Finite Group Databasebeta

← #26225542

Subgroups — #26225542

Order 12032.

1538 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 315 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#20245986C212yesmaximal, complemented
2#20244225C218yesmaximal, complemented
3#20243303C214yesmaximal, complemented
4C2×C47×#534C211yesmaximal, characteristic
5C2×#13093939C22116yescomplemented
6#15689016—28no—
7#15688983C2214yescomplemented
8#15688979C2214yes—
9C4×C22⋊Dic47C2214yescomplemented
10C23×D4×C47C2212yes—
11C47×#534—28no—
12C2×#123×C47C414yescomplemented
13C2×#121×C47C2214yes—
14C2×#120×C47C414yescomplemented
15C22×C22⋊C4×C47C2211yescharacteristic
16C2×C4⋊Dic47—24no—
17C2×C22⋊Dic47—24no—
18C2×C22⋊Dic47—24no—
19C2×C22⋊Dic47C2318yes—
20C2×C4×Dic47—24no—
21#13093939—416no—
22C47⋊#95—24no—
23C47⋊#95—24no—
24C25×C47C2×C412yes—
25C25×C47C2311yescharacteristic
26C22×D4×C47—28no—
27C22×D4×C47—48no—
28C22×D4×C47C2×C414yes—
29C22×D4×C47C2×C414yes—
30C22×D4×C47C2314yes—
31C23×C4×C47C2×C412yes—
32C23×C4×C47C2312yes—
33#123×C47—216no—
34#121×C47—28no—
35#121×C47—28no—
36#120×C47—216no—
37C2×C4⋊C4×C47C2×C414yes—
38C2×C22⋊C4×C47C2×C418yes—
39C2×C22⋊C4×C47C2×C414yes—
40C2×C22⋊C4×C47D418yes—
41C24×C47—24no—
42C24×C47—24no—
43C24×C47—44no—
44C24×C47—82no—
45C24×C47—84no—
46C24×C47C22×C414yes—
47C24×C47C22×C412yes—
48C24×C47C2×D414yes—
49C24×C47C2411yescharacteristic
50C2×D4×C47—28no—
51C2×D4×C47—28no—
52C2×D4×C47—216no—
53C2×D4×C47—28no—
54C2×D4×C47—48no—
55C2×D4×C47—432no—
56C2×D4×C47—84no—
57C22×C4×C47—24no—
58C22×C4×C47—44no—
59C22×C4×C47—82no—
60C22×C4×C47C22×C414yes—
61C22×C4×C47C22×C414yes—
62C22×C4×C47C2×D414yes—
63C22×C4×C47C2×D414yes—
64C22×Dic47—28no—
65C4⋊C4×C47—28no—
66C22⋊C4×C47—216no—
67C22⋊C4×C47C22⋊C4116yes—
68C22⋊Dic47—216no—
69C22⋊Dic47—84no—
70C22⋊Dic47—84no—
71C4⋊Dic47—84no—
72C4×Dic47—84no—
73C23×C47—24no—
74C23×C47—28no—
75C23×C47—24no—
76C23×C47—28no—
77C23×C47—42no—
78C23×C47—44no—
79C23×C47—44no—
80C23×C47—44no—
81C23×C47—48no—
82C23×C47—48no—
83C23×C47—88no—
84C23×C47—88no—
85C23×C47—816no—
86C23×C47C2×C22⋊C414yes—
87C23×C47C23×C411yescharacteristic
88C23×C47C22×D412yes—
89C23×C47H(5,𝔽2)14yes—
90D4×C47—416no—
91D4×C47—416no—
92D4×C47—88no—
93C2×C4×C47—24no—
94C2×C4×C47—28no—
95C2×C4×C47—44no—
96C2×C4×C47—44no—
97C2×C4×C47—44no—
98C2×C4×C47—48no—
99C2×C4×C47C2×C22⋊C418yes—
100C2×Dic47—48no—
101C2×Dic47—48no—
102C2×Dic47—48no—
103#21795—471nomaximal, Sylow
104C22×C47—22no—
105C22×C47—22no—
106C22×C47—22no—
107C22×C47—24no—
108C22×C47—28no—
109C22×C47—44no—
110C22×C47—48no—
111C22×C47—44no—
112C22×C47—48no—
113C22×C47—88no—
114C22×C47—88no—
115C22×C47—816no—
116C22×C47C22×C22⋊C411yescharacteristic
117C22×C47#51812yes—
118C22×C47#53514yes—
119C4×C47—28no—
120C4×C47—44no—
121Dic47—88no—
122C2×#534D4711yescharacteristic
123#2312—472no—
124#1964—478no—
125#1781—474no—
126C2×C47—24no—
127C2×C47—22no—
128C2×C47—44no—
129C2×C47—84no—
130C2×C47#230911yescharacteristic
131C2×C47#288312yes—
132C23×D4C2×D4712yes—
133#534—28no—
134C2×#123Dic4714yescomplemented
135C2×#121C2×D4714yes—
136C2×#120Dic4714yescomplemented
137C22×C22⋊C4C2×D4711yescharacteristic
138C2×#99—4716no—
139#388—948no—
140#380—474no—
141#379—474no—
142C4×C22⋊C4—474no—
143C47#2179511yescharacteristic, complemented, Sylow
144C25C2×Dic4712yes—
145C25C22×D4711yescharacteristic
146C22×D4—28no—
147C22×D4—48no—
148C22×D4C2×Dic4714yes—
149C22×D4C2×Dic4714yes—
150C22×D4C22×D4714yes—
151C23×C4C2×Dic4712yes—
152C23×C4C22×D4712yes—
153#123—216no—
154#121—28no—
155#121—28no—
156#120—216no—
157C2×C4⋊C4—944no—
158C2×C4⋊C4C2×Dic4714yes—
159C2×C22⋊C4—478no—
160C2×C22⋊C4—944no—
161C2×C22⋊C4—944no—
162C2×C22⋊C4C2×Dic4718yes—
163C2×C22⋊C4C2×Dic4714yes—
164C2×C22⋊C4C22⋊D4718yes—
165C2×C42—944no—
166#99—18816no—
167#95—944no—
168#95—944no—
169C24—24no—
170C24—24no—
171C24—44no—
172C24—82no—
173C24—84no—
174C24C22×Dic4714yes—
175C24C22×Dic4712yes—
176C24C2×C22⋊D4714yes—
177C24C23×D4711yescharacteristic
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—216no—
181C2×D4—28no—
182C2×D4—48no—
183C2×D4—432no—
184C2×D4—84no—
185C22×C4—24no—
186C22×C4—44no—
187C22×C4—82no—
188C22×C4—948no—
189C22×C4C22×Dic4714yes—
190C22×C4C22×Dic4714yes—
191C22×C4C2×C22⋊D4714yes—
192C22×C4C2×C22⋊D4714yes—
193C4⋊C4—28no—
194C4⋊C4—3764no—
195C22⋊C4—216no—
196C22⋊C4—9416no—
197C22⋊C4—3764no—
198C22⋊C4—3764no—
199C22⋊C4C22⋊Dic47116yes—
200C42—3764no—
201C23—24no—
202C23—28no—
203C23—24no—
204C23—28no—
205C23—42no—
206C23—44no—
207C23—44no—
208C23—44no—
209C23—48no—
210C23—48no—
211C23—88no—
212C23—88no—
213C23—816no—
214C23C2×C22⋊Dic4714yes—
215C23C23×Dic4711yescharacteristic
216C23C22×C22⋊D4712yes—
217C23#1309406914yes—
218D4—416no—
219D4—416no—
220D4—88no—
221C2×C4—24no—
222C2×C4—28no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—48no—
227C2×C4—1888no—
228C2×C4—1888no—
229C2×C4—1888no—
230C2×C4C2×C22⋊Dic4718yes—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—28no—
235C22—24no—
236C22—44no—
237C22—48no—
238C22—48no—
239C22—44no—
240C22—88no—
241C22—88no—
242C22—816no—
243C22C22×C22⋊Dic4711yescharacteristic
244C22#1568959812yes—
245C22#1568969014yes—
246C4—28no—
247C4—44no—
248C4—3768no—
249C2—24no—
250C2—22no—
251C2—44no—
252C2—84no—
253C2#2024597011yescharacteristic
254C2#2025184912yes—
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
Fitting, prime radicalC2×C47×#534C2
socleC22×C47C22×C22⋊C4
cosocleC24×C47C24
hypercenterC2×#534D47
hypocenterC47#21795
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #20245986C2semidirect
2. #20244225C2semidirect
3. #20243303C2semidirect
5. C2×#13093939C22semidirect
7. #15688983C22semidirect
9. C4×C22⋊Dic47C22semidirect
12. C2×#123×C47C4semidirect
14. C2×#120×C47C4semidirect
134. C2×#123Dic47semidirect
136. C2×#120Dic47semidirect
143. C47#21795semidirect

Maximal subgroups

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

subgroupindex
#202459862
#202442252
#202433032
C2×C47×#5342
#2179547