Finite Group Databasebeta

← #13101502

Subgroups — #13101502

Order 1600.

616 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C22×C4×C5⋊D5C211yesmaximal, characteristic
2C2×#87769C212yesmaximal, complemented
3C5×C22⋊#168C212yesmaximal, complemented
4#11716598C212yesmaximal, complemented
5C23×C5⋊D5C411yescharacteristic
6C22×C4×C52C2211yescharacteristic
7C22×#1039C411yescharacteristic
8C2×C4×C5⋊D5—24no—
9C2×C4×C5⋊D5C411yescharacteristic
10C2×C4×C5⋊D5C411yescharacteristic
11C2×C4×C5⋊D5C2211yescharacteristic
12C2×C4×C5⋊D5C2211yescharacteristic
13C2×C5×#168C2214yescomplemented
14#87769—24no—
15#65015—52nomaximal
16C23×C52C2×C411yescharacteristic
17C22×C5⋊D5—22no—
18C22×C5⋊D5—22no—
19C22×C5⋊D5—22no—
20C22×C5⋊D5C2×C411yescharacteristic
21C22×C5⋊D5C2×C411yescharacteristic
22C2×C4×C52—22no—
23C2×C4×C52C2×C411yescharacteristic
24C2×C4×C52C2311yescharacteristic
25C2×#1039—22no—
26C2×#1039C2×C411yescharacteristic
27C2×#1039C2×C411yescharacteristic
28C4×C5⋊D5—22no—
29C4×C5⋊D5—22no—
30C4×C5⋊D5—22no—
31C4×C5⋊D5D414yes—
32C5×#168—24no—
33C22×C4×D5—52no—
34C22×C4×D5—102no—
35C2×#676—54no—
36C5×C22⋊C8—52no—
37C22⋊#168D512yescomplemented
38#3952—54no—
39C22×C52—21no—
40C22×C52—21no—
41C22×C52M1612yescomplemented
42C22×C52C22×C411yescharacteristic
43C2×C5⋊D5—28no—
44C2×C5⋊D5—21no—
45C2×C5⋊D5—21no—
46C2×C5⋊D5—21no—
47C2×C5⋊D5—21no—
48C2×C5⋊D5C22⋊C412yes—
49C2×C5⋊D5M1612yescomplemented
50C4×C52—21no—
51C4×C52C2×D412yes—
52#1039—21no—
53#1039C22⋊C412yes—
54C23×D5—52no—
55C23×D5—102no—
56C22×C4×C5—22no—
57C22×C4×C5C2×D512yes—
58C22×Dic5—52no—
59C22×Dic5—102no—
60C2×C4×D5—52no—
61C2×C4×D5—52no—
62C2×C4×D5—52no—
63C2×C4×D5—52no—
64C2×C4×D5—102no—
65C2×C4×D5—102no—
66C2×C4×D5—102no—
67C2×C4×D5—102no—
68C2×C4×D5—108no—
69C2×C4×D5—1016no—
70C2×C5×C8—54no—
71C2×#168C2×D514yescomplemented
72#676—108no—
73#407—251noSylow
74C2×C52—22no—
75C2×C52C2×C22⋊C411yescharacteristic
76C2×C52C2×M1611yescharacteristic
77C2×C52C2×M1611yescharacteristic
78C5⋊D5—22no—
79C5⋊D5—22no—
80C23×C5—22no—
81C23×C5C4×D512yes—
82C22×D5—52no—
83C22×D5—52no—
84C22×D5—102no—
85C22×D5—102no—
86C22×D5—104no—
87C22×D5—104no—
88C22×D5—108no—
89C22×D5—108no—
90C22×D5—108no—
91C22×D5—104no—
92C2×C4×C5—28no—
93C2×C4×C5—24no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5C4×D512yes—
97C2×C4×C5C22×D512yes—
98C2×Dic5—52no—
99C2×Dic5—52no—
100C2×Dic5—102no—
101C2×Dic5—102no—
102C2×Dic5—104no—
103C2×Dic5—108no—
104C4×D5—58no—
105C4×D5—108no—
106C4×D5—104no—
107C4×D5—104no—
108C4×D5—108no—
109C4×D5—108no—
110C4×D5—108no—
111C4×D5—104no—
112C5×C8—104no—
113#168—24no—
114C23×C4—251no—
115C2×M16—252no—
116C22⋊C8—52no—
117C22⋊C8—252no—
118C52#40711yescharacteristic, complemented, Sylow
119C22×C5—24no—
120C22×C5—24no—
121C22×C5—22no—
122C22×C5—22no—
123C22×C5—22no—
124C22×C5—24no—
125C22×C5#67614yescomplemented
126C22×C5C2×C4×D512yes—
127C2×D5—54no—
128C2×D5—54no—
129C2×D5—104no—
130C2×D5—104no—
131C2×D5—104no—
132C2×D5—1032no—
133C2×D5—1016no—
134C2×D5—104no—
135C2×D5—104no—
136C2×D5—104no—
137C2×D5—102no—
138C2×D5—102no—
139C2×D5—102no—
140C2×D5—102no—
141C4×C5—24no—
142C4×C5—22no—
143C4×C5—24no—
144C4×C5D4×D514yes—
145Dic5—54no—
146Dic5—104no—
147Dic5—104no—
148Dic5—102no—
149C24—251no—
150C22×C4—251no—
151C22×C4—251no—
152C22×C4—251no—
153C22×C4—251no—
154C22×C4—251no—
155C22×C4—504no—
156C22×C4D5211yescharacteristic
157M16—504no—
158C2×C8—54no—
159C2×C5—28no—
160C2×C5—24no—
161C2×C5—22no—
162C2×C5—22no—
163C2×C5—22no—
164C2×C5D5×C22⋊C412yes—
165C2×C5C2×#67612yes—
166C2×C5D5×M1612yes—
167D5—108no—
168D5—108no—
169D5—104no—
170D5—104no—
171C23—251no—
172C23—251no—
173C23—502no—
174C23—502no—
175C23—502no—
176C23#603611yescharacteristic
177C2×C4—22no—
178C2×C4—251no—
179C2×C4—254no—
180C2×C4—251no—
181C2×C4—502no—
182C2×C4—502no—
183C2×C4—502no—
184C2×C4—502no—
185C2×C4#603611yescharacteristic
186C2×C4C2×D5211yescharacteristic
187C8—104no—
188C5—22no—
189C5#6501512yescomplemented
190C22—21no—
191C22—21no—
192C22—252no—
193C22—252no—
194C22—508no—
195C22—501no—
196C22—501no—
197C22—501no—
198C22—501no—
199C22#8776912yescomplemented
200C22C2×#603611yescharacteristic
201C4—21no—
202C4—252no—
203C4—501no—
204C4#8788612yes—
205C2—22no—
206C2—502no—
207C2—502no—
208C2#1171693611yescharacteristic
209C2C2×#8776911yescharacteristic
210C2#1171709211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D52
commutator, socleC22×C52C22×C4
FittingC22×C4×C52C22
prime radicalC23×C5⋊D5C4
cosocleC2×C4×C52C23
hypercenterC22×C4D52
hypocenterC52#407
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#87769C2semidirect
3. C5×C22⋊#168C2semidirect
4. #11716598C2semidirect
13. C2×C5×#168C22semidirect
37. C22⋊#168D5semidirect
41. C22×C52M16semidirect
49. C2×C5⋊D5M16semidirect
71. C2×#168C2×D5semidirect
118. C52#407semidirect
125. C22×C5#676semidirect
189. C5#65015semidirect
199. C22#87769semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5⋊D52
C2×#877692
C5×C22⋊#1682
#117165982
#650155