Finite Group Databasebeta

← #10613017

Subgroups — #10613017

Order 640.

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

normal subgroups only

This group has 462 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
2C2×#4043C211yesmaximal, characteristic, complemented
3C5×#480C211yesmaximal, characteristic, complemented
4#65329C211yesmaximal, characteristic, complemented
5#65256C211yesmaximal, characteristic, complemented
6#65175C212yesmaximal, complemented
7#65173C212yesmaximal, complemented
8#65069C212yesmaximal, complemented
9#65065C212yesmaximal, complemented
10#65027C211yesmaximal, characteristic, complemented
11#65015C211yesmaximal, characteristic, complemented
12C2×D4×D5—21no—
13C2×D4×D5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2211yescharacteristic, complemented
16C5×#121C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic, complemented
18#4085—21no—
19#4084C2211yescharacteristic
20C2×#687C2212yescomplemented
21#4076C2211yescharacteristic
22C2×#685—21no—
23C2×C4⋊Dic5—21no—
24C2×#677C2212yescomplemented
25C2×#676C2212yescomplemented
26#4043—21no—
27#4043—21no—
28#4043C2212yes—
29#4043C2212yescomplemented
30#4041C2211yescharacteristic
31D5×C4⋊C4C2211yescharacteristic
32#4030—21no—
33D5×C22⋊C4—21no—
34C5×#107C2212yescomplemented
35C5×#102C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic, complemented
37C22⋊#168C2211yescharacteristic, complemented
38#3952C2212yes—
39#3948C2212yescomplemented
40#3942C2212yescomplemented
41#3941C2212yescomplemented
42#3939C2212yescomplemented
43#3193—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—
52C2×C22⋊D5—21no—
53C22×Dic5—21no—
54C22×Dic5D411yescharacteristic
55D4×D5—42no—
56D4×D5—44no—
57C2×D20—21no—
58C2×D20C2311yescharacteristic
59C2×C4×D5—21no—
60C2×C4×D5—42no—
61C2×C4×D5D411yescharacteristic
62C2×C4×D5D411yescharacteristic
63C2×C4×D5C2311yescharacteristic
64C2×C4×D5C2311yescharacteristic
65C2×Dic10—21no—
66C2×Dic10—42no—
67C2×Dic10C2312yes—
68C2×C5×C8C2312yes—
69C5×C4⋊C4—21no—
70C5×C4⋊C4C2312yes—
71C5×C4⋊C4C2311yescharacteristic
72C5×C22⋊C4—21no—
73C22⋊Dic5—21no—
74#687—42no—
75#685—22no—
76#685—22no—
77#685—21no—
78#685—21no—
79C4⋊Dic5—22no—
80C4⋊Dic5—22no—
81C4⋊Dic5C2311yescharacteristic
82#683—21no—
83C2×#168C2312yes—
84#677—42no—
85#676—42no—
86C2×C22⋊Q8—51no—
87C2×#121—51no—
88#484—52no—
89#480—51no—
90#480D511yescharacteristic, complemented
91#468—52no—
92#468—52no—
93#450—52no—
94#448—51no—
95#448—51no—
96#407—51no—
97C23×C5—21no—
98C23×C5C2×D411yescharacteristic
99C22×D5—22no—
100C22×D5—22no—
101C22×D5—21no—
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—
118C22⋊D5—42no—
119C2×Dic5—21no—
120C2×Dic5—22no—
121C2×Dic5—41no—
122C2×Dic5—42no—
123C2×Dic5C2×D411yescharacteristic
124C2×Dic5C2×D411yescharacteristic
125D20—22no—
126D20—42no—
127C4×D5—22no—
128C4×D5—22no—
129C4×D5—41no—
130C4×D5—41no—
131C4×D5—42no—
132Dic10—22no—
133Dic10—22no—
134Dic10—42no—
135C5×C8—22no—
136#168—22no—
137C22×Q8—51no—
138C22×D4—51no—
139C22×D4—101no—
140C23×C4—51no—
141C2×SD16—52no—
142C2×SD16—52no—
143C2×M16—52no—
144C22⋊Q8—52no—
145C22⋊Q8—52no—
146C22⋊Q8—101no—
147C22⋊Q8—101no—
148#121—51no—
149#121—51no—
150#121—51no—
151#121—101no—
152#121—101no—
153#121C2×D511yescharacteristic
154C2×C4⋊C4—51no—
155C2×C4⋊C4—101no—
156C2×C4⋊C4C2×D511yescharacteristic, complemented
157C2×C22⋊C4—101no—
158C2×C22⋊C4—101no—
159#107—52no—
160#107C2×D512yescomplemented
161#103—52no—
162#103—52no—
163#102—52no—
164#102C2×D512yescomplemented
165C22⋊C8—51no—
166C22⋊C8—52no—
167C22⋊C8C2×D511yescharacteristic, complemented
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—41no—
171C22×C5—41no—
172C22×C5—41no—
173C22×C5AGL(1,ℤ/8)12yes—
174C22×C5C22×D411yescharacteristic
175C2×D5—21no—
176C2×D5—21no—
177C2×D5—21no—
178C2×D5—21no—
179C2×D5—21no—
180C2×D5—44no—
181C2×D5—41no—
182C2×D5—41no—
183C2×D5—41no—
184C2×D5—84no—
185C2×D5#13712yes—
186C4×C5—21no—
187C4×C5—41no—
188C4×C5—42no—
189C4×C5H(5,𝔽2)12yes—
190Dic5—21no—
191Dic5—21no—
192Dic5—42no—
193Dic5—41no—
194C24—51no—
195C24—101no—
196C2×Q8—52no—
197C2×Q8—101no—
198C2×Q8—202no—
199C2×D4—21no—
200C2×D4—51no—
201C2×D4—101no—
202C2×D4—101no—
203C2×D4—101no—
204C2×D4—101no—
205C2×D4—202no—
206C2×D4—204no—
207C2×D4C22×D511yescharacteristic
208C22×C4—21no—
209C22×C4—51no—
210C22×C4—51no—
211C22×C4—51no—
212C22×C4—51no—
213C22×C4—51no—
214C22×C4—101no—
215C22×C4—101no—
216C22×C4—202no—
217C22×C4C22×D511yescharacteristic
218SD16—202no—
219SD16—202no—
220M16—202no—
221C2×C8—52no—
222C2×C8C22×D512yes—
223C4⋊C4—21no—
224C4⋊C4—51no—
225C4⋊C4—101no—
226C4⋊C4—102no—
227C4⋊C4—102no—
228C4⋊C4C22×D512yes—
229C4⋊C4C22×D511yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—101no—
232C22⋊C4—102no—
233C22⋊C4—102no—
234C22⋊C4—101no—
235C22⋊C4—101no—
236C2×C5—22no—
237C2×C5—81no—
238C2×C5#53411yescharacteristic
239C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
240C2×C5C2×#13711yescharacteristic
241D5—22no—
242D5—41no—
243D5—81no—
244C23—21no—
245C23—51no—
246C23—51no—
247C23—102no—
248C23—102no—
249C23—101no—
250C23—201no—
251C23—202no—
252C23—202no—
253C23—202no—
254C23D4×D511yescharacteristic
255Q8—102no—
256Q8—102no—
257Q8—202no—
258D4—22no—
259D4—42no—
260D4—102no—
261D4—202no—
262D4—202no—
263D4—202no—
264D4—202no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—51no—
270C2×C4—51no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—101no—
274C2×C4—102no—
275C2×C4—201no—
276C2×C4—201no—
277C2×C4—202no—
278C2×C4—201no—
279C2×C4—202no—
280C2×C4D4×D511yescharacteristic
281C2×C4C23×D511yescharacteristic
282C8—22no—
283C8—102no—
284C5#319311yescharacteristic, complemented, Sylow
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—52no—
291C22—101no—
292C22—101no—
293C22—101no—
294C22—101no—
295C22—101no—
296C22—204no—
297C22—201no—
298C22—201no—
299C22—201no—
300C22—404no—
301C22#405712yes—
302C22C2×D4×D511yescharacteristic
303C4—21no—
304C4—42no—
305C4—41no—
306C4—101no—
307C4—101no—
308C4—201no—
309C4—202no—
310C4#414912yes—
311C2—22no—
312C2—81no—
313C2—102no—
314C2—201no—
315C2—401no—
316C2#6598711yescharacteristic
317C2C2×#405711yescharacteristic
318C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#480C2
socleC22×C5C22×D4
prime radicalD5×#121C2
hypercenter#480D5
hypocenterC5#3193
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
2. C2×#4043C2semidirect
3. C5×#480C2semidirect
4. #65329C2semidirect
5. #65256C2semidirect
6. #65175C2semidirect
7. #65173C2semidirect
8. #65069C2semidirect
9. #65065C2semidirect
10. #65027C2semidirect
11. #65015C2semidirect
15. C22×Dic10C22semidirect
17. C2×C5×C4⋊C4C22semidirect
20. C2×#687C22semidirect
24. C2×#677C22semidirect
25. C2×#676C22semidirect
29. #4043C22semidirect
34. C5×#107C22semidirect
35. C5×#102C22semidirect
36. C5×C22⋊C8C22semidirect
37. C22⋊#168C22semidirect
39. #3948C22semidirect
40. #3942C22semidirect
41. #3941C22semidirect
42. #3939C22semidirect
90. #480D5semidirect
156. C2×C4⋊C4C2×D5semidirect
160. #107C2×D5semidirect
164. #102C2×D5semidirect
167. C22⋊C8C2×D5semidirect
284. C5#3193semidirect

Maximal subgroups

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

subgroupindex
D5×#1212
C2×#40432
C5×#4802
#653292
#652562
#651752
#651732
#650692
#650652
#650272
#650152
#31935