Finite Group Databasebeta

← C22×C22⋊#1042 #13107799

Subgroups — C22×C22⋊#1042 #13107799

Order 1600.

1415 conjugacy classes of subgroups; only the 341 normal subgroups are shown.

normal subgroups only

This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 341 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
1C24×C5×D5C211yesmaximal, characteristic
2C23×#1042C212yesmaximal, complemented
3C2×C22⋊#1042C2112yesmaximal, direct product factor
4C24×C52C411yescharacteristic, complemented
5C23×C5×D5—28no—
6C23×C5×D5C416yescomplemented
7C23×C5×D5C411yescharacteristic, complemented
8C23×C5×D5C2216yes—
9C23×C5×D5C2211yescharacteristic
10C22×#1042—28no—
11C22×#1042C22112yescomplemented
12C22⋊#1042C22116yesdirect product factor
13C22×C22⋊AGL(1,𝔽5)—51nomaximal
14C22×C22⋊Dic5—51nomaximal
15C23×C52—24no—
16C23×C52C2×C411yescharacteristic
17C23×C52C2×C416yescomplemented
18C22×C5×D5—24no—
19C22×C5×D5—224no—
20C22×C5×D5—224no—
21C22×C5×D5C2×C413yes—
22C22×C5×D5C2×C4112yescomplemented
23C22×C5×D5C2×C416yescomplemented
24C22×C5×D5D418yes—
25C22×C5×D5C2314yes—
26C22×C5×D5C2313yes—
27C2×#1042—224no—
28C2×#1042C2318yescomplemented
29C25×C5—51no—
30C24×D5D511yescharacteristic
31C23×AGL(1,𝔽5)—52no—
32C23×Dic5—52no—
33C2×C22⋊AGL(1,𝔽5)—512no—
34C2×C22⋊Dic5—512no—
35C22×C52—212no—
36C22×C52C22⋊C414yescomplemented
37C22×C52C22×C413yes—
38C22×C52C22×C414yescomplemented
39C2×C5×D5—224no—
40C2×C5×D5—28no—
41C2×C5×D5—212no—
42C2×C5×D5C22⋊C4112yescomplemented
43C2×C5×D5C22×C413yes—
44C2×C5×D5C22×C414yescomplemented
45C2×C5×D5C2×D4112yes—
46C2×C5×D5C2411yescharacteristic
47#1042—28no—
48C24×C5—41no—
49C24×C5—51no—
50C24×C5—51no—
51C24×C5—56no—
52C24×C5—56no—
53C24×C5—108no—
54C24×C5Dic511yescharacteristic, complemented
55C24×C5AGL(1,𝔽5)11yescharacteristic, complemented
56C23×D5—28no—
57C23×D5Dic516yescomplemented
58C23×D5Dic511yescharacteristic, complemented
59C23×D5C2×D516yes—
60C23×D5C2×D511yescharacteristic
61C22×AGL(1,𝔽5)—512no—
62C22×AGL(1,𝔽5)—108no—
63C22×Dic5—512no—
64C22×Dic5—108no—
65C22⋊AGL(1,𝔽5)—516no—
66C22⋊Dic5—516no—
67C22×C22⋊C4—251noSylow
68C2×C52—24no—
69C2×C52C2×C22⋊C416yescomplemented
70C2×C52C23×C411yescharacteristic
71C5×D5—24no—
72C5×D5C2×C22⋊C416yescomplemented
73C5×D5C22×D412yes—
74C23×C5—24no—
75C23×C5—24no—
76C23×C5—48no—
77C23×C5—41no—
78C23×C5—46no—
79C23×C5—58no—
80C23×C5—53no—
81C23×C5—53no—
82C23×C5—56no—
83C23×C5—512no—
84C23×C5—54no—
85C23×C5—1024no—
86C23×C5—1024no—
87C23×C5—104no—
88C23×C5C2×Dic511yescharacteristic
89C23×C5C2×Dic516yescomplemented
90C23×C5C2×AGL(1,𝔽5)11yescharacteristic
91C23×C5C2×AGL(1,𝔽5)16yescomplemented
92C22×D5—24no—
93C22×D5—224no—
94C22×D5—224no—
95C22×D5C2×Dic513yes—
96C22×D5C2×Dic5112yescomplemented
97C22×D5C2×Dic516yescomplemented
98C22×D5C22⋊D518yes—
99C22×D5C22×D514yes—
100C22×D5C22×D513yes—
101C2×AGL(1,𝔽5)—58no—
102C2×AGL(1,𝔽5)—1024no—
103C2×Dic5—58no—
104C2×Dic5—1024no—
105C25—51no—
106C23×C4—252no—
107C2×C22⋊C4—2512no—
108C52C22×C22⋊C411yescharacteristic, complemented, Sylow
109C22×C5—212no—
110C22×C5—212no—
111C22×C5—424no—
112C22×C5—44no—
113C22×C5—43no—
114C22×C5—44no—
115C22×C5—512no—
116C22×C5—512no—
117C22×C5—53no—
118C22×C5—51no—
119C22×C5—54no—
120C22×C5—108no—
121C22×C5—1024no—
122C22×C5—1012no—
123C22×C5C22⋊Dic514yescomplemented
124C22×C5C22⋊AGL(1,𝔽5)14yescomplemented
125C22×C5C22×Dic513yes—
126C22×C5C22×Dic514yescomplemented
127C22×C5C22×AGL(1,𝔽5)13yes—
128C22×C5C22×AGL(1,𝔽5)14yescomplemented
129C2×D5—212no—
130C2×D5—224no—
131C2×D5—28no—
132C2×D5C22⋊Dic5112yescomplemented
133C2×D5C22×Dic513yes—
134C2×D5C22×Dic514yescomplemented
135C2×D5C2×C22⋊D5112yes—
136C2×D5C23×D511yescharacteristic
137AGL(1,𝔽5)—108no—
138Dic5—108no—
139C24—51no—
140C24—51no—
141C24—56no—
142C24—56no—
143C24—108no—
144C24#104211yescharacteristic, complemented
145C22×C4—2512no—
146C22×C4—508no—
147C22⋊C4—2516no—
148C2×C5—24no—
149C2×C5—24no—
150C2×C5—48no—
151C2×C5—46no—
152C2×C5—41no—
153C2×C5—52no—
154C2×C5—56no—
155C2×C5—104no—
156C2×C5C2×C22⋊Dic516yescomplemented
157C2×C5C2×C22⋊AGL(1,𝔽5)16yescomplemented
158C2×C5C23×Dic511yescharacteristic
159C2×C5C23×AGL(1,𝔽5)11yescharacteristic
160D5—24no—
161D5C2×C22⋊Dic516yescomplemented
162D5C22×C22⋊D512yes—
163C23—24no—
164C23—58no—
165C23—53no—
166C23—53no—
167C23—56no—
168C23—54no—
169C23—512no—
170C23—1024no—
171C23—1024no—
172C23—104no—
173C23C2×#104211yescharacteristic
174C23C2×#104216yescomplemented
175C2×C4—258no—
176C2×C4—5024no—
177C5—41no—
178C5C22×C22⋊Dic511yescharacteristic, complemented
179C5C22×C22⋊AGL(1,𝔽5)11yescharacteristic, complemented
180C22—212no—
181C22—512no—
182C22—512no—
183C22—51no—
184C22—53no—
185C22—54no—
186C22—108no—
187C22—1024no—
188C22—1012no—
189C22C22⋊#104214yesdirect product factor
190C22C22×#104213yes—
191C22C22×#104214yescomplemented
192C4—508no—
193C2—24no—
194C2—52no—
195C2—56no—
196C2—104no—
197C2C2×C22⋊#104216yesdirect product factor
198C2C23×#104211yescharacteristic
subgroupG/H
centerC23C2×#1042
commutatorC2×C52C23×C4
FrattiniC2C23×#1042
FittingC24×C52C4
socleC23×C52C2×C4
prime radicalC24×C5×D5C2
cosocleC2×C5×D5C24
hypercenterC24#1042
hypocenterC52C22×C22⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×#1042C2semidirect
3. C2×C22⋊#1042C2direct (N × H)
4. C24×C52C4semidirect
6. C23×C5×D5C4semidirect
7. C23×C5×D5C4semidirect
11. C22×#1042C22semidirect
12. C22⋊#1042C22direct (N × H)
17. C23×C52C2×C4semidirect
22. C22×C5×D5C2×C4semidirect
23. C22×C5×D5C2×C4semidirect
28. C2×#1042C23semidirect
36. C22×C52C22⋊C4semidirect
38. C22×C52C22×C4semidirect
42. C2×C5×D5C22⋊C4semidirect
44. C2×C5×D5C22×C4semidirect
54. C24×C5Dic5semidirect
55. C24×C5AGL(1,𝔽5)semidirect
57. C23×D5Dic5semidirect
58. C23×D5Dic5semidirect
69. C2×C52C2×C22⋊C4semidirect
72. C5×D5C2×C22⋊C4semidirect
89. C23×C5C2×Dic5semidirect
91. C23×C5C2×AGL(1,𝔽5)semidirect
96. C22×D5C2×Dic5semidirect
97. C22×D5C2×Dic5semidirect
108. C52C22×C22⋊C4semidirect
123. C22×C5C22⋊Dic5semidirect
124. C22×C5C22⋊AGL(1,𝔽5)semidirect
126. C22×C5C22×Dic5semidirect
128. C22×C5C22×AGL(1,𝔽5)semidirect
132. C2×D5C22⋊Dic5semidirect
134. C2×D5C22×Dic5semidirect
144. C24#1042semidirect
156. C2×C5C2×C22⋊Dic5semidirect
157. C2×C5C2×C22⋊AGL(1,𝔽5)semidirect
161. D5C2×C22⋊Dic5semidirect
174. C23C2×#1042semidirect
178. C5C22×C22⋊Dic5semidirect
179. C5C22×C22⋊AGL(1,𝔽5)semidirect
189. C22C22⋊#1042direct (N × H)
191. C22C22×#1042semidirect
197. C2C2×C22⋊#1042direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C5×D52
C23×#10422
C2×C22⋊#10422
C22×C22⋊AGL(1,𝔽5)5
C22×C22⋊Dic55