Finite Group Databasebeta

← #10613214

Subgroups — #10613214

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×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4041C211yesmaximal, characteristic
3C5×#484C211yesmaximal, characteristic, complemented
4#65335C211yesmaximal, characteristic, complemented
5#65259C211yesmaximal, characteristic, complemented
6#65175C212yesmaximal, complemented
7#65171C212yesmaximal, complemented
8#65101C212yesmaximal, complemented
9#65095C212yesmaximal, complemented
10#65020C211yesmaximal, characteristic, complemented
11#65015C211yesmaximal, characteristic, complemented
12C2×Q8×D5C2211yescharacteristic, complemented
13C22×D20—21no—
14C22×D20C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C5×C22⋊Q8C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18#4092C2211yescharacteristic, complemented
19C2×#688C2212yescomplemented
20C2×#685—21no—
21#4070C2211yescharacteristic
22C2×#677C2212yescomplemented
23C2×#676C2212yescomplemented
24#4043C2211yescharacteristic, complemented
25#4042—21no—
26#4041—21no—
27#4041—21no—
28#4041C2212yes—
29#4041C2212yes—
30D5×C4⋊C4—21no—
31D5×C4⋊C4C2211yescharacteristic, complemented
32D5×C22⋊C4—21no—
33#4024—21no—
34C5×#107C2212yescomplemented
35C5×#103C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic
37C22⋊#168C2211yescharacteristic
38#3953C2212yescomplemented
39#3952C2212yescomplemented
40#3942C2212yescomplemented
41#3941C2212yescomplemented
42#3939C2212yescomplemented
43#3193—51nomaximal, Sylow
44C23×D5—21no—
45C23×D5D411yescharacteristic
46C2×C5×Q8C2311yescharacteristic
47C22×C4×C5—21no—
48C22×C4×C5C2311yescharacteristic
49C22×Dic5D411yescharacteristic
50Q8×D5—42no—
51C2×D20—22no—
52C2×D20—22no—
53C2×D20—21no—
54C2×D20—42no—
55C2×D20—44no—
56C2×D20C2312yes—
57C2×C4×D5—21no—
58C2×C4×D5—21no—
59C2×C4×D5—42no—
60C2×C4×D5D411yescharacteristic
61C2×C4×D5D411yescharacteristic
62C2×C4×D5C2311yescharacteristic
63C2×C4×D5C2311yescharacteristic
64C2×Dic10—21no—
65C2×Dic10C2311yescharacteristic
66C2×C5×C8C2312yes—
67C5×C4⋊C4—21no—
68C5×C4⋊C4—21no—
69C5×C4⋊C4C2312yes—
70C5×C4⋊C4C2311yescharacteristic
71C5×C22⋊C4—21no—
72C22⋊Dic5—21no—
73#688—42no—
74#685—21no—
75#685—21no—
76#685—22no—
77#685—22no—
78C4⋊Dic5—21no—
79C4⋊Dic5C2311yescharacteristic
80#683—21no—
81#683—21no—
82#683—21no—
83C2×#168C2312yes—
84#677—42no—
85#676—42no—
86C2×C22⋊Q8—51no—
87C2×#121—51no—
88#484—51no—
89#484D511yescharacteristic, complemented
90#480—52no—
91#468—52no—
92#468—52no—
93#450—51no—
94#450—51no—
95#448—52no—
96#407—51no—
97C23×C5C2×D411yescharacteristic
98C22×D5—22no—
99C22×D5—22no—
100C22×D5—22no—
101C22×D5—41no—
102C22×D5—42no—
103C22×D5—42no—
104C22×D5—42no—
105C22×D5C2×D411yescharacteristic
106C22×D5C2×D411yescharacteristic
107C5×Q8—22no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—22no—
112C2×C4×C5—42no—
113C2×C4×C5C2×D411yescharacteristic
114C2×C4×C5C2411yescharacteristic
115C2×Dic5—21no—
116C2×Dic5—21no—
117C2×Dic5—41no—
118C2×Dic5C2×D411yescharacteristic
119C2×Dic5C2×D411yescharacteristic
120D20—22no—
121D20—22no—
122D20—42no—
123D20—44no—
124D20—44no—
125C4×D5—22no—
126C4×D5—22no—
127C4×D5—41no—
128C4×D5—41no—
129C4×D5—42no—
130C4×D5—42no—
131Dic10—22no—
132Dic10—42no—
133C5×C8—22no—
134#168—22no—
135C22×Q8—51no—
136C22×D4—51no—
137C22×D4—101no—
138C23×C4—51no—
139C2×SD16—52no—
140C2×SD16—52no—
141C2×M16—52no—
142C22⋊Q8—51no—
143C22⋊Q8—51no—
144C22⋊Q8—51no—
145C22⋊Q8—101no—
146C22⋊Q8—101no—
147C22⋊Q8C2×D511yescharacteristic
148#121—52no—
149#121—52no—
150#121—101no—
151#121—101no—
152C2×C4⋊C4—51no—
153C2×C4⋊C4—101no—
154C2×C4⋊C4C2×D511yescharacteristic
155C2×C22⋊C4—101no—
156C2×C22⋊C4—101no—
157#107—52no—
158#107C2×D512yescomplemented
159#103—52no—
160#103C2×D512yescomplemented
161#102—52no—
162#102—52no—
163C22⋊C8—51no—
164C22⋊C8—52no—
165C22⋊C8C2×D511yescharacteristic
166C22×C5—21no—
167C22×C5—21no—
168C22×C5#13712yes—
169C22×C5C22×D411yescharacteristic
170C2×D5—21no—
171C2×D5—21no—
172C2×D5—21no—
173C2×D5—21no—
174C2×D5—21no—
175C2×D5—44no—
176C2×D5—42no—
177C2×D5—42no—
178C2×D5—42no—
179C2×D5—84no—
180C2×D5AGL(1,ℤ/8)12yes—
181C4×C5—21no—
182C4×C5—41no—
183C4×C5—41no—
184C4×C5—42no—
185C4×C5H(5,𝔽2)12yes—
186Dic5—21no—
187Dic5—21no—
188Dic5—41no—
189Dic5—41no—
190C24—51no—
191C24—101no—
192C2×Q8—51no—
193C2×Q8—101no—
194C2×Q8—202no—
195C2×Q8C22×D511yescharacteristic
196C2×D4—52no—
197C2×D4—102no—
198C2×D4—102no—
199C2×D4—101no—
200C2×D4—202no—
201C2×D4—204no—
202C22×C4—21no—
203C22×C4—51no—
204C22×C4—51no—
205C22×C4—51no—
206C22×C4—51no—
207C22×C4—51no—
208C22×C4—101no—
209C22×C4—101no—
210C22×C4—202no—
211C22×C4C22×D511yescharacteristic
212SD16—202no—
213SD16—202no—
214M16—202no—
215C2×C8—52no—
216C2×C8C22×D512yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—51no—
220C4⋊C4—101no—
221C4⋊C4—101no—
222C4⋊C4—101no—
223C4⋊C4—101no—
224C4⋊C4C22×D511yescharacteristic
225C4⋊C4C22×D512yes—
226C22⋊C4—21no—
227C22⋊C4—102no—
228C22⋊C4—102no—
229C22⋊C4—101no—
230C22⋊C4—101no—
231C22⋊C4—101no—
232C2×C5—22no—
233C2×C5#53411yescharacteristic
234C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
235C2×C5C2×#13711yescharacteristic
236D5—22no—
237D5—41no—
238D5—82no—
239C23—51no—
240C23—51no—
241C23—102no—
242C23—102no—
243C23—102no—
244C23—201no—
245C23—202no—
246C23—202no—
247C23—202no—
248C23D4×D511yescharacteristic
249Q8—22no—
250Q8—102no—
251Q8—202no—
252D4—102no—
253D4—102no—
254D4—202no—
255D4—204no—
256D4—204no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—51no—
263C2×C4—51no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—101no—
267C2×C4—101no—
268C2×C4—201no—
269C2×C4—201no—
270C2×C4—202no—
271C2×C4—202no—
272C2×C4—201no—
273C2×C4D4×D511yescharacteristic
274C2×C4C23×D511yescharacteristic
275C8—22no—
276C8—102no—
277C5#319311yescharacteristic, complemented, Sylow
278C22—21no—
279C22—21no—
280C22—52no—
281C22—101no—
282C22—101no—
283C22—101no—
284C22—101no—
285C22—101no—
286C22—204no—
287C22—202no—
288C22—202no—
289C22—202no—
290C22—404no—
291C22#406412yes—
292C22C2×D4×D511yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—101no—
298C4—101no—
299C4—201no—
300C4—201no—
301C4#414912yes—
302C2—22no—
303C2—102no—
304C2—201no—
305C2—402no—
306C2#6598711yescharacteristic
307C2C2×#406411yescharacteristic
308C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#484C2
socleC22×C5C22×D4
prime radicalC2×#4041C2
hypercenter#484D5
hypocenterC5#3193
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. D5×C22⋊Q8C2semidirect
3. C5×#484C2semidirect
4. #65335C2semidirect
5. #65259C2semidirect
6. #65175C2semidirect
7. #65171C2semidirect
8. #65101C2semidirect
9. #65095C2semidirect
10. #65020C2semidirect
11. #65015C2semidirect
12. C2×Q8×D5C22semidirect
18. #4092C22semidirect
19. C2×#688C22semidirect
22. C2×#677C22semidirect
23. C2×#676C22semidirect
24. #4043C22semidirect
31. D5×C4⋊C4C22semidirect
34. C5×#107C22semidirect
35. C5×#103C22semidirect
38. #3953C22semidirect
39. #3952C22semidirect
40. #3942C22semidirect
41. #3941C22semidirect
42. #3939C22semidirect
89. #484D5semidirect
158. #107C2×D5semidirect
160. #103C2×D5semidirect
277. C5#3193semidirect

Maximal subgroups

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

subgroupindex
D5×C22⋊Q82
C2×#40412
C5×#4842
#653352
#652592
#651752
#651712
#651012
#650952
#650202
#650152
#31935