, complemented
9#65344C211yesmaximal, characteristic, complemented
10#65003C211yesmaximal, characteristic, complemented
11#64994C211yesmaximal, characteristic, complemented
12C2×Q8×D5C2211yescharacteristic
13C2×D4×D5C2211yescharacteristic
14C2×C5×Q16C2212yescomplemented
15C2×C5×SD16C2212yes—
16C5×C4⋊Q8C2211yescharacteristic, complemented
17C5×#124C2211yescharacteristic
18#4093C2211yescharacteristic
19#4092C2212yes—
20Q8×Dic5C2212yescomplemented
21C2×#689C2212yescomplemented
22C2×#688C2212yes—
23#4082C2211yescharacteristic
24#4043—21no—
25#4038—21no—
26#4036—21no—
27#4033—21no—
28#4031—21no—
29#4031—21no—
30D5×C22⋊C4—21no—
31D5×C22⋊C4—21no—
32#4021C2211yescharacteristic
33C42×D5C2211yescharacteristic
34#4015C2211yescharacteristic, complemented
35C5×#97C2211yescharacteristic, complemented
36#3967C2212yescomplemented
37#3964C2212yes—
38#3953C2212yes—
39#3952C2212yes—
40#3948C2212yescomplemented
41#3946C2212yescomplemented
42#3935C2211yescharacteristic, complemented
43#3215—51nomaximal, Sylow
44C23×D5—21no—
45C2×C5×Q8C2311yescharacteristic
46C2×C5×Q8C2312yes—
47C2×C5×D4C2311yescharacteristic
48C2×C22⋊D5—21no—
49Q8×D5—42no—
50D4×D5—42no—
51C2×D20C2311yescharacteristic
52C2×C4×D5—21no—
53C2×C4×D5D411yescharacteristic
54C2×C4×D5D411yescharacteristic
55C2×C4×D5C2311yescharacteristic
56C2×Dic10—21no—
57C2×Dic10—21no—
58C2×Dic10C2311yescharacteristic
59C5×Q16—42no—
60C5×SD16—42no—
61C2×C5×C8C2312yes—
62C5×C4⋊C4—21no—
63C5×C4⋊C4—21no—
64C5×C22⋊C4—21no—
65C5×C22⋊C4—21no—
66C42×C5C2311yescharacteristic
67C22⋊Dic5—21no—
68C22⋊Dic5—21no—
69#689—42no—
70#688—42no—
71#685—22no—
72#685—21no—
73#685—21no—
74#685—21no—
75#685—21no—
76C4⋊Dic5—21no—
77C4⋊Dic5—22no—
78C4⋊Dic5C2312yes—
79#683—22no—
80C4×Dic5—22no—
81C4×Dic5D411yescharacteristic
82C4×Dic5D411yescharacteristic
83C4×Dic5C2311yescharacteristic
84C2×#168C2312yes—
85#533—51no—
86C2×#124—51no—
87#497—51no—
88#497D511yescharacteristic, complemented
89#489—51no—
90#489—51no—
91#464—52no—
92#464—52no—
93#451—52no—
94#450—52no—
95#432—51no—
96C23×C5—21no—
97C22×D5—21no—
98C22×D5—42no—
99C22×D5—42no—
100C22×D5—42no—
101C22×D5C2×D411yescharacteristic
102C5×Q8—22no—
103C5×Q8—22no—
104C5×Q8—22no—
105C5×D4—22no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5C2×D411yescharacteristic
109C2×C4×C5C2×D411yescharacteristic
110C2×C4×C5C2411yescharacteristic
111C22⋊D5—42no—
112C2×Dic5—22no—
113C2×Dic5—21no—
114C2×Dic5C2×D411yescharacteristic
115C2×Dic5C2×D411yescharacteristic
116C2×Dic5C2×D411yescharacteristic
117D20—22no—
118C4×D5—22no—
119C4×D5—22no—
120C4×D5—41no—
121C4×D5—42no—
122Dic10—22no—
123Dic10—42no—
124Dic10—42no—
125C5×C8—22no—
126#168—22no—
127C22×Q8—51no—
128C22×D4—51no—
129C2×Q16—52no—
130C2×Q16C2×D512yescomplemented
131C2×SD16—52no—
132C2×SD16C2×D512yes—
133C4⋊Q8—51no—
134C4⋊Q8C2×D511yescharacteristic, complemented
135#125—101no—
136#124—51no—
137#124—51no—
138#124—51no—
139#124—101no—
140#124—101no—
141#124C2×D511yescharacteristic
142C22⋊Q8—52no—
143C22⋊Q8—101no—
144C4×Q8—52no—
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#102—52no—
155C22⋊C8—52no—
156#97—51no—
157#97C2×D511yescharacteristic, complemented
158C22×C5—41no—
159C22×C5—41no—
160C22×C5—41no—
161C22×C5C22×D411yescharacteristic
162C2×D5—21no—
163C2×D5—21no—
164C2×D5—41no—
165C2×D5—41no—
166C2×D5—41no—
167C2×D5—84no—
168C2×D5#13712yes—
169C4×C5—21no—
170C4×C5—21no—
171C4×C5—41no—
172C4×C5—42no—
173C4×C5H(5,𝔽2)12yes—
174Dic5—21no—
175Dic5—21no—
176Dic5—21no—
177Dic5—41no—
178Dic5—42no—
179C24—101no—
180C2×Q8—51no—
181C2×Q8—101no—
182C2×Q8—101no—
183C2×Q8—202no—
184C2×Q8C22×D511yescharacteristic
185C2×Q8C22×D512yes—
186C2×D4—51no—
187C2×D4—101no—
188C2×D4—202no—
189C2×D4C22×D511yescharacteristic
190C22×C4—51no—
191C22×C4—51no—
192C22×C4—51no—
193C22×C4—101no—
194Q16—42no—
195Q16—202no—
196SD16—42no—
197SD16—202no—
198C2×C8—52no—
199C2×C8C22×D512yes—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—52no—
203C4⋊C4—102no—
204C4⋊C4—102no—
205C4⋊C4—101no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—101no—
209C22⋊C4—101no—
210C22⋊C4—102no—
211C22⋊C4—101no—
212C22⋊C4—101no—
213C22⋊C4—101no—
214C22⋊C4—101no—
215C42—51no—
216C42—51no—
217C42—51no—
218C42—102no—
219C42C22×D511yescharacteristic
220C2×C5—81no—
221C2×C5#53511yescharacteristic
222C2×C5C2×#13711yescharacteristic
223C2×C5#57511yescharacteristic
224D5—22no—
225D5—81no—
226C23—21no—
227C23—51no—
228C23—101no—
229C23—202no—
230C23—202no—
231C23—202no—
232Q8—22no—
233Q8—22no—
234Q8—22no—
235Q8—102no—
236Q8—202no—
237Q8—202no—
238D4—22no—
239D4—102no—
240D4—202no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—51no—
244C2×C4—51no—
245C2×C4—51no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—102no—
249C2×C4—101no—
250C2×C4—201no—
251C2×C4—202no—
252C2×C4D4×D511yescharacteristic
253C2×C4D4×D511yescharacteristic
254C2×C4C23×D511yescharacteristic
255C8—22no—
256C8—102no—
257C5#321511yescharacteristic, complemented, Sylow
258C22—41no—
259C22—41no—
260C22—41no—
261C22—52no—
262C22—101no—
263C22—101no—
264C22—201no—
265C22—201no—
266C22—201no—
267C22—404no—
268C22C2×D4×D511yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—101no—
274C4—101no—
275C4—101no—
276C4—201no—
277C4—202no—
278C4#414412yes—
279C2—81no—
280C2—102no—
281C2—401no—
282C2#6604911yescharacteristic
283C2D5×#13711yescharacteristic
284C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#497C2
socleC22×C5C22×D4
prime radicalC2×D4×D5C22
hypercenter#497D5
hypocenterC5#3215
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #66062C2semidirect
3. C5×#497C2semidirect
4. #65475C2semidirect
5. #65474C2semidirect
7. #65455C2semidirect
8. #65375C2semidirect
9. #65344C2semidirect
10. #65003C2semidirect
11. #64994C2semidirect
14. C2×C5×Q16C22semidirect
16. C5×C4⋊Q8C22semidirect
20. Q8×Dic5C22semidirect
21. C2×#689C22semidirect
34. #4015C22semidirect
35. C5×#97C22semidirect
36. #3967C22semidirect
40. #3948C22semidirect
41. #3946C22semidirect
42. #3935C22semidirect
88. #497D5semidirect
130. C2×Q16C2×D5semidirect
134. C4⋊Q8C2×D5semidirect
157. #97C2×D5semidirect
257. C5#3215semidirect

Maximal subgroups

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

subgroupindex
#660622
D5×#1242
C5×#4972
#654752
#654742
#654582
#654552
#653752
#653442
#650032
#649942
#32155