Finite Group Databasebeta

← #10613451

Subgroups — #10613451

Order 640.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#66049C211yesmaximal, characteristic
2#66031C211yesmaximal, characteristic, complemented
3C5×#490C211yesmaximal, characteristic, complemented
4#65365C211yesmaximal, characteristic, complemented
5#65357C211yesmaximal, characteristic, complemented
6#65072C212yesmaximal, complemented
7#65065C212yesmaximal, complemented
8#65056C212yesmaximal, complemented
9#65051C212yesmaximal, complemented
10#65004C211yesmaximal, characteristic, complemented
11#64998C211yesmaximal, characteristic, complemented
12C2×#711C2211yescharacteristic
13C2×D4×D5C2211yescharacteristic
14C5×#127C2211yescharacteristic
15C5×#125C2211yescharacteristic, complemented
16#4086C2211yescharacteristic
17#4085—22no—
18#4083—22no—
19#4082C2211yescharacteristic
20C2×#687C2212yes—
21C2×#687C2212yescomplemented
22C2×Dic20C2212yescomplemented
23C2×#677C2212yes—
24#4044—22no—
25#4043C2212yes—
26#4036—22no—
27#4033C2212yescomplemented
28#4021C2211yescharacteristic
29#4018C2211yescharacteristic
30#4015C2211yescharacteristic, complemented
31C5×#102C2212yes—
32C5×#102C2212yescomplemented
33C5×#97C2211yescharacteristic, complemented
34#3952C2212yes—
35#3946C2212yescomplemented
36#3942C2212yescomplemented
37#3941C2212yes—
38C4×#168C2211yescharacteristic, complemented
39#3222—51nomaximal, Sylow
40C23×D5—21no—
41C2×C5×D4—22no—
42C2×C5×D4C2311yescharacteristic
43C2×C5×D4C2311yescharacteristic
44C2×C22⋊D5—21no—
45C2×C22⋊D5—21no—
46C2×C22⋊D5—22no—
47C22×Dic5—21no—
48#711—42no—
49D4×D5—42no—
50C2×D20C2311yescharacteristic
51C2×C4×D5C2311yescharacteristic
52C2×Dic10C2311yescharacteristic
53C2×Dic10C2312yes—
54C2×C5×C8C2312yes—
55C5×C4⋊C4—22no—
56C5×C4⋊C4C2312yes—
57C42×C5C2311yescharacteristic
58C22⋊Dic5—22no—
59#687—42no—
60#687—42no—
61#685—22no—
62#685—22no—
63#685D412yes—
64C4⋊Dic5—22no—
65C4⋊Dic5—22no—
66#683—22no—
67#683D412yes—
68C4×Dic5—22no—
69C4×Dic5C2311yescharacteristic
70C2×#168C2312yes—
71Dic20—42no—
72#677—42no—
73#541—51no—
74#535—51no—
75#497—51no—
76#492—51no—
77#490D511yescharacteristic, complemented
78#488—51no—
79#464—52no—
80#460—52no—
81#452—52no—
82#450—52no—
83#433—51no—
84C23×C5—21no—
85C23×C5—21no—
86C22×D5—21no—
87C22×D5—42no—
88C22×D5—82no—
89C22×D5C2×D411yescharacteristic
90C5×D4—22no—
91C5×D4—22no—
92C5×D4—82no—
93C2×C4×C5—22no—
94C2×C4×C5C2×D412yes—
95C2×C4×C5C2411yescharacteristic
96C22⋊D5—42no—
97C22⋊D5—42no—
98C22⋊D5—82no—
99C2×Dic5—22no—
100C2×Dic5—21no—
101C2×Dic5—42no—
102C2×Dic5C2×D412yes—
103C2×Dic5C2×D411yescharacteristic
104D20—22no—
105C4×D5—22no—
106Dic10—22no—
107Dic10—22no—
108Dic10—22no—
109C5×C8—22no—
110#168—22no—
111C2×#41—51no—
112C22×D4—51no—
113C2×Q16—52no—
114C2×SD16—52no—
115C2×SD16—52no—
116C2×SD16—52no—
117C4⋊Q8—51no—
118#127—51no—
119#127C2×D511yescharacteristic
120#126—102no—
121#125—102no—
122#125C2×D511yescharacteristic, complemented
123#124—51no—
124#124—51no—
125C22⋊Q8—52no—
126#121—102no—
127#120—102no—
128C4×Q8—52no—
129#117—51no—
130C4⋊C8—52no—
131#103—52no—
132#102—52no—
133#102C2×D512yes—
134#102C2×D512yescomplemented
135C22⋊C8—52no—
136#97C2×D511yescharacteristic, complemented
137C4×C8—51no—
138C22×C5—41no—
139C22×C5—41no—
140C22×C5—42no—
141C22×C5—42no—
142C22×C5C22×D411yescharacteristic
143C2×D5—21no—
144C2×D5—22no—
145C2×D5—41no—
146C2×D5—42no—
147C2×D5—84no—
148C4×C5—22no—
149C4×C5—42no—
150C4×C5H(5,𝔽2)12yes—
151Dic5—21no—
152Dic5—22no—
153Dic5—42no—
154Dic5—41no—
155C24—101no—
156#41—202no—
157C2×Q8—51no—
158C2×Q8—52no—
159C2×D4—22no—
160C2×D4—51no—
161C2×D4—101no—
162C2×D4—102no—
163C2×D4—101no—
164C2×D4—202no—
165C2×D4C22×D511yescharacteristic
166C2×D4C22×D511yescharacteristic
167C22×C4—51no—
168C22×C4—101no—
169Q16—202no—
170SD16—202no—
171SD16—202no—
172SD16—202no—
173C2×C8—52no—
174C2×C8C22×D512yes—
175C4⋊C4—22no—
176C4⋊C4—52no—
177C4⋊C4—102no—
178C4⋊C4—102no—
179C4⋊C4—102no—
180C4⋊C4C22×D512yes—
181C22⋊C4—52no—
182C22⋊C4—102no—
183C22⋊C4—102no—
184C22⋊C4—102no—
185C42—51no—
186C42—102no—
187C42C22×D511yescharacteristic
188C2×C5—81no—
189C2×C5—81no—
190C2×C5#53511yescharacteristic
191C2×C5#57712yes—
192D5—41no—
193D5—81no—
194C23—21no—
195C23—21no—
196C23—51no—
197C23—101no—
198C23—202no—
199C23—402no—
200Q8—102no—
201Q8—102no—
202Q8—102no—
203D4—22no—
204D4—22no—
205D4—82no—
206D4—102no—
207D4—202no—
208D4—202no—
209D4—402no—
210C2×C4—22no—
211C2×C4—52no—
212C2×C4—51no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—101no—
216C2×C4—202no—
217C2×C4D4×D512yes—
218C2×C4C23×D511yescharacteristic
219C8—22no—
220C8—102no—
221C5#322211yescharacteristic, complemented, Sylow
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—102no—
227C22—101no—
228C22—202no—
229C22—201no—
230C22—404no—
231C22C2×D4×D511yescharacteristic
232C4—22no—
233C4—42no—
234C4—102no—
235C4—101no—
236C4—201no—
237C4—202no—
238C4#414912yes—
239C2—81no—
240C2—81no—
241C2—201no—
242C2—401no—
243C2#6600811yescharacteristic
244C2#6610912yes—
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#490C2
socleC22×C5C22×D4
prime radical#66049C2
hypercenter#490D5
hypocenterC5#3222
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #66031C2semidirect
3. C5×#490C2semidirect
4. #65365C2semidirect
5. #65357C2semidirect
6. #65072C2semidirect
7. #65065C2semidirect
8. #65056C2semidirect
9. #65051C2semidirect
10. #65004C2semidirect
11. #64998C2semidirect
15. C5×#125C22semidirect
21. C2×#687C22semidirect
22. C2×Dic20C22semidirect
27. #4033C22semidirect
30. #4015C22semidirect
32. C5×#102C22semidirect
33. C5×#97C22semidirect
35. #3946C22semidirect
36. #3942C22semidirect
38. C4×#168C22semidirect
77. #490D5semidirect
122. #125C2×D5semidirect
134. #102C2×D5semidirect
136. #97C2×D5semidirect
221. C5#3222semidirect

Maximal subgroups

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

subgroupindex
#660492
#660312
C5×#4902
#653652
#653572
#650722
#650652
#650562
#650512
#650042
#649982
#32225