Finite Group Databasebeta

← #10611869

Subgroups — #10611869

Order 640.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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×#4086C211yesmaximal, characteristic
2#65956C211yesmaximal, characteristic, complemented
3C5×#450C211yesmaximal, characteristic, complemented
4#65507C211yesmaximal, characteristic, complemented
5#65464C212yesmaximal, complemented
6#65452C212yesmaximal, complemented
7#65335C211yesmaximal, characteristic, complemented
8#65051C212yesmaximal, complemented
9#65049C212yesmaximal, complemented
10#65020C211yesmaximal, characteristic, complemented
11#65007C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5—21no—
14C22×C22⋊D5—21no—
15C22×D20C2211yescharacteristic
16C2×C5×SD16C2212yescomplemented
17C5×C22⋊Q8C2211yescharacteristic
18Q8×Dic5C2211yescharacteristic, complemented
19#4090C2211yescharacteristic, complemented
20C2×#688C2212yescomplemented
21#4086—21no—
22#4086—21no—
23#4086C2212yes—
24#4086C2212yes—
25C2×#687C2212yescomplemented
26#4070C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28C2×#677C2212yescomplemented
29#4035—21no—
30#4034C2211yescharacteristic, complemented
31#4033—21no—
32#4033C2211yescharacteristic, complemented
33#4024—21no—
34#4023—21no—
35C5×#102C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic
37#3964C2212yescomplemented
38C22⋊#168C2211yescharacteristic
39#3953C2212yescomplemented
40#3946C2212yescomplemented
41C8×Dic5C2212yescomplemented
42#3941C2212yescomplemented
43#3208—51nomaximal, Sylow
44C24×C5—21no—
45C23×D5—21no—
46C2×C5×Q8C2311yescharacteristic
47C2×C5×D4—21no—
48C2×C5×D4—42no—
49C2×C5×D4C2312yes—
50C22×C4×C5C2311yescharacteristic
51C2×C22⋊D5—22no—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—22no—
54C2×C22⋊D5—22no—
55C2×C22⋊D5—44no—
56C2×C22⋊D5—44no—
57C22×Dic5D411yescharacteristic
58C22×Dic5D411yescharacteristic
59C2×D20—21no—
60C2×D20—42no—
61C2×D20C2312yes—
62C2×Dic10—21no—
63C2×Dic10C2311yescharacteristic
64C5×SD16—42no—
65C2×C5×C8C2312yes—
66C5×C4⋊C4—21no—
67C5×C4⋊C4C2311yescharacteristic
68C5×C22⋊C4—21no—
69C22⋊Dic5—21no—
70#688—42no—
71#687—42no—
72C4⋊Dic5—21no—
73C4⋊Dic5C2311yescharacteristic
74#683—21no—
75#683—21no—
76#683—21no—
77#683—21no—
78#683—21no—
79C4×Dic5—21no—
80C4×Dic5—21no—
81C4×Dic5D411yescharacteristic
82C4×Dic5D411yescharacteristic
83C4×Dic5C2311yescharacteristic
84C4×Dic5C2311yescharacteristic
85C2×#168C2312yes—
86#677—42no—
87#533—51no—
88C2×#127—51no—
89#492—52no—
90#486—52no—
91#460—52no—
92#460—52no—
93#450—51no—
94#450—51no—
95#450—51no—
96#450D511yescharacteristic, complemented
97#431—51no—
98C23×C5—22no—
99C23×C5—42no—
100C23×C5—42no—
101C23×C5—42no—
102C23×C5C2×D411yescharacteristic
103C22×D5—22no—
104C22×D5—42no—
105C22×D5—42no—
106C22×D5—42no—
107C5×Q8—22no—
108C5×D4—22no—
109C5×D4—22no—
110C5×D4—42no—
111C2×C4×C5—22no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5C2×D411yescharacteristic
115C2×C4×C5C2411yescharacteristic
116C22⋊D5—44no—
117C22⋊D5—44no—
118C22⋊D5—44no—
119C22⋊D5—44no—
120C2×Dic5—22no—
121C2×Dic5—21no—
122C2×Dic5—21no—
123C2×Dic5SD1614yescomplemented
124C2×Dic5C2×D411yescharacteristic
125C2×Dic5C2×D411yescharacteristic
126C2×Dic5C2×D411yescharacteristic
127C2×Dic5C2×D411yescharacteristic
128D20—22no—
129D20—22no—
130D20—42no—
131Dic10—22no—
132Dic10—42no—
133C5×C8—22no—
134#168—22no—
135C22×D4—51no—
136C22×D4—101no—
137C22×D4—101no—
138C22×D4C2×D511yescharacteristic
139C2×SD16—52no—
140C2×SD16—52no—
141C2×SD16—52no—
142C2×SD16C2×D512yescomplemented
143C4⋊Q8—51no—
144C4⋊Q8—51no—
145#127—52no—
146#127—52no—
147#127—101no—
148#127—101no—
149#125—101no—
150C22⋊Q8—51no—
151C22⋊Q8—101no—
152C22⋊Q8C2×D511yescharacteristic
153C4×Q8—51no—
154C4×Q8—51no—
155C4×Q8—101no—
156#117—101no—
157C2×C42—51no—
158C4⋊C8—52no—
159#102—52no—
160#102—52no—
161#102—52no—
162#102C2×D512yescomplemented
163C22⋊C8—51no—
164C22⋊C8C2×D511yescharacteristic
165C4×C8—52no—
166C22×C5—21no—
167C22×C5—21no—
168C22×C5—42no—
169C22×C5—42no—
170C22×C5—42no—
171C22×C5—84no—
172C22×C5C2×SD1612yes—
173C22×C5C22×D411yescharacteristic
174C2×D5—42no—
175C2×D5—42no—
176C2×D5—42no—
177C2×D5—84no—
178C4×C5—21no—
179C4×C5—41no—
180C4×C5—41no—
181C4×C5H(5,𝔽2)12yes—
182Dic5—21no—
183Dic5—41no—
184Dic5—41no—
185Dic5C2×SD1612yes—
186Dic5C2×SD1612yes—
187Dic5AGL(1,ℤ/8)12yes—
188C24—21no—
189C24—101no—
190C2×Q8—51no—
191C2×Q8—101no—
192C2×Q8C22×D511yescharacteristic
193C2×D4—21no—
194C2×D4—42no—
195C2×D4—52no—
196C2×D4—102no—
197C2×D4—102no—
198C2×D4—102no—
199C2×D4—102no—
200C2×D4—101no—
201C2×D4—202no—
202C2×D4—204no—
203C2×D4—204no—
204C2×D4C22×D512yes—
205C22×C4—51no—
206C22×C4—51no—
207C22×C4C22×D511yescharacteristic
208SD16—42no—
209SD16—202no—
210SD16—202no—
211SD16—202no—
212C2×C8—52no—
213C2×C8C22×D512yes—
214C4⋊C4—21no—
215C4⋊C4—51no—
216C4⋊C4—101no—
217C4⋊C4—101no—
218C4⋊C4—101no—
219C4⋊C4—101no—
220C4⋊C4—101no—
221C4⋊C4—101no—
222C4⋊C4C22×D511yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—101no—
225C42—51no—
226C42—51no—
227C42—51no—
228C42—51no—
229C42—101no—
230C42—101no—
231C2×C5—22no—
232C2×C5—82no—
233C2×C5#53511yescharacteristic
234C2×C5C22×SD1611yescharacteristic
235C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
236D5—82no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—102no—
242C23—202no—
243C23—202no—
244C23—202no—
245C23D4×D511yescharacteristic
246Q8—22no—
247Q8—102no—
248Q8—202no—
249D4—22no—
250D4—22no—
251D4—42no—
252D4—102no—
253D4—102no—
254D4—202no—
255D4—204no—
256D4—204no—
257D4—204no—
258D4—204no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—51no—
263C2×C4—51no—
264C2×C4—51no—
265C2×C4—54no—
266C2×C4—51no—
267C2×C4—102no—
268C2×C4—101no—
269C2×C4—101no—
270C2×C4D4×D511yescharacteristic
271C2×C4C23×D511yescharacteristic
272C8—22no—
273C8—102no—
274C5#320811yescharacteristic, complemented, Sylow
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—202no—
282C22—202no—
283C22—202no—
284C22—404no—
285C22D5×SD1612yes—
286C22C2×D4×D511yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—52no—
291C4—52no—
292C4—52no—
293C4—101no—
294C4—201no—
295C4—201no—
296C4#414412yes—
297C2—22no—
298C2—82no—
299C2—402no—
300C2#6592611yescharacteristic
301C2C2×D5×SD1611yescharacteristic
302C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#450C2
socleC22×C5C22×D4
prime radicalC2×#4086C2
hypercenter#450D5
hypocenterC5#3208
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
3. C5×#450C2semidirect
4. #65507C2semidirect
5. #65464C2semidirect
6. #65452C2semidirect
7. #65335C2semidirect
8. #65051C2semidirect
9. #65049C2semidirect
10. #65020C2semidirect
11. #65007C2semidirect
16. C2×C5×SD16C22semidirect
18. Q8×Dic5C22semidirect
19. #4090C22semidirect
20. C2×#688C22semidirect
25. C2×#687C22semidirect
28. C2×#677C22semidirect
30. #4034C22semidirect
32. #4033C22semidirect
35. C5×#102C22semidirect
37. #3964C22semidirect
39. #3953C22semidirect
40. #3946C22semidirect
41. C8×Dic5C22semidirect
42. #3941C22semidirect
96. #450D5semidirect
123. C2×Dic5SD16semidirect
142. C2×SD16C2×D5semidirect
162. #102C2×D5semidirect
274. C5#3208semidirect

Maximal subgroups

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

subgroupindex
C2×#40862
#659562
C5×#4502
#655072
#654642
#654522
#653352
#650512
#650492
#650202
#650072
#32085