Finite Group Databasebeta

← #26108780

Subgroups — #26108780

Order 12032.

818 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 131 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#20243471C211yesmaximal, characteristic, complemented
2#20243465C211yesmaximal, characteristic, complemented
3#20241917C214yesmaximal, complemented
4C2×C47×#540C211yesmaximal, characteristic
5#15689087C2212yes—
6#15688986C2212yes—
7#15688892—22no—
8#15688890—22no—
9#15688857—28no—
10C47×#540—28no—
11C2×#127×C47C411yescharacteristic, complemented
12C2×#125×C47C411yescharacteristic, complemented
13C2×#123×C47—22no—
14C2×#121×C47—22no—
15C2×#121×C47C2212yes—
16C2×C4×D4×C47C412yescomplemented
17C2×#117×C47C2211yescharacteristic
18C2×#10625329—22no—
19C2×C22⋊Dic47—22no—
20C22⋊(C47⋊C8)—24no—
21C47⋊#95—24no—
22C22×D4×C47—22no—
23C22×D4×C47—22no—
24C22×D4×C47C2×C412yes—
25C23×C4×C47—22no—
26C23×C4×C47C2311yescharacteristic
27#127×C47—24no—
28#125×C47—24no—
29#123×C47—216no—
30#121×C47—216no—
31#121×C47—24no—
32#121×C47D418yes—
33C4×D4×C47—24no—
34C4×D4×C47—24no—
35#117×C47—24no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47C2×C412yes—
38C2×C4⋊C4×C47D412yes—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47D412yes—
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C24×C47—22no—
45C24×C47C2×D411yescharacteristic
46C2×D4×C47—28no—
47C2×D4×C47—28no—
48C2×D4×C47—22no—
49C2×D4×C47—44no—
50C2×D4×C47—84no—
51C2×D4×C47—84no—
52C2×D4×C47C22⋊C414yes—
53C22×C4×C47—22no—
54C22×C4×C47—28no—
55C22×C4×C47—42no—
56C22×C4×C47—84no—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D414yes—
62C22×Dic47—42no—
63C4⋊C4×C47—28no—
64C4⋊C4×C47—22no—
65C4⋊C4×C47—24no—
66C4⋊C4×C47C22⋊C414yes—
67C22⋊C4×C47—28no—
68C22⋊C4×C47—24no—
69C22⋊C4×C47—28no—
70C42×C47—22no—
71C42×C47—22no—
72C22⋊Dic47—44no—
73#10625329—44no—
74C2×C47⋊C8—42no—
75C23×C47—22no—
76C23×C47—24no—
77C23×C47—42no—
78C23×C47—44no—
79C23×C47—44no—
80C23×C47—44no—
81C23×C47C2×C22⋊C411yescharacteristic
82C23×C47#12012yes—
83D4×C47—44no—
84D4×C47—44no—
85D4×C47—88no—
86D4×C47—88no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—22no—
91C2×C4×C47—24no—
92C2×C4×C47—44no—
93C2×C4×C47—44no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—88no—
97C2×C4×C47C2×C22⋊C412yes—
98C2×C4×C47#12012yes—
99C2×Dic47—82no—
100C2×Dic47—84no—
101C47⋊C8—82no—
102#8518—471nomaximal, Sylow
103C22×C47—22no—
104C22×C47—24no—
105C22×C47—22no—
106C22×C47—44no—
107C22×C47—44no—
108C22×C47—44no—
109C22×C47—82no—
110C22×C47—88no—
111C22×C47#37911yescharacteristic
112C22×C47#41012yes—
113C4×C47—22no—
114C4×C47—22no—
115C4×C47—44no—
116C4×C47—42no—
117C4×C47—42no—
118Dic47—162no—
119C2×#540D4711yescharacteristic
120#1822—471no—
121#1820—471no—
122#1510—474no—
123C2×C47—22no—
124C2×C47—42no—
125C2×C47—84no—
126C2×C47#178211yescharacteristic
127C2×C47#179512yes—
128#540—28no—
129C2×#127Dic4711yescharacteristic, complemented
130C2×#125Dic4711yescharacteristic, complemented
131C2×#123—22no—
132C2×#121—22no—
133C2×#121C2×D4712yes—
134C2×C4×D4Dic4712yescomplemented
135C2×#117C2×D4711yescharacteristic
136#407—472no—
137#381—472no—
138#343—942no—
139#342—942no—
140#327—948no—
141C47#851811yescharacteristic, complemented, Sylow
142C22×D4—22no—
143C22×D4—22no—
144C22×D4C2×Dic4712yes—
145C23×C4—22no—
146C23×C4C22×D4711yescharacteristic
147C2×M16—942no—
148#127—24no—
149#125—24no—
150#123—216no—
151#121—216no—
152#121—24no—
153#121C22⋊D4718yes—
154C4×D4—24no—
155C4×D4—24no—
156#117—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×Dic4712yes—
159C2×C4⋊C4C22⋊D4712yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—942no—
163C2×C22⋊C4C22⋊D4712yes—
164C2×C42C2×Dic4711yescharacteristic
165C2×C42C2×Dic4711yescharacteristic
166C22⋊C8—944no—
167#95—944no—
168C24—22no—
169C24C2×C22⋊D4711yescharacteristic
170C2×D4—28no—
171C2×D4—28no—
172C2×D4—22no—
173C2×D4—44no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4C22⋊Dic4714yes—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—42no—
180C22×C4—84no—
181C22×C4—1882no—
182C22×C4C22⋊Dic4712yes—
183C22×C4C22⋊Dic4712yes—
184C22×C4C22×Dic4711yescharacteristic
185C22×C4C2×C22⋊D4711yescharacteristic
186C22×C4C2×C22⋊D4714yes—
187M16—1884no—
188C2×C8—1882no—
189C4⋊C4—28no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4C22⋊Dic4714yes—
193C22⋊C4—28no—
194C22⋊C4—24no—
195C22⋊C4—28no—
196C22⋊C4—1884no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—24no—
201C23—42no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23C2×C22⋊Dic4711yescharacteristic
206C23#1309399312yes—
207D4—44no—
208D4—44no—
209D4—88no—
210D4—88no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—88no—
221C2×C4—3762no—
222C2×C4—3764no—
223C2×C4C2×C22⋊Dic4712yes—
224C2×C4#1309399312yes—
225C8—3762no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—82no—
233C22—88no—
234C22#1568897911yescharacteristic
235C22#1568909712yes—
236C4—22no—
237C4—22no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—7522no—
242C2—22no—
243C2—42no—
244C2—84no—
245C2#2024330611yescharacteristic
246C2#2024335812yes—
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC2×C47×#540C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC2×#540D47
hypocenterC47#8518
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. #20243471C2semidirect
2. #20243465C2semidirect
3. #20241917C2semidirect
11. C2×#127×C47C4semidirect
12. C2×#125×C47C4semidirect
16. C2×C4×D4×C47C4semidirect
129. C2×#127Dic47semidirect
130. C2×#125Dic47semidirect
134. C2×C4×D4Dic47semidirect
141. C47#8518semidirect

Maximal subgroups

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

subgroupindex
#202434712
#202434652
#202419172
C2×C47×#5402
#851847