Finite Group Databasebeta

← #10611727

Subgroups — #10611727

Order 640.

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

normal subgroups only

This group has 444 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
1C2×#4082C211yesmaximal, characteristic, complemented
2#65930C211yesmaximal, characteristic, complemented
3C5×#447C211yesmaximal, characteristic, complemented
4#65506C211yesmaximal, characteristic, complemented
5#65443C212yesmaximal, complemented
6#65442C212yesmaximal, complemented
7#65329C211yesmaximal, characteristic, complemented
8#65076C212yesmaximal, complemented
9#65054C212yesmaximal, complemented
10#65027C211yesmaximal, characteristic, complemented
11#65008C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C2×#711—21no—
14C22×Dic10C2211yescharacteristic
15C2×C5×D8C2212yes—
16C5×#121C2211yescharacteristic
17C2×C22⋊Dic5—21no—
18C2×C22⋊Dic5—21no—
19#4086C2211yescharacteristic, complemented
20#4085—21no—
21#4082—21no—
22#4082—21no—
23#4082C2212yes—
24#4082C2212yes—
25D4×Dic5C2211yescharacteristic, complemented
26C2×#687C2212yescomplemented
27C2×#686C2212yes—
28#4076C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30C2×#677C2212yescomplemented
31#4039C2211yescharacteristic, complemented
32#4034C2211yescharacteristic, complemented
33#4031—21no—
34#4027—21no—
35C5×#102C2212yes—
36C5×C22⋊C8C2211yescharacteristic
37#3964C2212yes—
38C22⋊#168C2211yescharacteristic
39#3948C2212yescomplemented
40#3947C2212yescomplemented
41#3946C2212yescomplemented
42#3942C2212yescomplemented
43#3213—51nomaximal, Sylow
44C24×C5—21no—
45C2×C5×D4—21no—
46C2×C5×D4—21no—
47C2×C5×D4—42no—
48C2×C5×D4C2311yescharacteristic
49C2×C5×D4C2312yes—
50C22×C4×C5C2311yescharacteristic
51C2×C22⋊D5—21no—
52C2×C22⋊D5—21no—
53C2×C22⋊D5—21no—
54C22×Dic5—21no—
55C22×Dic5—21no—
56C22×Dic5D411yescharacteristic
57C22×Dic5D411yescharacteristic
58#711—44no—
59C2×D20C2311yescharacteristic
60C2×C4×D5—21no—
61C2×Dic10—21no—
62C2×Dic10—21no—
63C2×Dic10—42no—
64C2×Dic10C2312yes—
65C5×D8—42no—
66C2×C5×C8C2312yes—
67C5×C4⋊C4C2311yescharacteristic
68C5×C22⋊C4—21no—
69C22⋊Dic5—22no—
70C22⋊Dic5—22no—
71C22⋊Dic5—22no—
72C22⋊Dic5—22no—
73C22⋊Dic5—21no—
74#687—42no—
75#686—42no—
76#685—21no—
77#685—21no—
78C4⋊Dic5C2311yescharacteristic
79#683—21no—
80C4×Dic5D411yescharacteristic
81C4×Dic5D411yescharacteristic
82C4×Dic5C2311yescharacteristic
83C4×Dic5C2311yescharacteristic
84C2×#168C2312yes—
85#677—42no—
86#532—51no—
87C2×#124—51no—
88#496—52no—
89#489—52no—
90#463—52no—
91#463—52no—
92#448—51no—
93#448—51no—
94#447—51no—
95#447D511yescharacteristic, complemented
96#432—51no—
97C23×C5—22no—
98C23×C5—21no—
99C23×C5—42no—
100C23×C5—42no—
101C23×C5—42no—
102C23×C5C2×D411yescharacteristic
103C22×D5—21no—
104C5×D4—22no—
105C5×D4—22no—
106C5×D4—22no—
107C5×D4—42no—
108C5×D4—42no—
109C2×C4×C5—22no—
110C2×C4×C5—21no—
111C2×C4×C5C2×D411yescharacteristic
112C2×C4×C5C2411yescharacteristic
113C22⋊D5—42no—
114C22⋊D5—42no—
115C22⋊D5—42no—
116C2×Dic5—22no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5—41no—
120C2×Dic5—42no—
121C2×Dic5—42no—
122C2×Dic5C2×D411yescharacteristic
123C2×Dic5C2×D411yescharacteristic
124C2×Dic5C2×D411yescharacteristic
125C2×Dic5C2×D411yescharacteristic
126D20—22no—
127C4×D5—42no—
128Dic10—22no—
129Dic10—22no—
130Dic10—42no—
131Dic10—42no—
132C5×C8—22no—
133#168—22no—
134C2×#41—101no—
135C22×Q8—51no—
136C22×D4C2×D511yescharacteristic
137C2×SD16—52no—
138C2×SD16—52no—
139C2×D8—52no—
140C2×D8C2×D512yes—
141C4⋊Q8—51no—
142#127—51no—
143#124—52no—
144#124—52no—
145#124—101no—
146#124—101no—
147#124—101no—
148#121—51no—
149#121—101no—
150#121C2×D511yescharacteristic
151C4×D4—51no—
152C4×D4—51no—
153C4×D4—101no—
154C2×C22⋊C4—101no—
155C2×C22⋊C4—101no—
156C2×C42—51no—
157C4⋊C8—52no—
158#103—52no—
159#103—52no—
160#102—52no—
161#102C2×D512yes—
162C22⋊C8—51no—
163C22⋊C8C2×D511yescharacteristic
164#97—52no—
165C22×C5—21no—
166C22×C5—21no—
167C22×C5—42no—
168C22×C5—41no—
169C22×C5—41no—
170C22×C5—42no—
171C22×C5—41no—
172C22×C5—42no—
173C22×C5—84no—
174C22×C5AGL(1,ℤ/8)12yes—
175C22×C5C22×D411yescharacteristic
176C2×D5—41no—
177C2×D5—41no—
178C2×D5—41no—
179C4×C5—21no—
180C4×C5—41no—
181C4×C5H(5,𝔽2)12yes—
182Dic5—21no—
183Dic5—21no—
184Dic5—21no—
185Dic5—21no—
186Dic5—42no—
187Dic5—41no—
188C24—21no—
189#41—204no—
190C2×Q8—52no—
191C2×Q8—101no—
192C2×Q8—101no—
193C2×Q8—202no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—42no—
197C2×D4—51no—
198C2×D4—101no—
199C2×D4—101no—
200C2×D4—101no—
201C2×D4C22×D512yes—
202C2×D4C22×D511yescharacteristic
203C22×C4—51no—
204C22×C4—51no—
205C22×C4—101no—
206C22×C4—101no—
207C22×C4—101no—
208C22×C4C22×D511yescharacteristic
209SD16—202no—
210SD16—202no—
211D8—42no—
212D8—202no—
213C2×C8—52no—
214C2×C8C22×D512yes—
215C4⋊C4—51no—
216C4⋊C4—101no—
217C4⋊C4C22×D511yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—101no—
220C22⋊C4—102no—
221C22⋊C4—102no—
222C22⋊C4—102no—
223C22⋊C4—102no—
224C22⋊C4—101no—
225C22⋊C4—101no—
226C42—51no—
227C42—51no—
228C42—51no—
229C42—51no—
230C2×C5—22no—
231C2×C5—81no—
232C2×C5—82no—
233C2×C5#53511yescharacteristic
234C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
235C2×C5#57511yescharacteristic
236D5—81no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—101no—
243C23D4×D511yescharacteristic
244Q8—102no—
245Q8—102no—
246Q8—202no—
247Q8—202no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—42no—
252D4—42no—
253D4—102no—
254D4—202no—
255D4—202no—
256D4—202no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—51no—
260C2×C4—51no—
261C2×C4—51no—
262C2×C4—51no—
263C2×C4—102no—
264C2×C4—102no—
265C2×C4—101no—
266C2×C4—202no—
267C2×C4—201no—
268C2×C4—202no—
269C2×C4—202no—
270C2×C4D4×D511yescharacteristic
271C2×C4C23×D511yescharacteristic
272C8—22no—
273C8—102no—
274C5#321311yescharacteristic, complemented, Sylow
275C22—21no—
276C22—21no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—41no—
283C22—84no—
284C22—201no—
285C22—201no—
286C22—201no—
287C22#405712yes—
288C22C2×D4×D511yescharacteristic
289C4—21no—
290C4—41no—
291C4—101no—
292C4—101no—
293C4—101no—
294C4—101no—
295C4—201no—
296C4—202no—
297C4#414412yes—
298C2—22no—
299C2—82no—
300C2—81no—
301C2—401no—
302C2#6592611yescharacteristic
303C2C2×#405711yescharacteristic
304C2#6610711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#447C2
socleC22×C5C22×D4
hypercenter#447D5
hypocenterC5#3213
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#4082C2semidirect
2. #65930C2semidirect
3. C5×#447C2semidirect
4. #65506C2semidirect
5. #65443C2semidirect
6. #65442C2semidirect
7. #65329C2semidirect
8. #65076C2semidirect
9. #65054C2semidirect
10. #65027C2semidirect
11. #65008C2semidirect
19. #4086C22semidirect
25. D4×Dic5C22semidirect
26. C2×#687C22semidirect
30. C2×#677C22semidirect
31. #4039C22semidirect
32. #4034C22semidirect
39. #3948C22semidirect
40. #3947C22semidirect
41. #3946C22semidirect
42. #3942C22semidirect
95. #447D5semidirect
274. C5#3213semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
#659302
C5×#4472
#655062
#654432
#654422
#653292
#650762
#650542
#650272
#650082
#32135