Finite Group Databasebeta

← #10611872

Subgroups — #10611872

Order 640.

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

normal subgroups only

This group has 490 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
1C2×#4086C211yesmaximal, characteristic, complemented
2#65958C211yesmaximal, characteristic, complemented
3C5×#450C211yesmaximal, characteristic, complemented
4#65506C211yesmaximal, characteristic, complemented
5#65465C212yesmaximal, complemented
6#65452C212yesmaximal, complemented
7#65335C211yesmaximal, characteristic, complemented
8#65075C212yesmaximal, complemented
9#65053C212yesmaximal, complemented
10#65021C211yesmaximal, characteristic, complemented
11#65008C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5—21no—
14C22×C22⋊D5—21no—
15C22×D20C2211yescharacteristic
16C2×C5×SD16C2212yescomplemented
17C5×C22⋊Q8C2211yescharacteristic
18#4093C2211yescharacteristic, complemented
19Q8×Dic5C2211yescharacteristic, complemented
20C2×#688C2212yescomplemented
21#4086—21no—
22#4086—21no—
23#4086C2212yes—
24#4086C2212yes—
25C2×#686C2212yescomplemented
26#4076C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28C2×D40C2212yescomplemented
29#4044—21no—
30#4039C2211yescharacteristic, complemented
31#4038—21no—
32#4036C2211yescharacteristic, complemented
33#4032—21no—
34#4027—21no—
35C5×#102C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic
37#3964C2212yescomplemented
38C22⋊#168C2211yescharacteristic
39#3953C2212yescomplemented
40#3947C2212yescomplemented
41#3946C2212yescomplemented
42#3941C2212yescomplemented
43#3217—51nomaximal, Sylow
44C24×C5—21no—
45C23×D5—21no—
46C2×C5×Q8C2311yescharacteristic
47C2×C5×D4—21no—
48C2×C5×D4—42no—
49C2×C5×D4C2312yes—
50C22×C4×C5C2311yescharacteristic
51C2×C22⋊D5—21no—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—22no—
54C2×C22⋊D5—22no—
55C2×C22⋊D5—22no—
56C2×C22⋊D5—44no—
57C2×C22⋊D5—44no—
58C22×Dic5D411yescharacteristic
59C22×Dic5D411yescharacteristic
60C2×D20—21no—
61C2×D20—42no—
62C2×D20C2312yes—
63C2×D20C2311yescharacteristic
64C2×C4×D5—21no—
65C5×SD16—42no—
66C2×C5×C8C2312yes—
67C5×C4⋊C4—21no—
68C5×C4⋊C4C2311yescharacteristic
69C5×C22⋊C4—21no—
70#688—42no—
71#686—42no—
72#685—21no—
73#685—21no—
74#685—21no—
75#685—21no—
76C4⋊Dic5—21no—
77C4⋊Dic5—21no—
78C4⋊Dic5C2311yescharacteristic
79#683—21no—
80C4×Dic5—21no—
81C4×Dic5D411yescharacteristic
82C4×Dic5D411yescharacteristic
83C4×Dic5C2311yescharacteristic
84C4×Dic5C2311yescharacteristic
85C2×#168C2312yes—
86D40—42no—
87C2×#127—51no—
88#529—51no—
89#496—52no—
90#490—52no—
91#460—52no—
92#459—52no—
93#450—51no—
94#450D511yescharacteristic, complemented
95#447—51no—
96#447—51no—
97#432—51no—
98C23×C5—22no—
99C23×C5—42no—
100C23×C5—42no—
101C23×C5—42no—
102C23×C5C2×D411yescharacteristic
103C22×D5—21no—
104C22×D5—22no—
105C22×D5—42no—
106C22×D5—42no—
107C22×D5—42no—
108C5×Q8—22no—
109C5×D4—22no—
110C5×D4—22no—
111C5×D4—42no—
112C2×C4×C5—22no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5C2×D411yescharacteristic
116C2×C4×C5C2411yescharacteristic
117C22⋊D5—44no—
118C22⋊D5—44no—
119C22⋊D5—44no—
120C22⋊D5—42no—
121C22⋊D5—44no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—21no—
125C2×Dic5C2×D411yescharacteristic
126C2×Dic5C2×D411yescharacteristic
127C2×Dic5C2×D411yescharacteristic
128C2×Dic5C2×D411yescharacteristic
129D20—22no—
130D20—22no—
131D20—22no—
132D20—42no—
133C4×D5—42no—
134C5×C8—22no—
135#168—22no—
136C22×D4—51no—
137C22×D4—101no—
138C22×D4—101no—
139C22×D4C2×D511yescharacteristic
140C2×SD16—52no—
141C2×SD16C2×D512yescomplemented
142C2×D8—52no—
143C2×D8—52no—
144#127—52no—
145#127—52no—
146#127—101no—
147#127—101no—
148#126—101no—
149#125—51no—
150#124—51no—
151#123—101no—
152C22⋊Q8C2×D511yescharacteristic
153#121—51no—
154C4×Q8—51no—
155C4×D4—51no—
156C4×D4—101no—
157#117—101no—
158C2×C42—51no—
159C4⋊C8—52no—
160#102—52no—
161#102—52no—
162#102—52no—
163#102C2×D512yescomplemented
164C22⋊C8—51no—
165C22⋊C8C2×D511yescharacteristic
166#97—52no—
167C22×C5—21no—
168C22×C5—21no—
169C22×C5—42no—
170C22×C5—42no—
171C22×C5—42no—
172C22×C5—84no—
173C22×C5AGL(1,ℤ/8)12yes—
174C22×C5C22×D411yescharacteristic
175C2×D5—42no—
176C2×D5—41no—
177C2×D5—41no—
178C2×D5—42no—
179C2×D5—42no—
180C2×D5—41no—
181C2×D5—84no—
182C4×C5—21no—
183C4×C5—41no—
184C4×C5—41no—
185C4×C5H(5,𝔽2)12yes—
186Dic5—21no—
187Dic5—21no—
188Dic5—21no—
189Dic5—41no—
190Dic5AGL(1,ℤ/8)12yes—
191C24—21no—
192C24—101no—
193C2×Q8C22×D511yescharacteristic
194C2×D4—21no—
195C2×D4—42no—
196C2×D4—52no—
197C2×D4—51no—
198C2×D4—102no—
199C2×D4—102no—
200C2×D4—102no—
201C2×D4—102no—
202C2×D4—101no—
203C2×D4—101no—
204C2×D4—202no—
205C2×D4—204no—
206C2×D4—204no—
207C2×D4C22×D512yes—
208C22×C4—51no—
209C22×C4—51no—
210C22×C4—101no—
211C22×C4C22×D511yescharacteristic
212SD16—42no—
213SD16—202no—
214D8—202no—
215D8—202no—
216C2×C8—52no—
217C2×C8C22×D512yes—
218C4⋊C4—21no—
219C4⋊C4—51no—
220C4⋊C4—101no—
221C4⋊C4—101no—
222C4⋊C4—101no—
223C4⋊C4C22×D511yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—101no—
226C22⋊C4—101no—
227C22⋊C4—101no—
228C22⋊C4—101no—
229C42—51no—
230C42—51no—
231C42—51no—
232C42—51no—
233C42—101no—
234C2×C5—22no—
235C2×C5—82no—
236C2×C5#53511yescharacteristic
237C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
238C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
239D5—81no—
240D5—82no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—101no—
246C23—102no—
247C23—202no—
248C23—202no—
249C23—202no—
250C23D4×D511yescharacteristic
251Q8—22no—
252D4—22no—
253D4—22no—
254D4—42no—
255D4—102no—
256D4—102no—
257D4—102no—
258D4—202no—
259D4—204no—
260D4—204no—
261D4—204no—
262D4—204no—
263D4—202no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—51no—
268C2×C4—51no—
269C2×C4—51no—
270C2×C4—51no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—101no—
274C2×C4—202no—
275C2×C4D4×D511yescharacteristic
276C2×C4C23×D511yescharacteristic
277C8—22no—
278C8—102no—
279C5#321711yescharacteristic, complemented, Sylow
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—202no—
287C22—201no—
288C22—201no—
289C22—202no—
290C22—202no—
291C22—201no—
292C22—404no—
293C22#406012yes—
294C22C2×D4×D511yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—52no—
299C4—101no—
300C4—101no—
301C4—101no—
302C4—201no—
303C4#414412yes—
304C2—22no—
305C2—82no—
306C2—402no—
307C2—401no—
308C2#6592611yescharacteristic
309C2C2×#406011yescharacteristic
310C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#450C2
socleC22×C5C22×D4
hypercenter#450D5
hypocenterC5#3217
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#4086C2semidirect
2. #65958C2semidirect
3. C5×#450C2semidirect
4. #65506C2semidirect
5. #65465C2semidirect
6. #65452C2semidirect
7. #65335C2semidirect
8. #65075C2semidirect
9. #65053C2semidirect
10. #65021C2semidirect
11. #65008C2semidirect
16. C2×C5×SD16C22semidirect
18. #4093C22semidirect
19. Q8×Dic5C22semidirect
20. C2×#688C22semidirect
25. C2×#686C22semidirect
28. C2×D40C22semidirect
30. #4039C22semidirect
32. #4036C22semidirect
35. C5×#102C22semidirect
37. #3964C22semidirect
39. #3953C22semidirect
40. #3947C22semidirect
41. #3946C22semidirect
42. #3941C22semidirect
94. #450D5semidirect
141. C2×SD16C2×D5semidirect
163. #102C2×D5semidirect
279. C5#3217semidirect

Maximal subgroups

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

subgroupindex
C2×#40862
#659582
C5×#4502
#655062
#654652
#654522
#653352
#650752
#650532
#650212
#650082
#32175