Finite Group Databasebeta

← #10613071

Subgroups — #10613071

Order 640.

514 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 115 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
1D5×#121C211yesmaximal, characteristic, complemented
2C2×#4041C211yesmaximal, characteristic, complemented
3C5×#481C211yesmaximal, characteristic, complemented
4#65325C211yesmaximal, characteristic, complemented
5#65257C211yesmaximal, characteristic, complemented
6#65154C212yesmaximal, complemented
7#65152C212yesmaximal, complemented
8#65071C212yesmaximal, complemented
9#65062C212yesmaximal, complemented
10#65021C211yesmaximal, characteristic, complemented
11#65015C211yesmaximal, characteristic, complemented
12C2×D4×D5—21no—
13C2×D4×D5C2211yescharacteristic, complemented
14C22×D20—21no—
15C22×D20C2211yescharacteristic, complemented
16C22×C4×D5C2211yescharacteristic
17C5×#121C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic, complemented
19#4085—21no—
20#4084C2211yescharacteristic, complemented
21C2×#686C2212yescomplemented
22#4076C2211yescharacteristic
23C2×#685—21no—
24C2×D40C2212yescomplemented
25C2×#676C2212yescomplemented
26#4041—21no—
27#4041—21no—
28#4041C2212yes—
29#4041C2212yescomplemented
30#4041C2211yescharacteristic, complemented
31D5×C4⋊C4C2211yescharacteristic, complemented
32#4030—21no—
33D5×C22⋊C4—21no—
34C5×#106C2212yescomplemented
35C5×#102C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic, complemented
37C22⋊#168C2211yescharacteristic, complemented
38#3953C2212yescomplemented
39#3952C2212yescomplemented
40#3941C2212yescomplemented
41#3941C2212yescomplemented
42#3940C2212yescomplemented
43#3192—51nomaximal, Sylow
44C23×D5—21no—
45C23×D5—21no—
46C23×D5D411yescharacteristic
47C2×C5×D4—21no—
48C2×C5×D4C2311yescharacteristic
49C22×C4×C5—21no—
50C22×C4×C5C2311yescharacteristic
51C2×C22⋊D5—21no—
52C2×C22⋊D5—21no—
53C2×C22⋊D5—21no—
54C22×Dic5D411yescharacteristic
55D4×D5—42no—
56D4×D5—44no—
57C2×D20—22no—
58C2×D20—22no—
59C2×D20—21no—
60C2×D20—21no—
61C2×D20—42no—
62C2×D20—44no—
63C2×D20C2312yes—
64C2×D20C2311yescharacteristic
65C2×C4×D5—21no—
66C2×C4×D5—42no—
67C2×C4×D5D411yescharacteristic
68C2×C4×D5D411yescharacteristic
69C2×C4×D5C2311yescharacteristic
70C2×C4×D5C2311yescharacteristic
71C2×C5×C8C2312yes—
72C5×C4⋊C4—21no—
73C5×C4⋊C4C2312yes—
74C5×C4⋊C4C2311yescharacteristic
75C5×C22⋊C4—21no—
76C22⋊Dic5—21no—
77#686—42no—
78#685—21no—
79#685—21no—
80#685—22no—
81#685—22no—
82C4⋊Dic5C2311yescharacteristic
83#683—21no—
84C2×#168C2312yes—
85D40—42no—
86#676—42no—
87C2×#121—51no—
88C2×#121—51no—
89#481—51no—
90#481—52no—
91#481D511yescharacteristic, complemented
92#469—52no—
93#469—52no—
94#447—51no—
95#447—52no—
96#447—51no—
97#407—51no—
98C23×C5—21no—
99C23×C5C2×D411yescharacteristic
100C22×D5—22no—
101C22×D5—22no—
102C22×D5—22no—
103C22×D5—21no—
104C22×D5—41no—
105C22×D5—42no—
106C22×D5—42no—
107C22×D5—42no—
108C22×D5—42no—
109C22×D5—42no—
110C22×D5—42no—
111C22×D5C2×D411yescharacteristic
112C22×D5C2×D411yescharacteristic
113C5×D4—22no—
114C5×D4—42no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5—22no—
118C2×C4×C5—42no—
119C2×C4×C5C2×D411yescharacteristic
120C2×C4×C5C2411yescharacteristic
121C22⋊D5—42no—
122C22⋊D5—42no—
123C22⋊D5—42no—
124C2×Dic5—21no—
125C2×Dic5—41no—
126C2×Dic5C2×D411yescharacteristic
127C2×Dic5C2×D411yescharacteristic
128D20—22no—
129D20—22no—
130D20—22no—
131D20—42no—
132D20—44no—
133D20—42no—
134D20—44no—
135C4×D5—22no—
136C4×D5—22no—
137C4×D5—41no—
138C4×D5—41no—
139C4×D5—42no—
140C5×C8—22no—
141#168—22no—
142C22×D4—51no—
143C22×D4—51no—
144C22×D4—101no—
145C22×D4—101no—
146C23×C4—51no—
147C2×D8—52no—
148C2×D8—52no—
149C2×M16—52no—
150#121—51no—
151#121—51no—
152#121—52no—
153#121—51no—
154#121—52no—
155#121—101no—
156#121—101no—
157#121—101no—
158#121—101no—
159#121C2×D511yescharacteristic
160C2×C4⋊C4—51no—
161C2×C4⋊C4C2×D511yescharacteristic, complemented
162C2×C22⋊C4—101no—
163C2×C22⋊C4—101no—
164#106—52no—
165#106C2×D512yescomplemented
166#102—52no—
167#102—52no—
168#102—52no—
169#102C2×D512yescomplemented
170C22⋊C8—51no—
171C22⋊C8—52no—
172C22⋊C8C2×D511yescharacteristic, complemented
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—41no—
176C22×C5—41no—
177C22×C5—41no—
178C22×C5AGL(1,ℤ/8)12yes—
179C22×C5C22×D411yescharacteristic
180C2×D5—21no—
181C2×D5—21no—
182C2×D5—21no—
183C2×D5—21no—
184C2×D5—21no—
185C2×D5—44no—
186C2×D5—42no—
187C2×D5—41no—
188C2×D5—42no—
189C2×D5—41no—
190C2×D5—42no—
191C2×D5—41no—
192C2×D5—84no—
193C2×D5—84no—
194C2×D5AGL(1,ℤ/8)12yes—
195C4×C5—21no—
196C4×C5—41no—
197C4×C5—42no—
198C4×C5H(5,𝔽2)12yes—
199Dic5—21no—
200Dic5—21no—
201Dic5—41no—
202C24—51no—
203C24—101no—
204C24—101no—
205C2×D4—21no—
206C2×D4—52no—
207C2×D4—51no—
208C2×D4—101no—
209C2×D4—102no—
210C2×D4—102no—
211C2×D4—101no—
212C2×D4—101no—
213C2×D4—101no—
214C2×D4—101no—
215C2×D4—202no—
216C2×D4—204no—
217C2×D4—202no—
218C2×D4—204no—
219C2×D4C22×D511yescharacteristic
220C22×C4—21no—
221C22×C4—51no—
222C22×C4—51no—
223C22×C4—51no—
224C22×C4—51no—
225C22×C4—51no—
226C22×C4—101no—
227C22×C4—202no—
228C22×C4C22×D511yescharacteristic
229D8—202no—
230D8—202no—
231M16—202no—
232C2×C8—52no—
233C2×C8C22×D512yes—
234C4⋊C4—21no—
235C4⋊C4—51no—
236C4⋊C4—101no—
237C4⋊C4C22×D511yescharacteristic
238C4⋊C4C22×D512yes—
239C22⋊C4—21no—
240C22⋊C4—101no—
241C22⋊C4—102no—
242C22⋊C4—102no—
243C22⋊C4—101no—
244C22⋊C4—101no—
245C2×C5—22no—
246C2×C5—81no—
247C2×C5#53411yescharacteristic
248C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
249C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
250D5—22no—
251D5—41no—
252D5—82no—
253D5—81no—
254C23—21no—
255C23—51no—
256C23—51no—
257C23—102no—
258C23—102no—
259C23—101no—
260C23—102no—
261C23—201no—
262C23—202no—
263C23—202no—
264C23—202no—
265C23—202no—
266C23—202no—
267C23—202no—
268C23D4×D511yescharacteristic
269D4—22no—
270D4—42no—
271D4—102no—
272D4—102no—
273D4—102no—
274D4—202no—
275D4—204no—
276D4—202no—
277D4—204no—
278D4—202no—
279D4—202no—
280D4—202no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—51no—
286C2×C4—51no—
287C2×C4—102no—
288C2×C4—102no—
289C2×C4—101no—
290C2×C4—201no—
291C2×C4—201no—
292C2×C4—202no—
293C2×C4—201no—
294C2×C4D4×D511yescharacteristic
295C2×C4C23×D511yescharacteristic
296C8—22no—
297C8—102no—
298C5#319211yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—52no—
305C22—101no—
306C22—101no—
307C22—101no—
308C22—101no—
309C22—101no—
310C22—204no—
311C22—201no—
312C22—202no—
313C22—202no—
314C22—201no—
315C22—202no—
316C22—201no—
317C22—404no—
318C22—404no—
319C22#406012yes—
320C22C2×D4×D511yescharacteristic
321C4—21no—
322C4—41no—
323C4—42no—
324C4—101no—
325C4—101no—
326C4—201no—
327C4#414912yes—
328C2—22no—
329C2—81no—
330C2—102no—
331C2—201no—
332C2—401no—
333C2—402no—
334C2#6598711yescharacteristic
335C2C2×#406011yescharacteristic
336C2D5×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#3192
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. D5×#121C2semidirect
2. C2×#4041C2semidirect
3. C5×#481C2semidirect
4. #65325C2semidirect
5. #65257C2semidirect
6. #65154C2semidirect
7. #65152C2semidirect
8. #65071C2semidirect
9. #65062C2semidirect
10. #65021C2semidirect
11. #65015C2semidirect
13. C2×D4×D5C22semidirect
15. C22×D20C22semidirect
18. C2×C5×C4⋊C4C22semidirect
20. #4084C22semidirect
21. C2×#686C22semidirect
24. C2×D40C22semidirect
25. C2×#676C22semidirect
29. #4041C22semidirect
30. #4041C22semidirect
31. D5×C4⋊C4C22semidirect
34. C5×#106C22semidirect
35. C5×#102C22semidirect
36. C5×C22⋊C8C22semidirect
37. C22⋊#168C22semidirect
38. #3953C22semidirect
39. #3952C22semidirect
40. #3941C22semidirect
41. #3941C22semidirect
42. #3940C22semidirect
91. #481D5semidirect
161. C2×C4⋊C4C2×D5semidirect
165. #106C2×D5semidirect
169. #102C2×D5semidirect
172. C22⋊C8C2×D5semidirect
298. C5#3192semidirect

Maximal subgroups

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

subgroupindex
D5×#1212
C2×#40412
C5×#4812
#653252
#652572
#651542
#651522
#650712
#650622
#650212
#650152
#31925