Finite Group Databasebeta

← #13104890

Subgroups — #13104890

Order 1600.

616 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 84 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×#87863C212yesmaximal, complemented
3C22⋊#6034C212yesmaximal, complemented
4#11716896C212yesmaximal, complemented
5C23×C5⋊D5C411yescharacteristic
6C22×C4×C52C411yescharacteristic
7C22×#1039C2211yescharacteristic
8C2×C4×C5⋊D5—24no—
9C2×C4×C5⋊D5C411yescharacteristic
10C2×C4×C5⋊D5C411yescharacteristic
11C2×C4×C5⋊D5C2211yescharacteristic
12C2×C4×C5⋊D5C2211yescharacteristic
13C2×#6034C2214yescomplemented
14#87863—24no—
15#65756—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×C52C2×C411yescharacteristic
25C2×#1039—22no—
26C2×#1039C2×C411yescharacteristic
27C2×#1039C2311yescharacteristic
28C4×C5⋊D5—22no—
29C4×C5⋊D5—22no—
30C4×C5⋊D5—22no—
31C4×C5⋊D5D414yes—
32#6034—24no—
33C22×C4×D5—52no—
34C22×C4×D5—102no—
35C2×#700—54no—
36C22⋊#170—54no—
37#4003—54no—
38C22×C52—21no—
39C22×C52—21no—
40C22×C52M1612yescomplemented
41C22×C52C22×C411yescharacteristic
42C2×C5⋊D5—28no—
43C2×C5⋊D5—21no—
44C2×C5⋊D5—21no—
45C2×C5⋊D5—21no—
46C2×C5⋊D5—21no—
47C2×C5⋊D5C22⋊C412yes—
48C2×C5⋊D5M1612yescomplemented
49C4×C52—21no—
50C4×C52C22⋊C412yes—
51#1039—21no—
52#1039C2×D412yes—
53C23×D5—52no—
54C23×D5—102no—
55C22×C4×C5—22no—
56C22×C4×C5AGL(1,𝔽5)12yes—
57C22×Dic5—52no—
58C22×Dic5—102no—
59C2×C4×D5—52no—
60C2×C4×D5—52no—
61C2×C4×D5—52no—
62C2×C4×D5—52no—
63C2×C4×D5—102no—
64C2×C4×D5—102no—
65C2×C4×D5—102no—
66C2×C4×D5—102no—
67C2×C4×D5—108no—
68C2×C4×D5—1016no—
69C2×#170—58no—
70#700—108no—
71#407—251noSylow
72C2×C52—22no—
73C2×C52C2×C22⋊C411yescharacteristic
74C2×C52C2×M1611yescharacteristic
75C2×C52C2×M1611yescharacteristic
76C5⋊D5—22no—
77C5⋊D5—22no—
78C23×C5—22no—
79C23×C5C2×AGL(1,𝔽5)12yes—
80C22×D5—52no—
81C22×D5—52no—
82C22×D5—102no—
83C22×D5—102no—
84C22×D5—104no—
85C22×D5—104no—
86C22×D5—108no—
87C22×D5—108no—
88C22×D5—108no—
89C22×D5—104no—
90C2×C4×C5—28no—
91C2×C4×C5—24no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5C2×AGL(1,𝔽5)12yes—
95C2×C4×C5C2×AGL(1,𝔽5)12yes—
96C2×Dic5—52no—
97C2×Dic5—52no—
98C2×Dic5—102no—
99C2×Dic5—102no—
100C2×Dic5—104no—
101C2×Dic5—108no—
102C4×D5—58no—
103C4×D5—108no—
104C4×D5—104no—
105C4×D5—104no—
106C4×D5—108no—
107C4×D5—108no—
108C4×D5—108no—
109C4×D5—104no—
110#170—108no—
111C23×C4—251no—
112C2×M16—252no—
113C22⋊C8—252no—
114C22⋊C8—252no—
115C52#40711yescharacteristic, complemented, Sylow
116C22×C5—24no—
117C22×C5—24no—
118C22×C5—22no—
119C22×C5—22no—
120C22×C5—24no—
121C22×C5—22no—
122C22×C5#70014yescomplemented
123C22×C5C22×AGL(1,𝔽5)12yes—
124C2×D5—54no—
125C2×D5—54no—
126C2×D5—104no—
127C2×D5—104no—
128C2×D5—104no—
129C2×D5—1032no—
130C2×D5—1016no—
131C2×D5—104no—
132C2×D5—104no—
133C2×D5—104no—
134C2×D5—102no—
135C2×D5—102no—
136C2×D5—102no—
137C2×D5—102no—
138C4×C5—24no—
139C4×C5—22no—
140C4×C5—24no—
141C4×C5C22⋊AGL(1,𝔽5)14yes—
142Dic5—54no—
143Dic5—104no—
144Dic5—104no—
145Dic5—102no—
146C24—251no—
147C22×C4—251no—
148C22×C4—251no—
149C22×C4—251no—
150C22×C4—251no—
151C22×C4—251no—
152C22×C4—504no—
153C22×C4#104411yescharacteristic
154M16—504no—
155C2×C8—254no—
156C2×C5—28no—
157C2×C5—24no—
158C2×C5—22no—
159C2×C5—22no—
160C2×C5—22no—
161C2×C5C2×#70012yes—
162C2×C5D5⋊M1612yes—
163C2×C5C2×C22⋊AGL(1,𝔽5)12yes—
164D5—108no—
165D5—108no—
166D5—104no—
167D5—104no—
168C23—251no—
169C23—251no—
170C23—502no—
171C23—502no—
172C23—502no—
173C23C2×#104411yescharacteristic
174C2×C4—22no—
175C2×C4—251no—
176C2×C4—254no—
177C2×C4—251no—
178C2×C4—502no—
179C2×C4—502no—
180C2×C4—502no—
181C2×C4—502no—
182C2×C4C2×#104411yescharacteristic
183C2×C4C2×#104411yescharacteristic
184C8—504no—
185C5—22no—
186C5#6575612yescomplemented
187C22—21no—
188C22—21no—
189C22—252no—
190C22—252no—
191C22—508no—
192C22—501no—
193C22—501no—
194C22—501no—
195C22—501no—
196C22#8786312yescomplemented
197C22C22×#104411yescharacteristic
198C4—21no—
199C4—252no—
200C4—501no—
201C4C22⋊#104412yes—
202C2—22no—
203C2—502no—
204C2—502no—
205C2C2×#8786311yescharacteristic
206C2#1171744811yescharacteristic
207C2C2×C22⋊#104411yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1044
commutator, socleC22×C52C22×C4
FittingC22×C4×C52C4
prime radicalC23×C5⋊D5C4
cosocleC2×#1039C23
hypercenterC22×C4#1044
hypocenterC52#407
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#87863C2semidirect
3. C22⋊#6034C2semidirect
4. #11716896C2semidirect
13. C2×#6034C22semidirect
40. C22×C52M16semidirect
48. C2×C5⋊D5M16semidirect
115. C52#407semidirect
122. C22×C5#700semidirect
186. C5#65756semidirect
196. C22#87863semidirect

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×#878632
C22⋊#60342
#117168962
#657565