Finite Group Databasebeta

← #10612701

Subgroups — #10612701

Order 640.

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

normal subgroups only

This group has 332 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
1#66062C211yesmaximal, characteristic, complemented
2#65891C211yesmaximal, characteristic
3C5×#474C211yesmaximal, characteristic, complemented
4#65383C211yesmaximal, characteristic, complemented
5#65297C211yesmaximal, characteristic, complemented
6#65177C212yesmaximal, complemented
7#65167C212yesmaximal, complemented
8#65139C211yesmaximal, characteristic, complemented
9#65122C211yesmaximal, characteristic, complemented
10#65073C212yesmaximal
11#65057C212yesmaximal, complemented
12C5×C4⋊Q8C2211yescharacteristic
13C4×C5×D4C2211yescharacteristic
14#4092—21no—
15Q8×Dic5—21no—
16#4084C2211yescharacteristic
17#4082C2211yescharacteristic
18C4×C22⋊D5—21no—
19#4072—21no—
20#4043—21no—
21#4043C2211yescharacteristic
22#4038C2212yes—
23#4036C2212yes—
24#4036C2211yescharacteristic
25#4033—21no—
26#4029—21no—
27#4025—21no—
28C42×D5C2211yescharacteristic
29#4015C2211yescharacteristic
30C4×Dic10C2211yescharacteristic
31C5×#107C2212yes—
32C5×C4⋊C8C2211yescharacteristic
33C5×#102C2212yes—
34#3964C2212yes—
35#3952C2212yes—
36#3950C2212yes—
37#3948C2212yes—
38C8×Dic5C2212yes—
39#3942C2212yes—
40#3940C2212yes—
41#3940C2212yes—
42C4⋊#168C2211yescharacteristic
43#3233—51nomaximal, Sylow
44C2×C5×Q8—21no—
45C2×C5×D4C2311yescharacteristic
46C22×C4×C5—21no—
47C2×C22⋊D5—21no—
48C2×C4×D5D411yescharacteristic
49C2×C4×D5D411yescharacteristic
50C2×C4×D5C2311yescharacteristic
51C2×Dic10—21no—
52C2×Dic10C2311yescharacteristic
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4C2311yescharacteristic
56C5×C4⋊C4C2312yes—
57C5×C22⋊C4—21no—
58C42×C5C2311yescharacteristic
59C22⋊Dic5—21no—
60C22⋊Dic5—21no—
61C22⋊Dic5—21no—
62#685—21no—
63#685—22no—
64C4⋊Dic5—21no—
65C4⋊Dic5—21no—
66C4⋊Dic5—21no—
67C4⋊Dic5—22no—
68C4⋊Dic5C2311yescharacteristic
69C4⋊Dic5C2312yes—
70#683—21no—
71#683—21no—
72#683—22no—
73C4×Dic5—21no—
74C4×Dic5—22no—
75C4×Dic5D411yescharacteristic
76C4×Dic5D411yescharacteristic
77C4×Dic5C2311yescharacteristic
78C2×#168C2312yes—
79#533—51no—
80#529—51no—
81#499—52no—
82#488—52no—
83#485—52no—
84#482—52no—
85#477—51no—
86#476—51no—
87#475—51no—
88#474D511yescharacteristic, complemented
89#431—51no—
90C23×C5—21no—
91C22×D5C2×D411yescharacteristic
92C5×Q8—42no—
93C5×D4—21no—
94C5×D4—21no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—42no—
98C2×C4×C5C2×D411yescharacteristic
99C2×C4×C5C2×D411yescharacteristic
100C2×C4×C5C2411yescharacteristic
101C22⋊D5—42no—
102C2×Dic5—21no—
103C2×Dic5—22no—
104C2×Dic5—21no—
105C2×Dic5C2×D411yescharacteristic
106C2×Dic5C2×D411yescharacteristic
107C2×Dic5C2×D411yescharacteristic
108C4×D5—22no—
109C4×D5—21no—
110C4×D5—21no—
111C4×D5—41no—
112Dic10—21no—
113Dic10—21no—
114Dic10—42no—
115C5×C8—22no—
116#168—22no—
117C4⋊Q8—51no—
118C4⋊Q8C2×D511yescharacteristic
119#126—101no—
120#125—51no—
121#125—52no—
122#124—51no—
123#123—101no—
124C22⋊Q8—51no—
125C22⋊Q8—101no—
126C22⋊Q8—101no—
127#121—51no—
128C4×Q8—51no—
129C4×Q8—101no—
130C4×Q8—101no—
131C4×D4—101no—
132C4×D4C2×D511yescharacteristic
133#117—52no—
134#117—101no—
135C2×C42—51no—
136#107—52no—
137#107C2×D512yes—
138#106—52no—
139#106—52no—
140C4⋊C8—51no—
141C4⋊C8C2×D511yescharacteristic
142#103—52no—
143#103—52no—
144#102—52no—
145#102C2×D512yes—
146C22⋊C8—52no—
147C4×C8—52no—
148C22×C5—41no—
149C22×C5—41no—
150C22×C5—41no—
151C22×C5C22×D411yescharacteristic
152C2×D5—21no—
153C2×D5—21no—
154C2×D5—21no—
155C4×C5—21no—
156C4×C5—41no—
157C4×C5—42no—
158C4×C5#13512yes—
159C4×C52−1+411yescharacteristic
160C4×C52−1+411yescharacteristic
161Dic5—21no—
162Dic5—21no—
163Dic5—21no—
164Dic5—41no—
165Dic5—42no—
166Dic5—41no—
167C2×Q8—21no—
168C2×Q8—51no—
169C2×Q8—101no—
170C2×D4—101no—
171C2×D4C22×D511yescharacteristic
172C22×C4—21no—
173C22×C4—51no—
174C22×C4—51no—
175C22×C4—51no—
176C2×C8—52no—
177C2×C8C22×D512yes—
178C4⋊C4—21no—
179C4⋊C4—52no—
180C4⋊C4—51no—
181C4⋊C4—101no—
182C4⋊C4—101no—
183C4⋊C4—101no—
184C4⋊C4—102no—
185C4⋊C4—102no—
186C4⋊C4—101no—
187C4⋊C4—101no—
188C4⋊C4C22×D512yes—
189C4⋊C4C22×D511yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—102no—
192C22⋊C4—101no—
193C22⋊C4—101no—
194C22⋊C4—101no—
195C22⋊C4—101no—
196C42—51no—
197C42—51no—
198C42—51no—
199C42—102no—
200C42—101no—
201C42C22×D511yescharacteristic
202C2×C5—81no—
203C2×C5#53611yescharacteristic
204C2×C5C2×#13511yescharacteristic
205C2×C5#57511yescharacteristic
206D5—41no—
207C23—21no—
208C23—51no—
209Q8—42no—
210Q8—101no—
211Q8—101no—
212Q8—202no—
213D4—21no—
214D4—21no—
215D4—202no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—51no—
220C2×C4—51no—
221C2×C4—51no—
222C2×C4—101no—
223C2×C4—101no—
224C2×C4—102no—
225C2×C4—101no—
226C2×C4—101no—
227C2×C4—102no—
228C2×C4—201no—
229C2×C4D4×D511yescharacteristic
230C2×C4D4×D511yescharacteristic
231C2×C4C23×D511yescharacteristic
232C8—22no—
233C8—102no—
234C5#323311yescharacteristic, complemented, Sylow
235C22—41no—
236C22—41no—
237C22—41no—
238C22—101no—
239C22—101no—
240C22—101no—
241C22C2×D4×D511yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—101no—
246C4—101no—
247C4—101no—
248C4—201no—
249C4—202no—
250C4—201no—
251C4#405812yes—
252C4#414711yescharacteristic
253C4#415011yescharacteristic
254C2—81no—
255C2—201no—
256C2#6596211yescharacteristic
257C2C2×#405811yescharacteristic
258C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#474C2
socleC22×C5C22×D4
prime radical#4084C22
hypercenter#474D5
hypocenterC5#3233
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #66062C2semidirect
3. C5×#474C2semidirect
4. #65383C2semidirect
5. #65297C2semidirect
6. #65177C2semidirect
7. #65167C2semidirect
8. #65139C2semidirect
9. #65122C2semidirect
11. #65057C2semidirect
88. #474D5semidirect
234. C5#3233semidirect

Maximal subgroups

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

subgroupindex
#660622
#658912
C5×#4742
#653832
#652972
#651772
#651672
#651392
#651222
#650732
#650572
#32335