Finite Group Databasebeta

← #26125638

Subgroups — #26125638

Order 12032.

706 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 107 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#20241715C212yesmaximal
2#20241318C214yesmaximal, complemented
3C2×C47×#381C211yesmaximal, characteristic
4#15689087—22no—
5#15688847—22no—
6#15688847C2214yes—
7C24×C4×C47C411yescharacteristic
8C22×C22⋊C4×C47C411yescharacteristic
9C22×C22⋊C4×C47C2211yescharacteristic
10C47×#381—22no—
11C47×#381C412yes—
12C47×#381C2212yes—
13C2×#95×C47—22no—
14C2×#10625329—42no—
15C22⋊(C47⋊C8)—24no—
16C22⋊(C47⋊C8)—42no—
17C25×C47C2×C411yescharacteristic
18C23×C4×C47—21no—
19C23×C4×C47—21no—
20C23×C4×C47—24no—
21C23×C4×C47—21no—
22C23×C4×C47—42no—
23C23×C4×C47—42no—
24C23×C4×C47C2×C412yes—
25C23×C4×C47D412yes—
26C2×C22⋊C4×C47—21no—
27C2×C22⋊C4×C47—42no—
28C2×C22⋊C4×C47—42no—
29C2×C22⋊C4×C47C2×C412yes—
30C2×C22⋊C4×C47C2×C411yescharacteristic
31C2×C22⋊C4×C47D412yes—
32C2×C22⋊C4×C47C2311yescharacteristic
33#95×C47—24no—
34#95×C47—24no—
35C24×C47—22no—
36C24×C47—22no—
37C24×C47—22no—
38C24×C47—44no—
39C24×C47C22⋊C411yescharacteristic
40C24×C47C22⋊C411yescharacteristic
41C24×C47C22×C411yescharacteristic
42C22×C4×C47—24no—
43C22×C4×C47—24no—
44C22×C4×C47—22no—
45C22×C4×C47—22no—
46C22×C4×C47—22no—
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47—416no—
50C22×C4×C47—42no—
51C22×C4×C47—42no—
52C22×C4×C47—42no—
53C22×C4×C47—42no—
54C22×C4×C47—42no—
55C22×C4×C47—42no—
56C22×C4×C47—44no—
57C22×C4×C47—42no—
58C22×C4×C47—44no—
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47M1614yescomplemented
61C22×C4×C47C2×D412yes—
62C22⋊C4×C47—41no—
63C22⋊C4×C47—42no—
64C22⋊C4×C47—41no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67#10625329—84no—
68C2×C47⋊C8—44no—
69C23×C47—24no—
70C23×C47—21no—
71C23×C47—21no—
72C23×C47—28no—
73C23×C47—21no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47—21no—
77C23×C47—42no—
78C23×C47—42no—
79C23×C47—48no—
80C23×C47—48no—
81C23×C47—44no—
82C23×C47—44no—
83C23×C47#9912yes—
84C23×C47#10012yes—
85C23×C47C2×C22⋊C411yescharacteristic
86C23×C47C2×M1612yes—
87C2×C4×C47—24no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—48no—
91C2×C4×C47—48no—
92C2×C4×C47—44no—
93C2×C4×C47—44no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—44no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—44no—
100C2×C4×C47—42no—
101C2×C4×C47—82no—
102C2×C4×C47—84no—
103C47⋊C8—84no—
104#11500—471nomaximal, Sylow
105C22×C47—24no—
106C22×C47—24no—
107C22×C47—22no—
108C22×C47—22no—
109C22×C47—22no—
110C22×C47—22no—
111C22×C47—22no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—42no—
115C22×C47—416no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47—42no—
119C22×C47#35214yes—
120C22×C47#40711yescharacteristic
121C22×C47C2×#9911yescharacteristic
122C22×C47C2×#10011yescharacteristic
123C4×C47—42no—
124C4×C47—42no—
125C4×C47—44no—
126C4×C47—82no—
127C2×#381D4711yescharacteristic
128#1462—472no—
129#1317—474no—
130C2×C47—21no—
131C2×C47—24no—
132C2×C47—21no—
133C2×C47—42no—
134C2×C47—42no—
135C2×C47#146211yescharacteristic
136C2×C47C2×#35212yes—
137C24×C4Dic4711yescharacteristic
138C22×C22⋊C4Dic4711yescharacteristic
139C22×C22⋊C4C2×D4711yescharacteristic
140#407—942no—
141#381—22no—
142#381Dic4712yes—
143#381C2×D4712yes—
144C2×#95—22no—
145#323—474no—
146#323—942no—
147C47#1150011yescharacteristic, complemented, Sylow
148C25C2×Dic4711yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×Dic4712yes—
156C23×C4C22⋊D4712yes—
157C2×M16—1882no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4C2×Dic4712yes—
162C2×C22⋊C4C2×Dic4711yescharacteristic
163C2×C22⋊C4C22⋊D4712yes—
164C2×C22⋊C4C22×D4711yescharacteristic
165C22⋊C8—944no—
166C22⋊C8—1882no—
167#95—24no—
168#95—24no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—44no—
173C24C22⋊Dic4711yescharacteristic
174C24C22⋊Dic4711yescharacteristic
175C24C22×Dic4711yescharacteristic
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—416no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4#1062532914yescomplemented
194C22×C4C22⋊Dic4712yes—
195C22×C4C2×C22⋊D4712yes—
196M16—3764no—
197C2×C8—1884no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C23—21no—
204C23—21no—
205C23—24no—
206C23—28no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—42no—
212C23—42no—
213C23—48no—
214C23—48no—
215C23—44no—
216C23—44no—
217C23#1309393912yes—
218C23#1309394112yes—
219C23C2×C22⋊Dic4711yescharacteristic
220C23C2×#1062532912yes—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—48no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—84no—
237C8—3764no—
238C22—24no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—416no—
250C22—42no—
251C22—42no—
252C22#1568891314yes—
253C22#1568908711yescharacteristic
254C22C2×#1309393911yescharacteristic
255C22C2×#1309394111yescharacteristic
256C4—42no—
257C4—42no—
258C4—44no—
259C4—82no—
260C2—21no—
261C2—21no—
262C2—24no—
263C2—42no—
264C2—42no—
265C2#2024171511yescharacteristic
266C2C2×#1568891312yes—
subgroupG/H
centerC22C2×#13093939
commutatorC24×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC2×C47×#381C2
socleC22×C47C2×#99
prime radicalC25×C47C2×C4
cosocleC2×C22⋊C4×C47C23
hypercenterC2×#381D47
hypocenterC47#11500
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. #20241318C2semidirect
60. C22×C4×C47M16semidirect
147. C47#11500semidirect
193. C22×C4#10625329semidirect

Maximal subgroups

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

subgroupindex
#202417152
#202413182
C2×C47×#3812
#1150047