Finite Group Databasebeta

← #10612770

Subgroups — #10612770

Order 640.

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

normal subgroups only

This group has 370 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
1#66061C211yesmaximal, characteristic
2Q8×Dic10C211yesmaximal, characteristic, complemented
3C5×#475C211yesmaximal, characteristic, complemented
4#65368C211yesmaximal, characteristic, complemented
5C4×#688C211yesmaximal, characteristic
6#65159C212yesmaximal
7#65144C212yesmaximal, complemented
8#65131C211yesmaximal, characteristic
9#65119C211yesmaximal, characteristic, complemented
10#65080C212yesmaximal, complemented
11#65077C212yesmaximal
12C2×Q8×D5—21no—
13C5×C4⋊Q8C2211yescharacteristic
14C4×C5×Q8C2211yescharacteristic
15#4092—21no—
16Q8×Dic5C2212yes—
17C2×#688C2211yescharacteristic
18#4042—22no—
19#4039C2212yes—
20#4034—22no—
21#4034C2212yes—
22#4034C2212yes—
23C4×D20C2211yescharacteristic
24#4015—21no—
25#4015C2211yescharacteristic
26C4×Dic10—21no—
27C4×Dic10C2211yescharacteristic
28C5×#106C2212yes—
29C5×C4⋊C8C2211yescharacteristic
30C5×#103C2212yes—
31#3967C2211yescharacteristic
32#3953C2212yes—
33#3949C2212yes—
34#3946C2212yes—
35C8×Dic5C2212yes—
36#3941C2211yescharacteristic
37#3940C2211yescharacteristic
38#3940C2212yes—
39#3940C2212yes—
40C4×#168C2211yescharacteristic
41Q8×SD16—51nomaximal, Sylow
42C2×C5×Q8—21no—
43C2×C5×Q8C2311yescharacteristic
44Q8×D5—44no—
45C2×D20C2311yescharacteristic
46C2×C4×D5—21no—
47C2×C4×D5—22no—
48C2×Dic10—22no—
49C2×Dic10—21no—
50C2×Dic10—22no—
51C2×Dic10D411yescharacteristic
52C2×C5×C8C2312yes—
53C5×C4⋊C4—21no—
54C5×C4⋊C4—21no—
55C5×C4⋊C4C2311yescharacteristic
56C5×C4⋊C4C2312yes—
57C42×C5—21no—
58C42×C5C2311yescharacteristic
59#688Q814yescomplemented
60#685—21no—
61#685—22no—
62C4⋊Dic5—21no—
63C4⋊Dic5—22no—
64C4⋊Dic5D411yescharacteristic
65C4⋊Dic5C2312yes—
66C4⋊Dic5C2311yescharacteristic
67#683—22no—
68#683—22no—
69#683D412yes—
70C4×Dic5—22no—
71C4×Dic5C2312yes—
72C2×#168C2311yescharacteristic
73C2×#168C2311yescharacteristic
74Q82—51no—
75D4×Q8—51no—
76#498—51no—
77#498—52no—
78#476—52no—
79#476—51no—
80#475—52no—
81#475D511yescharacteristic, complemented
82C8×Q8—51no—
83C4×SD16—51no—
84C4×SD16—52no—
85C22×D5—21no—
86C5×Q8—21no—
87C5×Q8—42no—
88C5×Q8C2×Q812yescomplemented
89C2×C4×C5—21no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5C2×D411yescharacteristic
93C2×C4×C5C2×D411yescharacteristic
94C2×C4×C5C2411yescharacteristic
95C2×Dic5—22no—
96C2×Dic5—21no—
97C2×Dic5C2×D412yes—
98C2×Dic5C2×D412yes—
99D20—21no—
100D20C2×Q812yes—
101C4×D5—42no—
102C4×D5—44no—
103Dic10—44no—
104Dic10—82no—
105Dic10—81no—
106Dic10SD1614yescomplemented
107C5×C8—22no—
108#168—21no—
109#168C2×Q812yescomplemented
110C22×Q8—101no—
111C2×SD16—51no—
112C4⋊Q8—52no—
113C4⋊Q8—52no—
114C4⋊Q8—51no—
115C4⋊Q8—101no—
116C4⋊Q8—102no—
117C4⋊Q8C2×D511yescharacteristic
118C22⋊Q8—102no—
119C22⋊Q8—101no—
120C4×Q8—51no—
121C4×Q8—52no—
122C4×Q8—101no—
123C4×Q8C2×D511yescharacteristic
124C4×D4—52no—
125C4×D4—51no—
126#106—52no—
127#106—52no—
128#106—51no—
129#106—52no—
130#106C2×D512yes—
131C4⋊C8—52no—
132C4⋊C8C2×D511yescharacteristic
133#103—51no—
134#103C2×D512yes—
135#102—52no—
136#102—51no—
137C4×C8—52no—
138C4×C8—51no—
139C22×C5C22×D411yescharacteristic
140C2×D5—22no—
141C2×D5—41no—
142C2×D5—41no—
143C4×C5—22no—
144C4×C5—21no—
145C4×C5—42no—
146C4×C5—41no—
147C4×C5C2×SD1612yes—
148C4×C5C22×Q811yescharacteristic
149C4×C52−1+411yescharacteristic
150Dic5—22no—
151Dic5—41no—
152Dic5—42no—
153Dic5C2×SD1614yes—
154C2×Q8—21no—
155C2×Q8—51no—
156C2×Q8—102no—
157C2×Q8—102no—
158C2×Q8—101no—
159C2×Q8—204no—
160C2×Q8C22×D511yescharacteristic
161C2×D4—51no—
162C22×C4—102no—
163C22×C4—101no—
164SD16—54no—
165C2×C8—51no—
166C2×C8—51no—
167C2×C8C22×D512yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—52no—
171C4⋊C4—51no—
172C4⋊C4—52no—
173C4⋊C4—51no—
174C4⋊C4—102no—
175C4⋊C4—102no—
176C4⋊C4—102no—
177C4⋊C4—101no—
178C4⋊C4C22×D511yescharacteristic
179C4⋊C4C22×D512yes—
180C22⋊C4—102no—
181C22⋊C4—101no—
182C42—21no—
183C42—52no—
184C42—102no—
185C42C22×D511yescharacteristic
186C2×C5D4×Q811yescharacteristic
187C2×C5C22×SD1611yescharacteristic
188C2×C5#57711yescharacteristic
189D5—42no—
190C23—101no—
191Q8—21no—
192Q8—42no—
193Q8—54no—
194Q8—204no—
195Q8—401no—
196Q8—402no—
197Q8Q8×D512yescomplemented
198D4—52no—
199D4—101no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—52no—
204C2×C4—52no—
205C2×C4—102no—
206C2×C4—101no—
207C2×C4—204no—
208C2×C4—202no—
209C2×C4D4×D511yescharacteristic
210C2×C4D4×D511yescharacteristic
211C2×C4C23×D511yescharacteristic
212C8—22no—
213C8—52no—
214C8—101no—
215C5Q8×SD1611yescharacteristic, complemented, Sylow
216C22—102no—
217C22—201no—
218C22—201no—
219C22C2×D4×D511yescharacteristic
220C4—21no—
221C4—22no—
222C4—42no—
223C4—41no—
224C4—54no—
225C4—102no—
226C4—201no—
227C4—202no—
228C4D5×SD1612yes—
229C4C2×Q8×D511yescharacteristic
230C4#415011yescharacteristic
231C2—202no—
232C2#6596611yescharacteristic
233C2C2×D5×SD1611yescharacteristic
234C2#6611211yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#475C2
socleC22×C5C22×D4
prime radicalC2×D20C23
hypercenter#475D5
hypocenterC5Q8×SD16
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. Q8×Dic10C2semidirect
3. C5×#475C2semidirect
4. #65368C2semidirect
7. #65144C2semidirect
9. #65119C2semidirect
10. #65080C2semidirect
59. #688Q8semidirect
81. #475D5semidirect
88. C5×Q8C2×Q8semidirect
106. Dic10SD16semidirect
109. #168C2×Q8semidirect
197. Q8Q8×D5semidirect
215. C5Q8×SD16semidirect

Maximal subgroups

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

subgroupindex
#660612
Q8×Dic102
C5×#4752
#653682
C4×#6882
#651592
#651442
#651312
#651192
#650802
#650772
Q8×SD165