Finite Group Databasebeta

← #10614803

Subgroups — #10614803

Order 640.

556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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×C5×#125C211yesmaximal, characteristic, complemented
2C2×C42×D5C211yesmaximal, characteristic
3#65278C214yesmaximal
4#65276C212yesmaximal
5#65267C214yesmaximal, complemented
6#65262C212yesmaximal, complemented
7#65225C211yesmaximal, characteristic, complemented
8C22×C4×D5C2212yes—
9C22×C4×D5C2211yescharacteristic
10C5×#125—24no—
11C2×C5×C4⋊C4C2212yes—
12C2×C5×C4⋊C4C2214yes—
13C2×C42×C5C2211yescharacteristic
14C2×#685C2214yes—
15C2×C4⋊Dic5C2212yes—
16C2×#683C2214yes—
17C2×C4×Dic5C2211yescharacteristic
18C2×C4×Dic5C2212yes—
19C42×D5—24no—
20#3963C2214yes—
21#3963C2214yes—
22#3963C2214yes—
23#2851—51nomaximal, Sylow
24C23×D5C2311yescharacteristic
25C22×C4×C5C2314yes—
26C22×C4×C5C2312yes—
27C22×C4×C5C2311yescharacteristic
28C22×Dic5C2314yes—
29C22×Dic5C2311yescharacteristic
30C22×Dic5C2312yes—
31C2×C4×D5—24no—
32C2×C4×D5—24no—
33C2×C4×D5—22no—
34C2×C4×D5—22no—
35C2×C4×D5—24no—
36C2×C4×D5Q814yes—
37C5×C4⋊C4—28no—
38C5×C4⋊C4—24no—
39C42×C5D414yes—
40#685—28no—
41C4⋊Dic5—24no—
42#683—28no—
43C4×Dic5—24no—
44C4×Dic5—22no—
45C2×#125D511yescharacteristic, complemented
46C22×C42—51no—
47#389—52no—
48#387—54no—
49#385—54no—
50#382—51no—
51#380—52no—
52C23×C5C2411yescharacteristic
53C22×D5—21no—
54C22×D5—21no—
55C22×D5—22no—
56C22×D5C2×Q812yes—
57C22×D5#4114yes—
58C2×C4×C5—22no—
59C2×C4×C5—22no—
60C2×C4×C5—24no—
61C2×C4×C5—24no—
62C2×C4×C5—24no—
63C2×C4×C5C2×D414yes—
64C2×C4×C5C2×D412yes—
65C2×C4×C5C2×Q812yes—
66C2×C4×C5#4112yes—
67C2×Dic5—21no—
68C2×Dic5—22no—
69C2×Dic5—21no—
70C2×Dic5—22no—
71C2×Dic5—24no—
72C2×Dic5—24no—
73C2×Dic5—24no—
74C2×Dic5C2×Q812yes—
75C2×Dic5#4114yes—
76C4×D5—216no—
77C4×D5—28no—
78C23×C4—51no—
79C23×C4—52no—
80#125—24no—
81C2×C4⋊C4—52no—
82C2×C4⋊C4—54no—
83C2×C4⋊C4C2×D512yes—
84C2×C4⋊C4C2×D514yes—
85C2×C22⋊C4—54no—
86C2×C42—52no—
87C2×C42—51no—
88C2×C42—104no—
89C2×C42C2×D511yescharacteristic
90#95—54no—
91#95—54no—
92#95—54no—
93C22×C5C22×D411yescharacteristic
94C22×C5C22×Q811yescharacteristic
95C22×C5C2×#4111yescharacteristic
96C22×C5C2×#4112yes—
97C22×C5C2×#4112yes—
98C2×D5—22no—
99C2×D5—24no—
100C2×D5—24no—
101C2×D5—22no—
102C2×D5#12514yescomplemented
103C4×C5—24no—
104C4×C5—44no—
105C4×C5C22⋊Q814yes—
106Dic5—22no—
107Dic5—24no—
108Dic5—44no—
109C24—51no—
110C22×C4—51no—
111C22×C4—52no—
112C22×C4—54no—
113C22×C4—54no—
114C22×C4—104no—
115C22×C4—104no—
116C22×C4—102no—
117C22×C4—104no—
118C22×C4—102no—
119C22×C4C22×D514yes—
120C22×C4C22×D512yes—
121C22×C4C22×D511yescharacteristic
122C4⋊C4—28no—
123C4⋊C4—24no—
124C4⋊C4—108no—
125C4⋊C4—104no—
126C22⋊C4—108no—
127C42—104no—
128C42—102no—
129C42C22⋊D514yes—
130C2×C5C2×C22⋊Q811yescharacteristic
131C2×C5#52512yes—
132C2×C5C2×#12511yescharacteristic
133C2×C5#52912yes—
134C2×C5#53311yescharacteristic
135D5—24no—
136C23—52no—
137C23—54no—
138C23—101no—
139C23—102no—
140C23—101no—
141C23C23×D511yescharacteristic
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—52no—
148C2×C4—54no—
149C2×C4—1016no—
150C2×C4—101no—
151C2×C4—102no—
152C2×C4—101no—
153C2×C4—102no—
154C2×C4—108no—
155C2×C4—104no—
156C2×C4—104no—
157C2×C4—104no—
158C2×C4Q8×D512yes—
159C2×C4#71312yes—
160C2×C4C2×C22⋊D514yes—
161C2×C4C2×C22⋊D512yes—
162C5#285111yescharacteristic, complemented, Sylow
163C22—54no—
164C22—102no—
165C22—104no—
166C22—102no—
167C22—104no—
168C22C2×Q8×D511yescharacteristic
169C22C2×#71311yescharacteristic
170C22D5×#4112yes—
171C22D5×#4112yes—
172C22C22×C22⋊D511yescharacteristic
173C4—24no—
174C4—44no—
175C4—102no—
176C4—104no—
177C4—204no—
178C4#409214yes—
179C2—104no—
180C2D5×#12511yescharacteristic
181C2#6602211yescharacteristic
182C2#6602512yes—
183C2C2×#409211yescharacteristic
184C2#6616212yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#125C2
prime radicalC23×D5C23
hypercenterC2×#125D5
hypocenterC5#2851
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C5×#125C2semidirect
5. #65267C2semidirect
6. #65262C2semidirect
7. #65225C2semidirect
45. C2×#125D5semidirect
102. C2×D5#125semidirect
162. C5#2851semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1252
C2×C42×D52
#652782
#652762
#652672
#652622
#652252
#28515