Finite Group Databasebeta

← #10613300

Subgroups — #10613300

Order 640.

394 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 117 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#66050C211yesmaximal, characteristic
3C5×#486C211yesmaximal, characteristic, complemented
4#65382C211yesmaximal, characteristic, complemented
5#65365C211yesmaximal, characteristic, complemented
6#65072C214yesmaximal, complemented
7#65056C214yesmaximal, complemented
8#64988C211yesmaximal, characteristic, complemented
9#64974C211yesmaximal, characteristic, complemented
10C2×#711C2212yes—
11C5×C4⋊Q8C2211yescharacteristic
12C5×#127C2211yescharacteristic
13#4092—21no—
14Q8×Dic5—21no—
15#4084—21no—
16#4084—21no—
17#4082C2212yes—
18D4×Dic5—21no—
19D4×Dic5—21no—
20C2×#687C2214yes—
21C2×Dic20C2214yes—
22#4043C2212yes—
23#4038—21no—
24#4036—21no—
25#4033C2212yes—
26C42×D5C2211yescharacteristic
27#4015C2211yescharacteristic
28#4015C2211yescharacteristic
29C5×#102C2214yes—
30C4×C5×C8C2211yescharacteristic
31#3952C2212yes—
32#3946C2212yes—
33#3942C2214yes—
34C4×#168C2211yescharacteristic
35#3210—51nomaximal, Sylow
36C2×C5×Q8—21no—
37C2×C5×D4—21no—
38C2×C5×D4—21no—
39C2×C5×D4C2312yes—
40C2×C22⋊D5—22no—
41C22×Dic5—22no—
42#711—44no—
43C2×C4×D5D411yescharacteristic
44C2×C4×D5D411yescharacteristic
45C2×C4×D5C2311yescharacteristic
46C2×Dic10C2312yes—
47C2×Dic10C2312yes—
48C2×C5×C8C2312yes—
49C5×C4⋊C4—21no—
50C5×C4⋊C4C2312yes—
51C42×C5C2311yescharacteristic
52C22⋊Dic5—22no—
53C22⋊Dic5—22no—
54#687—44no—
55#685—22no—
56C4⋊Dic5—21no—
57C4⋊Dic5—21no—
58C4⋊Dic5—21no—
59C4⋊Dic5—21no—
60C4⋊Dic5—22no—
61#683—22no—
62C4×Dic5—22no—
63C4×Dic5D411yescharacteristic
64C4×Dic5D411yescharacteristic
65C4×Dic5C2311yescharacteristic
66C2×#168C2312yes—
67Dic20—44no—
68#533—51no—
69#532—51no—
70#494—51no—
71#492—51no—
72#487—51no—
73#486D511yescharacteristic, complemented
74#464—54no—
75#452—54no—
76#431—51no—
77C23×C5—22no—
78C22×D5C2×D411yescharacteristic
79C5×Q8—42no—
80C5×D4—24no—
81C5×D4—42no—
82C5×D4—42no—
83C2×C4×C5—22no—
84C2×C4×C5C2×D411yescharacteristic
85C2×C4×C5C2×D411yescharacteristic
86C2×C4×C5C2411yescharacteristic
87C22⋊D5—44no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—44no—
91C2×Dic5C2×D411yescharacteristic
92C2×Dic5C2×D411yescharacteristic
93C2×Dic5C2×D411yescharacteristic
94C4×D5—22no—
95C4×D5—22no—
96C4×D5—22no—
97Dic10—24no—
98Dic10—24no—
99C5×C8—22no—
100#168—22no—
101C2×#41—52no—
102C2×Q16—54no—
103C2×SD16—54no—
104C4⋊Q8—51no—
105C4⋊Q8—51no—
106C4⋊Q8C2×D511yescharacteristic
107#127C2×D511yescharacteristic
108#125—101no—
109#124—52no—
110C22⋊Q8—52no—
111C22⋊Q8—101no—
112#121—101no—
113#121—101no—
114C4×Q8—52no—
115C4×Q8—101no—
116C4×D4—101no—
117C4×D4—101no—
118#117—101no—
119C2×C42—51no—
120C4⋊C8—52no—
121#103—54no—
122#102C2×D514yes—
123C22⋊C8—52no—
124C4×C8—51no—
125C4×C8C2×D511yescharacteristic
126C22×C5—42no—
127C22×C5—42no—
128C22×C5—42no—
129C22×C5C22×D411yescharacteristic
130C2×D5—21no—
131C2×D5—21no—
132C2×D5—21no—
133C4×C5—42no—
134C4×C5#13512yes—
135C4×C5#13712yes—
136C4×C5H(5,𝔽2)12yes—
137Dic5—21no—
138Dic5—21no—
139Dic5—21no—
140Dic5—42no—
141Dic5—42no—
142#41—204no—
143C2×Q8—21no—
144C2×Q8—52no—
145C2×Q8—52no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—102no—
149C2×D4C22×D512yes—
150C22×C4—51no—
151C22×C4—51no—
152C22×C4—51no—
153C22×C4—102no—
154Q16—204no—
155SD16—204no—
156C2×C8—52no—
157C2×C8C22×D512yes—
158C4⋊C4—21no—
159C4⋊C4—102no—
160C4⋊C4—102no—
161C4⋊C4—101no—
162C4⋊C4—101no—
163C4⋊C4—101no—
164C4⋊C4—101no—
165C4⋊C4C22×D512yes—
166C22⋊C4—102no—
167C22⋊C4—102no—
168C22⋊C4—102no—
169C42—51no—
170C42—51no—
171C42—51no—
172C42—102no—
173C42C22×D511yescharacteristic
174C2×C5—82no—
175C2×C5#53511yescharacteristic
176C2×C5C2×#13511yescharacteristic
177C2×C5C2×#13711yescharacteristic
178D5—41no—
179C23—22no—
180C23—51no—
181Q8—42no—
182Q8—104no—
183Q8—104no—
184D4—24no—
185D4—42no—
186D4—42no—
187D4—204no—
188C2×C4—22no—
189C2×C4—51no—
190C2×C4—51no—
191C2×C4—51no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—204no—
198C2×C4D4×D511yescharacteristic
199C2×C4D4×D511yescharacteristic
200C2×C4C23×D511yescharacteristic
201C8—22no—
202C8—102no—
203C5#321011yescharacteristic, complemented, Sylow
204C22—42no—
205C22—42no—
206C22—42no—
207C22—101no—
208C22—101no—
209C22—101no—
210C22C2×D4×D511yescharacteristic
211C4—42no—
212C4—101no—
213C4—101no—
214C4—101no—
215C4—202no—
216C4—202no—
217C4#405812yes—
218C4#406112yes—
219C4#414912yes—
220C2—82no—
221C2—201no—
222C2#6600811yescharacteristic
223C2C2×#405811yescharacteristic
224C2C2×#406111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#486C2
socleC22×C5C22×D4
prime radical#66050C2
hypercenter#486D5
hypocenterC5#3210
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #66062C2semidirect
3. C5×#486C2semidirect
4. #65382C2semidirect
5. #65365C2semidirect
6. #65072C2semidirect
7. #65056C2semidirect
8. #64988C2semidirect
9. #64974C2semidirect
73. #486D5semidirect
203. C5#3210semidirect

Maximal subgroups

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

subgroupindex
#660622
#660502
C5×#4862
#653822
#653652
#650722
#650562
#649882
#649742
#32105