Finite Group Databasebeta

← #10611401

Subgroups — #10611401

Order 640.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#4060C212yesmaximal, complemented
2#65911C211yesmaximal, characteristic, complemented
3D4×D20C211yesmaximal, characteristic, complemented
4C4×C5×SD16C211yesmaximal, characteristic, complemented
5#65316C211yesmaximal, characteristic, complemented
6#65304C211yesmaximal, characteristic, complemented
7#65154C212yesmaximal, complemented
8#65100C212yesmaximal, complemented
9#65062C212yesmaximal, complemented
10#64984C211yesmaximal, characteristic, complemented
11#64977C211yesmaximal, characteristic, complemented
12C2×#713C2212yescomplemented
13C2×D4×D5C2212yes—
14C22×D20—21no—
15C2×C5×SD16C2211yescharacteristic
16C4×C5×Q8C2211yescharacteristic, complemented
17C4×C5×D4C2211yescharacteristic
18C2×#688C2212yescomplemented
19C2×#686C2212yes—
20#4076—21no—
21#4060—28no—
22C2×D40C2212yescomplemented
23C2×D40C2212yescomplemented
24C2×#676C2212yescomplemented
25#4041—22no—
26#4041C2212yes—
27#4041C2212yescomplemented
28#4028—22no—
29#4020—21no—
30#4020C2211yescharacteristic
31#4020C2211yescharacteristic, complemented
32C4×D20—21no—
33C4×D20C2211yescharacteristic
34C5×#106C2211yescharacteristic, complemented
35C5×#103C2211yescharacteristic, complemented
36C5×#102C2211yescharacteristic
37C4×C5×C8C2211yescharacteristic, complemented
38#3952C2212yescomplemented
39#3941C2212yescomplemented
40#3941C2212yescomplemented
41C4⋊#168C2211yescharacteristic, complemented
42#3274—51nomaximal, Sylow
43C23×D5—22no—
44C2×C5×Q8C2311yescharacteristic
45C2×C5×D4C2311yescharacteristic
46C22×C4×C5—21no—
47C2×C22⋊D5—22no—
48#713—24no—
49#713—42no—
50D4×D5—24no—
51D4×D5—42no—
52C2×D20—22no—
53C2×D20—22no—
54C2×D20—21no—
55C2×D20—21no—
56C2×D20—22no—
57C2×D20—21no—
58C2×D20—21no—
59C2×D20—44no—
60C2×D20D411yescharacteristic
61C2×D20C2312yes—
62C2×D20C2312yes—
63C2×C4×D5—22no—
64C2×C4×D5C2312yes—
65C5×SD16D414yescomplemented
66C2×C5×C8C2311yescharacteristic
67C2×C5×C8C2311yescharacteristic
68C5×C4⋊C4—21no—
69C5×C4⋊C4C2311yescharacteristic
70C5×C4⋊C4C2311yescharacteristic
71C5×C22⋊C4—21no—
72C42×C5—21no—
73C42×C5C2311yescharacteristic
74#688—24no—
75#686—24no—
76#685—22no—
77#685—22no—
78#685D412yes—
79C4⋊Dic5D411yescharacteristic
80C2×#168C2312yes—
81D40—24no—
82D40—42no—
83#676—24no—
84C2×AGL(1,ℤ/8)—52no—
85#550—51no—
86D42—51no—
87#493—51no—
88#469—52no—
89#460—51no—
90#459—51no—
91#449—52no—
92#447—52no—
93C4×SD16D511yescharacteristic, complemented
94#436—51no—
95C23×C5—21no—
96C22×D5—22no—
97C22×D5—22no—
98C22×D5—44no—
99C22×D5—44no—
100C22×D5—82no—
101C22×D5C2×D412yes—
102C5×Q8—21no—
103C5×Q8C2×D412yescomplemented
104C5×D4—21no—
105C5×D4C2×D412yes—
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—42no—
110C2×C4×C5C2×D411yescharacteristic
111C2×C4×C5C2×D411yescharacteristic
112C2×C4×C5C2411yescharacteristic
113C22⋊D5—44no—
114C2×Dic5C2×D412yes—
115D20—22no—
116D20—22no—
117D20—22no—
118D20—22no—
119D20—22no—
120D20—44no—
121D20—44no—
122D20—42no—
123D20—42no—
124D20—81no—
125D20—81no—
126D20—82no—
127C4×D5—22no—
128C4×D5—22no—
129C4×D5—44no—
130C5×C8—21no—
131C5×C8C2×D412yescomplemented
132#168—22no—
133C2×#41—52no—
134C22×D4—52no—
135C22×D4—101no—
136AGL(1,ℤ/8)—108no—
137C2×SD16—52no—
138C2×SD16C2×D511yescharacteristic
139C2×D8—52no—
140C2×D8—52no—
141C2×D8—52no—
142C2×M16—52no—
143#127—51no—
144#127—51no—
145#127—101no—
146#121—52no—
147#121—52no—
148#121—102no—
149#121—101no—
150#120—102no—
151C4×Q8C2×D511yescharacteristic, complemented
152C4×D4—51no—
153C4×D4—101no—
154C4×D4C2×D511yescharacteristic
155#106C2×D511yescharacteristic, complemented
156C4⋊C8—51no—
157#103C2×D511yescharacteristic, complemented
158#102—52no—
159#102—52no—
160#102C2×D511yescharacteristic
161C22⋊C8—52no—
162C4×C8C2×D511yescharacteristic, complemented
163C22×C5—22no—
164C22×C5—41no—
165C22×C5—41no—
166C22×C5C22×D411yescharacteristic
167C2×D5—22no—
168C2×D5—22no—
169C2×D5—22no—
170C2×D5—42no—
171C2×D5—42no—
172C2×D5—42no—
173C2×D5—42no—
174C2×D5—42no—
175C2×D5—42no—
176C2×D5—88no—
177C4×C5—22no—
178C4×C5—21no—
179C4×C5—41no—
180C4×C5—41no—
181C4×C5AGL(1,ℤ/8)12yescomplemented
182C4×C5C22×D411yescharacteristic
183C4×C5H(5,𝔽2)11yescharacteristic
184Dic5—22no—
185C24—102no—
186#41—104no—
187#41—202no—
188C2×Q8C22×D511yescharacteristic
189C2×D4—51no—
190C2×D4—52no—
191C2×D4—52no—
192C2×D4—102no—
193C2×D4—102no—
194C2×D4—102no—
195C2×D4—102no—
196C2×D4—104no—
197C2×D4—101no—
198C2×D4—101no—
199C2×D4—101no—
200C2×D4—101no—
201C2×D4—204no—
202C2×D4—202no—
203C2×D4C22×D511yescharacteristic
204C22×C4—21no—
205C22×C4—52no—
206C22×C4—102no—
207SD16—104no—
208SD16D2014yescomplemented
209D8—104no—
210D8—104no—
211D8—202no—
212M16—104no—
213C2×C8—52no—
214C2×C8C22×D511yescharacteristic
215C2×C8C22×D511yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—51no—
218C4⋊C4C22×D511yescharacteristic
219C4⋊C4C22×D511yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—52no—
222C22⋊C4—102no—
223C22⋊C4—102no—
224C42—21no—
225C42C22×D511yescharacteristic
226C2×C5—42no—
227C2×C5D4211yescharacteristic
228C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C5#57611yescharacteristic
230D5—42no—
231D5—82no—
232D5—82no—
233C23—21no—
234C23—52no—
235C23—102no—
236C23—102no—
237C23—204no—
238C23—204no—
239C23—402no—
240Q8—21no—
241Q8C2×D2012yescomplemented
242D4—21no—
243D4—102no—
244D4—102no—
245D4—102no—
246D4—102no—
247D4—102no—
248D4—204no—
249D4—204no—
250D4—204no—
251D4—202no—
252D4—202no—
253D4—401no—
254D4—401no—
255D4—402no—
256D4C2×D2012yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—52no—
262C2×C4—102no—
263C2×C4—102no—
264C2×C4—204no—
265C2×C4D4×D511yescharacteristic
266C2×C4D4×D511yescharacteristic
267C2×C4C23×D511yescharacteristic
268C8—21no—
269C8—102no—
270C8C2×D2012yescomplemented
271C5#327411yescharacteristic, complemented, Sylow
272C22—22no—
273C22—41no—
274C22—41no—
275C22—102no—
276C22—102no—
277C22—102no—
278C22—202no—
279C22—202no—
280C22—202no—
281C22—202no—
282C22—202no—
283C22—202no—
284C22—408no—
285C22C2×D4×D511yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—102no—
291C4#406012yescomplemented
292C4C22×D2011yescharacteristic
293C4#414911yescharacteristic
294C2—42no—
295C2—202no—
296C2—402no—
297C2—402no—
298C2D4×D2011yescharacteristic
299C2C2×#406011yescharacteristic
300C2#6610311yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC4×C5×SD16C2
socleC22×C5C22×D4
hypercenterC4×SD16D5
hypocenterC5#3274
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#4060C2semidirect
2. #65911C2semidirect
3. D4×D20C2semidirect
4. C4×C5×SD16C2semidirect
5. #65316C2semidirect
6. #65304C2semidirect
7. #65154C2semidirect
8. #65100C2semidirect
9. #65062C2semidirect
10. #64984C2semidirect
11. #64977C2semidirect
12. C2×#713C22semidirect
16. C4×C5×Q8C22semidirect
18. C2×#688C22semidirect
22. C2×D40C22semidirect
23. C2×D40C22semidirect
24. C2×#676C22semidirect
27. #4041C22semidirect
31. #4020C22semidirect
34. C5×#106C22semidirect
35. C5×#103C22semidirect
37. C4×C5×C8C22semidirect
38. #3952C22semidirect
39. #3941C22semidirect
40. #3941C22semidirect
41. C4⋊#168C22semidirect
65. C5×SD16D4semidirect
93. C4×SD16D5semidirect
103. C5×Q8C2×D4semidirect
131. C5×C8C2×D4semidirect
151. C4×Q8C2×D5semidirect
155. #106C2×D5semidirect
157. #103C2×D5semidirect
162. C4×C8C2×D5semidirect
181. C4×C5AGL(1,ℤ/8)semidirect
208. SD16D20semidirect
241. Q8C2×D20semidirect
270. C8C2×D20semidirect
271. C5#3274semidirect
291. C4#4060semidirect

Maximal subgroups

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

subgroupindex
C2×#40602
#659112
D4×D202
C4×C5×SD162
#653162
#653042
#651542
#651002
#650622
#649842
#649772
#32745