Finite Group Databasebeta

← #10612090

Subgroups — #10612090

Order 640.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#4057C212yesmaximal, complemented
2#66051C211yesmaximal, characteristic, complemented
3#65889C211yesmaximal, characteristic, complemented
4C5×#459C211yesmaximal, characteristic, complemented
5#65447C212yesmaximal, complemented
6#65365C211yesmaximal, characteristic, complemented
7C4×#686C211yesmaximal, characteristic, complemented
8#65130C211yesmaximal, characteristic, complemented
9#65127C211yesmaximal, characteristic, complemented
10#65069C212yesmaximal, complemented
11#65065C212yesmaximal, complemented
12C2×#711C2212yescomplemented
13C2×D4×D5—21no—
14C2×D4×D5C2212yes—
15C2×C5×D8C2212yescomplemented
16C5×#127C2211yescharacteristic
17C4×C5×D4C2211yescharacteristic, complemented
18#4084—21no—
19#4084C2212yes—
20#4083—22no—
21C2×#687C2212yescomplemented
22C2×#687C2212yescomplemented
23C2×#686C2211yescharacteristic
24#4076—21no—
25C2×C4⋊Dic5—21no—
26#4057—28no—
27C2×#677C2212yes—
28C2×#676C2212yescomplemented
29#4043C2212yescomplemented
30#4032—22no—
31C4×D20C2211yescharacteristic
32C4×D20C2211yescharacteristic
33#4015C2211yescharacteristic, complemented
34C5×C4⋊C8C2211yescharacteristic, complemented
35C5×#102C2212yescomplemented
36#3964C2211yescharacteristic, complemented
37#3952C2212yescomplemented
38#3948C2212yescomplemented
39#3941C2211yescharacteristic
40#3939C2211yescharacteristic, complemented
41C4×#168C2211yescharacteristic, complemented
42#3272—51nomaximal, Sylow
43C23×D5—22no—
44C2×C5×D4—21no—
45C2×C5×D4—21no—
46C2×C5×D4C2312yes—
47C2×C5×D4C2311yescharacteristic
48C22×C4×C5—21no—
49C2×C22⋊D5—22no—
50C2×C22⋊D5—22no—
51C2×C22⋊D5—22no—
52C22×Dic5—22no—
53#711—24no—
54#711—42no—
55D4×D5—24no—
56D4×D5—42no—
57D4×D5—44no—
58C2×D20D411yescharacteristic
59C2×D20C2311yescharacteristic
60C2×C4×D5—21no—
61C2×C4×D5C2312yes—
62C2×Dic10C2312yes—
63C5×D8—24no—
64C2×C5×C8C2312yes—
65C5×C4⋊C4C2311yescharacteristic
66C5×C22⋊C4—21no—
67C42×C5C2311yescharacteristic
68C22⋊Dic5—22no—
69#687—24no—
70#687—42no—
71#686D414yescomplemented
72#685—22no—
73#685—21no—
74#685D412yes—
75C4⋊Dic5—21no—
76C4⋊Dic5—21no—
77C4⋊Dic5—22no—
78C4⋊Dic5D411yescharacteristic
79C4⋊Dic5C2311yescharacteristic
80C2×#168C2311yescharacteristic
81C2×#168C2311yescharacteristic
82#677—24no—
83#676—24no—
84C2×AGL(1,ℤ/8)—52no—
85#547—51no—
86D42—51no—
87#492—51no—
88#468—52no—
89#461—51no—
90#459D511yescharacteristic, complemented
91#450—52no—
92#449—52no—
93C4×D8—51no—
94#436—51no—
95C23×C5—21no—
96C23×C5—22no—
97C22×D5—21no—
98C22×D5—44no—
99C22×D5—44no—
100C22×D5—82no—
101C22×D5C2×D412yes—
102C5×D4—22no—
103C5×D4—21no—
104C5×D4—22no—
105C5×D4—42no—
106C5×D4—81no—
107C5×D4C2×D412yescomplemented
108C2×C4×C5—21no—
109C2×C4×C5—42no—
110C2×C4×C5C2×D411yescharacteristic
111C2×C4×C5C2×D411yescharacteristic
112C2×C4×C5C2411yescharacteristic
113C22⋊D5—44no—
114C22⋊D5—44no—
115C22⋊D5—44no—
116C2×Dic5—22no—
117C2×Dic5—21no—
118C2×Dic5—44no—
119C2×Dic5C2×D412yes—
120D20—22no—
121D20—21no—
122D20C2×D412yes—
123C4×D5—22no—
124C4×D5—22no—
125C4×D5—42no—
126Dic10—22no—
127Dic10—22no—
128C5×C8—22no—
129#168—21no—
130#168C2×D412yescomplemented
131C2×#41—52no—
132C22×D4—52no—
133C22×D4—101no—
134AGL(1,ℤ/8)—108no—
135C2×SD16—52no—
136C2×SD16—52no—
137C2×SD16—52no—
138C2×D8—51no—
139C2×D8C2×D512yescomplemented
140C2×M16—52no—
141C4⋊Q8—51no—
142#127C2×D511yescharacteristic
143#123—102no—
144C22⋊Q8—52no—
145#121—52no—
146#121—101no—
147#121—101no—
148#120—102no—
149C4×D4—51no—
150C4×D4—51no—
151C4×D4C2×D511yescharacteristic, complemented
152C2×C4⋊C4—101no—
153#107—51no—
154C4⋊C8C2×D511yescharacteristic, complemented
155#103—52no—
156#102—51no—
157#102—51no—
158#102C2×D512yescomplemented
159C22⋊C8—52no—
160C4×C8—51no—
161C22×C5—22no—
162C22×C5—41no—
163C22×C5—41no—
164C22×C5—42no—
165C22×C5—42no—
166C22×C5—42no—
167C22×C5C22×D411yescharacteristic
168C2×D5—22no—
169C2×D5—22no—
170C2×D5—22no—
171C2×D5—22no—
172C2×D5—41no—
173C2×D5—41no—
174C2×D5—88no—
175C4×C5—21no—
176C4×C5—41no—
177C4×C5AGL(1,ℤ/8)12yescomplemented
178C4×C5C22×D411yescharacteristic
179C4×C5H(5,𝔽2)11yescharacteristic
180Dic5—22no—
181Dic5—42no—
182Dic5—41no—
183C24—102no—
184#41—104no—
185#41—202no—
186C2×Q8—52no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—51no—
190C2×D4—51no—
191C2×D4—102no—
192C2×D4—104no—
193C2×D4—102no—
194C2×D4—102no—
195C2×D4—202no—
196C2×D4—204no—
197C2×D4C22×D512yes—
198C2×D4C22×D511yescharacteristic
199C22×C4—21no—
200C22×C4—52no—
201C22×C4—102no—
202C22×C4—101no—
203SD16—104no—
204SD16—104no—
205SD16—202no—
206D8—24no—
207D8—54no—
208M16—104no—
209C2×C8—51no—
210C2×C8—51no—
211C2×C8C22×D512yes—
212C4⋊C4—51no—
213C4⋊C4—51no—
214C4⋊C4—102no—
215C4⋊C4—101no—
216C4⋊C4—101no—
217C4⋊C4C22×D511yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—52no—
220C22⋊C4—102no—
221C22⋊C4—102no—
222C22⋊C4—101no—
223C42C22×D511yescharacteristic
224C2×C5—42no—
225C2×C5—82no—
226C2×C5D4211yescharacteristic
227C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
228C2×C5#57711yescharacteristic
229D5—42no—
230D5—42no—
231C23—21no—
232C23—22no—
233C23—52no—
234C23—101no—
235C23—204no—
236C23—204no—
237C23—402no—
238Q8—102no—
239Q8—102no—
240D4—22no—
241D4—21no—
242D4—22no—
243D4—42no—
244D4—52no—
245D4—81no—
246D4—101no—
247D4—102no—
248D4—204no—
249D4—204no—
250D4—204no—
251D4D4×D512yescomplemented
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—52no—
255C2×C4—102no—
256C2×C4—102no—
257C2×C4—101no—
258C2×C4—102no—
259C2×C4—204no—
260C2×C4—202no—
261C2×C4D4×D511yescharacteristic
262C2×C4D4×D511yescharacteristic
263C2×C4C23×D511yescharacteristic
264C8—22no—
265C8—52no—
266C8—101no—
267C5#327211yescharacteristic, complemented, Sylow
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—102no—
275C22—102no—
276C22—102no—
277C22—102no—
278C22—201no—
279C22—201no—
280C22—408no—
281C22C2×D4×D511yescharacteristic
282C4—21no—
283C4—41no—
284C4—102no—
285C4—201no—
286C4—202no—
287C4#405712yescomplemented
288C4C2×D4×D511yescharacteristic
289C4#414911yescharacteristic
290C2—42no—
291C2—82no—
292C2—202no—
293C2—202no—
294C2#6594311yescharacteristic
295C2C2×#405711yescharacteristic
296C2#6610911yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#459C2
socleC22×C5C22×D4
hypercenter#459D5
hypocenterC5#3272
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#4057C2semidirect
2. #66051C2semidirect
3. #65889C2semidirect
4. C5×#459C2semidirect
5. #65447C2semidirect
6. #65365C2semidirect
7. C4×#686C2semidirect
8. #65130C2semidirect
9. #65127C2semidirect
10. #65069C2semidirect
11. #65065C2semidirect
12. C2×#711C22semidirect
15. C2×C5×D8C22semidirect
17. C4×C5×D4C22semidirect
21. C2×#687C22semidirect
22. C2×#687C22semidirect
28. C2×#676C22semidirect
29. #4043C22semidirect
33. #4015C22semidirect
34. C5×C4⋊C8C22semidirect
35. C5×#102C22semidirect
36. #3964C22semidirect
37. #3952C22semidirect
38. #3948C22semidirect
40. #3939C22semidirect
41. C4×#168C22semidirect
71. #686D4semidirect
90. #459D5semidirect
107. C5×D4C2×D4semidirect
130. #168C2×D4semidirect
139. C2×D8C2×D5semidirect
151. C4×D4C2×D5semidirect
154. C4⋊C8C2×D5semidirect
158. #102C2×D5semidirect
177. C4×C5AGL(1,ℤ/8)semidirect
251. D4D4×D5semidirect
267. C5#3272semidirect
287. C4#4057semidirect

Maximal subgroups

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

subgroupindex
C2×#40572
#660512
#658892
C5×#4592
#654472
#653652
C4×#6862
#651302
#651272
#650692
#650652
#32725