Finite Group Databasebeta

← #10613041

Subgroups — #10613041

Order 640.

488 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 119 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#65930C211yesmaximal, characteristic, complemented
2C2×#4039C211yesmaximal, characteristic, complemented
3C5×#481C211yesmaximal, characteristic, complemented
4#65325C211yesmaximal, characteristic, complemented
5#65257C211yesmaximal, characteristic, complemented
6#65159C212yesmaximal, complemented
7#65158C212yesmaximal, complemented
8#65075C212yesmaximal, complemented
9#65074C212yesmaximal, complemented
10#65021C211yesmaximal, characteristic, complemented
11#65008C211yesmaximal, characteristic, complemented
12C2×#711—21no—
13C22×D20C2211yescharacteristic
14C22×C4×D5—21no—
15C5×#121C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17#4086C2211yescharacteristic, complemented
18#4085—21no—
19D4×Dic5C2211yescharacteristic, complemented
20C2×#686C2212yes—
21#4076C2211yescharacteristic
22C2×#685—21no—
23C2×C4×Dic5C2211yescharacteristic
24C2×D40C2212yes—
25#4039—21no—
26#4039—21no—
27#4039C2212yes—
28#4039C2212yes—
29#4039C2211yescharacteristic, complemented
30#4034C2211yescharacteristic, complemented
31#4031—21no—
32#4027—21no—
33C5×#106C2212yescomplemented
34C5×#102C2212yescomplemented
35C5×C22⋊C8C2211yescharacteristic
36C22⋊#168C2211yescharacteristic
37#3953C2212yes—
38#3947C2212yescomplemented
39#3946C2212yescomplemented
40#3941C2212yes—
41#3941C2212yescomplemented
42#3940C2212yescomplemented
43#3109—51nomaximal, Sylow
44C23×D5—21no—
45C2×C5×D4—21no—
46C2×C5×D4C2311yescharacteristic
47C22×C4×C5—21no—
48C22×C4×C5C2311yescharacteristic
49C2×C22⋊D5—21no—
50C2×C22⋊D5—21no—
51C2×C22⋊D5—21no—
52C22×Dic5—21no—
53C22×Dic5D411yescharacteristic
54C22×Dic5D411yescharacteristic
55#711—44no—
56C2×D20—21no—
57C2×D20—24no—
58C2×D20C2311yescharacteristic
59C2×D20C2312yes—
60C2×C4×D5—22no—
61C2×C4×D5—22no—
62C2×C4×D5—21no—
63C2×C4×D5—44no—
64C2×Dic10—21no—
65C2×C5×C8C2312yes—
66C5×C4⋊C4—21no—
67C5×C4⋊C4C2312yes—
68C5×C4⋊C4C2311yescharacteristic
69C5×C22⋊C4—21no—
70C22⋊Dic5—21no—
71#686—24no—
72#685—21no—
73#685—21no—
74#685—22no—
75#685—22no—
76C4⋊Dic5C2311yescharacteristic
77#683—21no—
78C4×Dic5D411yescharacteristic
79C4×Dic5D411yescharacteristic
80C4×Dic5C2311yescharacteristic
81C4×Dic5C2311yescharacteristic
82C2×#168C2312yes—
83D40—24no—
84#532—51no—
85C2×C4×D4—51no—
86#481—51no—
87#481D511yescharacteristic, complemented
88#476—52no—
89#459—52no—
90#447—51no—
91#447—51no—
92#442—52no—
93#442—52no—
94#432—51no—
95C23×C5—21no—
96C23×C5C2×D411yescharacteristic
97C22×D5—24no—
98C22×D5—22no—
99C22×D5—21no—
100C22×D5—42no—
101C22×D5—42no—
102C5×D4—22no—
103C5×D4—42no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—42no—
108C2×C4×C5C2×D411yescharacteristic
109C2×C4×C5C2411yescharacteristic
110C22⋊D5—42no—
111C22⋊D5—42no—
112C22⋊D5—42no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—21no—
116C2×Dic5—42no—
117C2×Dic5C2×D411yescharacteristic
118C2×Dic5C2×D411yescharacteristic
119C2×Dic5C2×D411yescharacteristic
120C2×Dic5C2×D411yescharacteristic
121D20—24no—
122D20—22no—
123D20—22no—
124D20#4114yescomplemented
125C4×D5—44no—
126C4×D5—42no—
127C4×D5—44no—
128Dic10—42no—
129C5×C8—22no—
130#168—22no—
131C2×#41—101no—
132C22×D4—51no—
133C23×C4—101no—
134C2×D8—52no—
135C2×D8—52no—
136C4⋊Q8—51no—
137#127—51no—
138#124—101no—
139#121—51no—
140#121—101no—
141#121C2×D511yescharacteristic
142C4×D4—51no—
143C4×D4—52no—
144C4×D4—52no—
145C4×D4—51no—
146C4×D4—101no—
147C4×D4—101no—
148C4×D4—101no—
149C2×C4⋊C4C2×D511yescharacteristic
150C2×C22⋊C4—101no—
151C2×C42—51no—
152#106—52no—
153#106C2×D512yescomplemented
154C4⋊C8—52no—
155#102—52no—
156#102—52no—
157#102—52no—
158#102C2×D512yescomplemented
159C22⋊C8—51no—
160C22⋊C8C2×D511yescharacteristic
161#97—52no—
162C22×C5—21no—
163C22×C5—21no—
164C22×C5—41no—
165C22×C5—41no—
166C22×C5—41no—
167C22×C5AGL(1,ℤ/8)12yes—
168C22×C5C22×D411yescharacteristic
169C2×D5—24no—
170C2×D5—48no—
171C2×D5—42no—
172C2×D5—42no—
173C2×D5—41no—
174C2×D5—41no—
175C2×D5—41no—
176C4×C5—21no—
177C4×C5—41no—
178C4×C5—42no—
179C4×C5C2×#4112yes—
180Dic5—21no—
181Dic5—21no—
182Dic5—21no—
183Dic5—41no—
184Dic5AGL(1,ℤ/8)12yes—
185C24—101no—
186#41—204no—
187C2×Q8—101no—
188C2×D4—21no—
189C2×D4—52no—
190C2×D4—51no—
191C2×D4—104no—
192C2×D4—101no—
193C2×D4—101no—
194C2×D4—101no—
195C2×D4—101no—
196C2×D4C22×D511yescharacteristic
197C22×C4—21no—
198C22×C4—51no—
199C22×C4—51no—
200C22×C4—102no—
201C22×C4—102no—
202C22×C4—101no—
203C22×C4—101no—
204C22×C4—204no—
205C22×C4C22×D511yescharacteristic
206D8—104no—
207D8—104no—
208C2×C8—52no—
209C2×C8C22×D512yes—
210C4⋊C4—21no—
211C4⋊C4—51no—
212C4⋊C4—101no—
213C4⋊C4C22×D511yescharacteristic
214C4⋊C4C22×D512yes—
215C22⋊C4—21no—
216C22⋊C4—102no—
217C22⋊C4—102no—
218C22⋊C4—101no—
219C22⋊C4—101no—
220C22⋊C4—101no—
221C42—51no—
222C42—51no—
223C42—51no—
224C42—51no—
225C2×C5—22no—
226C2×C5—81no—
227C2×C5#52511yescharacteristic
228C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
230D5—44no—
231D5—81no—
232C23—21no—
233C23—104no—
234C23—102no—
235C23—101no—
236C23—202no—
237C23—202no—
238C23D4×D511yescharacteristic
239Q8—202no—
240D4—22no—
241D4—42no—
242D4—54no—
243D4—104no—
244D4—102no—
245D4—102no—
246D4—202no—
247D4—202no—
248D4—202no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—51no—
254C2×C4—51no—
255C2×C4—51no—
256C2×C4—51no—
257C2×C4—102no—
258C2×C4—102no—
259C2×C4—101no—
260C2×C4—204no—
261C2×C4—204no—
262C2×C4—202no—
263C2×C4—202no—
264C2×C4D4×D511yescharacteristic
265C2×C4C23×D511yescharacteristic
266C8—22no—
267C8—102no—
268C5#310911yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—104no—
275C22—208no—
276C22—202no—
277C22—202no—
278C22—201no—
279C22—201no—
280C22—201no—
281C22#406012yes—
282C22C2×D4×D511yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—52no—
287C4—101no—
288C4—101no—
289C4—101no—
290C4—201no—
291C4D5×#4112yes—
292C2—22no—
293C2—81no—
294C2—204no—
295C2—401no—
296C2#6599011yescharacteristic
297C2C2×#406011yescharacteristic
298C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#481C2
socleC22×C5C22×D4
hypercenter#481D5
hypocenterC5#3109
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #65930C2semidirect
2. C2×#4039C2semidirect
3. C5×#481C2semidirect
4. #65325C2semidirect
5. #65257C2semidirect
6. #65159C2semidirect
7. #65158C2semidirect
8. #65075C2semidirect
9. #65074C2semidirect
10. #65021C2semidirect
11. #65008C2semidirect
17. #4086C22semidirect
19. D4×Dic5C22semidirect
29. #4039C22semidirect
30. #4034C22semidirect
33. C5×#106C22semidirect
34. C5×#102C22semidirect
38. #3947C22semidirect
39. #3946C22semidirect
41. #3941C22semidirect
42. #3940C22semidirect
87. #481D5semidirect
124. D20#41semidirect
153. #106C2×D5semidirect
158. #102C2×D5semidirect
268. C5#3109semidirect

Maximal subgroups

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

subgroupindex
#659302
C2×#40392
C5×#4812
#653252
#652572
#651592
#651582
#650752
#650742
#650212
#650082
#31095