Finite Group Databasebeta

← #10613031

Subgroups — #10613031

Order 640.

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

normal subgroups only

This group has 466 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
1#65940C211yesmaximal, characteristic, complemented
2C2×#4041C211yesmaximal, characteristic, complemented
3C5×#480C211yesmaximal, characteristic, complemented
4#65325C211yesmaximal, characteristic, complemented
5#65260C211yesmaximal, characteristic, complemented
6#65175C212yesmaximal, complemented
7#65174C212yesmaximal, complemented
8#65071C212yesmaximal, complemented
9#65070C212yesmaximal, complemented
10#65020C211yesmaximal, characteristic, complemented
11#65015C211yesmaximal, characteristic, complemented
12C2×#711C2211yescharacteristic, complemented
13C22×D20—21no—
14C22×D20C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C5×#121C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18#4084C2211yescharacteristic, complemented
19#4083—21no—
20D4×Dic5—21no—
21C2×#686C2212yescomplemented
22C2×#685—21no—
23#4070C2211yescharacteristic
24C2×#677C2212yescomplemented
25C2×#676C2212yescomplemented
26#4043C2211yescharacteristic, complemented
27#4041—21no—
28#4041—21no—
29#4041C2212yes—
30#4041C2212yes—
31#4038C2211yescharacteristic, complemented
32#4029—21no—
33#4027—21no—
34C5×#107C2212yescomplemented
35C5×#102C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic
37C22⋊#168C2211yescharacteristic
38#3953C2212yescomplemented
39#3952C2212yescomplemented
40#3942C2212yescomplemented
41#3941C2212yescomplemented
42#3940C2212yescomplemented
43#3194—51nomaximal, Sylow
44C23×D5—21no—
45C23×D5D411yescharacteristic
46C2×C5×D4—21no—
47C2×C5×D4C2311yescharacteristic
48C22×C4×C5—21no—
49C22×C4×C5C2311yescharacteristic
50C2×C22⋊D5—21no—
51C2×C22⋊D5—21no—
52C22×Dic5—21no—
53C22×Dic5D411yescharacteristic
54#711—42no—
55C2×D20—22no—
56C2×D20—22no—
57C2×D20—21no—
58C2×D20—42no—
59C2×D20—44no—
60C2×D20C2312yes—
61C2×C4×D5—42no—
62C2×C4×D5D411yescharacteristic
63C2×C4×D5D411yescharacteristic
64C2×C4×D5C2311yescharacteristic
65C2×C4×D5C2311yescharacteristic
66C2×Dic10C2311yescharacteristic
67C2×C5×C8C2312yes—
68C5×C4⋊C4—21no—
69C5×C4⋊C4C2312yes—
70C5×C4⋊C4C2311yescharacteristic
71C5×C22⋊C4—21no—
72C22⋊Dic5—21no—
73C22⋊Dic5—21no—
74#686—42no—
75#685—21no—
76#685—21no—
77#685—22no—
78#685—22no—
79C4⋊Dic5—21no—
80C4⋊Dic5C2311yescharacteristic
81#683—21no—
82C4×Dic5—21no—
83C2×#168C2312yes—
84#677—42no—
85#676—42no—
86#525—51no—
87C2×#121—51no—
88#483—51no—
89#482—52no—
90#480D511yescharacteristic, complemented
91#469—52no—
92#468—52no—
93#450—51no—
94#449—52no—
95#447—51no—
96#407—51no—
97C23×C5—21no—
98C23×C5C2×D411yescharacteristic
99C22×D5—22no—
100C22×D5—22no—
101C22×D5—41no—
102C22×D5—41no—
103C22×D5—42no—
104C22×D5—42no—
105C22×D5—42no—
106C22×D5C2×D411yescharacteristic
107C22×D5C2×D411yescharacteristic
108C5×D4—22no—
109C5×D4—42no—
110C2×C4×C5—22no—
111C2×C4×C5—21no—
112C2×C4×C5—22no—
113C2×C4×C5—42no—
114C2×C4×C5C2×D411yescharacteristic
115C2×C4×C5C2411yescharacteristic
116C22⋊D5—42no—
117C22⋊D5—42no—
118C2×Dic5—21no—
119C2×Dic5—21no—
120C2×Dic5—41no—
121C2×Dic5—42no—
122C2×Dic5C2×D411yescharacteristic
123C2×Dic5C2×D411yescharacteristic
124D20—22no—
125D20—22no—
126D20—42no—
127D20—44no—
128D20—44no—
129C4×D5—22no—
130C4×D5—22no—
131C4×D5—41no—
132C4×D5—41no—
133Dic10—22no—
134C5×C8—22no—
135#168—22no—
136C2×#41—51no—
137C22×D4—51no—
138C22×D4—101no—
139C23×C4—51no—
140C2×SD16—52no—
141C2×D8—52no—
142C2×M16—52no—
143#123—101no—
144C22⋊Q8—51no—
145C22⋊Q8—51no—
146#121—51no—
147#121—52no—
148#121—52no—
149#121—101no—
150#121—101no—
151#121C2×D511yescharacteristic
152#120—101no—
153C4×D4—101no—
154C4×D4—101no—
155#117—51no—
156C2×C4⋊C4C2×D511yescharacteristic
157C2×C22⋊C4—101no—
158#107C2×D512yescomplemented
159#106—52no—
160#103—52no—
161#102—52no—
162#102—52no—
163#102C2×D512yescomplemented
164C22⋊C8—51no—
165C22⋊C8—52no—
166C22⋊C8C2×D511yescharacteristic
167C22×C5—21no—
168C22×C5—21no—
169C22×C5—41no—
170C22×C5—41no—
171C22×C5—41no—
172C22×C5AGL(1,ℤ/8)12yes—
173C22×C5C22×D411yescharacteristic
174C2×D5—21no—
175C2×D5—21no—
176C2×D5—21no—
177C2×D5—21no—
178C2×D5—21no—
179C2×D5—21no—
180C2×D5—44no—
181C2×D5—42no—
182C2×D5—42no—
183C2×D5—42no—
184C2×D5—84no—
185C4×C5—21no—
186C4×C5—42no—
187C4×C5—41no—
188C4×C5H(5,𝔽2)12yes—
189Dic5—21no—
190Dic5—21no—
191Dic5—41no—
192Dic5—41no—
193C24—51no—
194C24—101no—
195#41—202no—
196C2×Q8—51no—
197C2×D4—21no—
198C2×D4—52no—
199C2×D4—102no—
200C2×D4—102no—
201C2×D4—101no—
202C2×D4—101no—
203C2×D4—101no—
204C2×D4—202no—
205C2×D4—204no—
206C2×D4C22×D511yescharacteristic
207C22×C4—21no—
208C22×C4—51no—
209C22×C4—51no—
210C22×C4—51no—
211C22×C4—51no—
212C22×C4—51no—
213C22×C4—101no—
214C22×C4—202no—
215C22×C4C22×D511yescharacteristic
216SD16—202no—
217D8—202no—
218M16—202no—
219C2×C8—52no—
220C2×C8C22×D512yes—
221C4⋊C4—21no—
222C4⋊C4—51no—
223C4⋊C4—101no—
224C4⋊C4—101no—
225C4⋊C4C22×D511yescharacteristic
226C4⋊C4C22×D512yes—
227C22⋊C4—21no—
228C22⋊C4—102no—
229C22⋊C4—102no—
230C22⋊C4—101no—
231C22⋊C4—101no—
232C22⋊C4—101no—
233C22⋊C4—101no—
234C42—101no—
235C2×C5—22no—
236C2×C5—81no—
237C2×C5#53411yescharacteristic
238C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
239C2×C5#57511yescharacteristic
240D5—41no—
241D5—41no—
242D5—82no—
243C23—21no—
244C23—51no—
245C23—51no—
246C23—102no—
247C23—102no—
248C23—201no—
249C23—201no—
250C23—202no—
251C23—202no—
252C23—202no—
253C23D4×D511yescharacteristic
254Q8—102no—
255D4—22no—
256D4—42no—
257D4—102no—
258D4—102no—
259D4—202no—
260D4—204no—
261D4—204no—
262D4—202no—
263D4—202no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—51no—
269C2×C4—51no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—101no—
273C2×C4—101no—
274C2×C4—201no—
275C2×C4—201no—
276C2×C4—201no—
277C2×C4—202no—
278C2×C4D4×D511yescharacteristic
279C2×C4C23×D511yescharacteristic
280C8—22no—
281C8—102no—
282C5#319411yescharacteristic, complemented, Sylow
283C22—21no—
284C22—21no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—101no—
289C22—101no—
290C22—101no—
291C22—101no—
292C22—101no—
293C22—101no—
294C22—204no—
295C22—202no—
296C22—202no—
297C22—202no—
298C22—404no—
299C22#405712yes—
300C22C2×D4×D511yescharacteristic
301C4—21no—
302C4—42no—
303C4—41no—
304C4—101no—
305C4—101no—
306C4—201no—
307C4—201no—
308C4#414912yes—
309C2—22no—
310C2—81no—
311C2—201no—
312C2—201no—
313C2—402no—
314C2#6598711yescharacteristic
315C2C2×#405711yescharacteristic
316C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#480C2
socleC22×C5C22×D4
hypercenter#480D5
hypocenterC5#3194
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #65940C2semidirect
2. C2×#4041C2semidirect
3. C5×#480C2semidirect
4. #65325C2semidirect
5. #65260C2semidirect
6. #65175C2semidirect
7. #65174C2semidirect
8. #65071C2semidirect
9. #65070C2semidirect
10. #65020C2semidirect
11. #65015C2semidirect
12. C2×#711C22semidirect
18. #4084C22semidirect
21. C2×#686C22semidirect
24. C2×#677C22semidirect
25. C2×#676C22semidirect
26. #4043C22semidirect
31. #4038C22semidirect
34. C5×#107C22semidirect
35. C5×#102C22semidirect
38. #3953C22semidirect
39. #3952C22semidirect
40. #3942C22semidirect
41. #3941C22semidirect
42. #3940C22semidirect
90. #480D5semidirect
158. #107C2×D5semidirect
163. #102C2×D5semidirect
282. C5#3194semidirect

Maximal subgroups

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

subgroupindex
#659402
C2×#40412
C5×#4802
#653252
#652602
#651752
#651742
#650712
#650702
#650202
#650152
#31945