Finite Group Databasebeta

← #26125391

Subgroups — #26125391

Order 12032.

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

normal subgroups only

This group has 686 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#20241702C212yesmaximal, complemented
2#20241320C214yesmaximal, complemented
3C47×#2309C211yesmaximal, characteristic
4#15689176—22no—
5#15688849C2214yescomplemented
6#15688847—22no—
7C23×D4×C47C411yescharacteristic
8C2×#117×C47C411yescharacteristic
9C2×#117×C47C2211yescharacteristic
10C47×#390—22no—
11C47×#390C412yes—
12C47×#390C2212yes—
13C47×#379—22no—
14C4⋊(C47⋊C8)—42no—
15C22⋊(C47⋊C8)—24no—
16C4×C47⋊C8—42no—
17C25×C47—21no—
18C22×D4×C47—21no—
19C22×D4×C47—22no—
20C22×D4×C47—42no—
21C22×D4×C47—42no—
22C22×D4×C47C814yescomplemented
23C22×D4×C47C2×C412yes—
24C23×C4×C47C2×C411yescharacteristic
25#117×C47—42no—
26#117×C47—42no—
27C2×C4⋊C4×C47—21no—
28C2×C4⋊C4×C47C2×C411yescharacteristic
29C2×C4⋊C4×C47C2311yescharacteristic
30C2×C22⋊C4×C47—21no—
31C2×C22⋊C4×C47—24no—
32C2×C22⋊C4×C47—24no—
33C2×C22⋊C4×C47D412yes—
34C2×C42×C47C2×C412yes—
35C2×C42×C47D412yes—
36C24×C47—22no—
37C24×C47—24no—
38C24×C47—42no—
39C24×C47—42no—
40C24×C47—82no—
41C24×C47—82no—
42C24×C47C22⋊C411yescharacteristic
43C2×D4×C47—24no—
44C2×D4×C47—22no—
45C2×D4×C47—48no—
46C2×D4×C47—42no—
47C2×D4×C47—42no—
48C2×D4×C47—48no—
49C2×D4×C47—42no—
50C2×D4×C47—42no—
51C2×D4×C47C2×C814yes—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C2×C812yes—
58C22×C4×C47M1612yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D412yes—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—41no—
63C4⋊C4×C47—42no—
64C22⋊C4×C47—42no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—84no—
67C22⋊C4×C47—84no—
68C42×C47—22no—
69C42×C47—22no—
70C42×C47—42no—
71C2×C47⋊C8—44no—
72C23×C47—21no—
73C23×C47—21no—
74C23×C47—24no—
75C23×C47—41no—
76C23×C47—41no—
77C23×C47—44no—
78C23×C47—44no—
79C23×C47—48no—
80C23×C47—42no—
81C23×C47—42no—
82C23×C47—88no—
83C23×C47—88no—
84C23×C47—84no—
85C23×C47—84no—
86C23×C47C22⋊C812yes—
87C23×C47C2×C22⋊C411yescharacteristic
88D4×C47—44no—
89D4×C47—44no—
90D4×C47—48no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—24no—
94C2×C4×C47—24no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47C22⋊C812yes—
103C2×C4×C47#9912yes—
104C2×C4×C47#10012yes—
105C2×C4×C47C22×C811yescharacteristic
106C2×C4×C47C2×M1611yescharacteristic
107C47⋊C8—84no—
108#11433—471nomaximal, Sylow
109C22×C47—21no—
110C22×C47—21no—
111C22×C47—22no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—48no—
115C22×C47—41no—
116C22×C47—41no—
117C22×C47—44no—
118C22×C47—84no—
119C22×C47—816no—
120C22×C47—82no—
121C22×C47—82no—
122C22×C47C2×C22⋊C811yescharacteristic
123C22×C47C2×#9911yescharacteristic
124C22×C47C2×#10011yescharacteristic
125C4×C47—21no—
126C4×C47—21no—
127C4×C47—42no—
128C4×C47—42no—
129C4×C47—82no—
130C4×C47#32314yes—
131#2309D4711yescharacteristic
132#1458—472no—
133#1318—474no—
134C2×C47—21no—
135C2×C47—21no—
136C2×C47—42no—
137C2×C47—84no—
138C2×C47C2×#32311yescharacteristic
139C2×C47#212811yescharacteristic
140C2×C47#213411yescharacteristic
141C23×D4Dic4711yescharacteristic
142C2×#117Dic4711yescharacteristic
143C2×#117C2×D4711yescharacteristic
144#431—942no—
145#390—22no—
146#390Dic4712yes—
147#390C2×D4712yes—
148#379—22no—
149#324—474no—
150#323—942no—
151C47#1143311yescharacteristic, complemented, Sylow
152C25—21no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4C47⋊C814yescomplemented
158C22×D4C2×Dic4712yes—
159C23×C4C2×Dic4711yescharacteristic
160#117—42no—
161#117—42no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Dic4711yescharacteristic
164C2×C4⋊C4C22×D4711yescharacteristic
165C2×C22⋊C4—21no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4C22⋊D4712yes—
169C2×C42C2×Dic4712yes—
170C2×C42C22⋊D4712yes—
171C4⋊C8—1882no—
172C22⋊C8—944no—
173C4×C8—1882no—
174C24—22no—
175C24—24no—
176C24—42no—
177C24—42no—
178C24—82no—
179C24—82no—
180C24C22⋊Dic4711yescharacteristic
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—48no—
186C2×D4—48no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4C2×C47⋊C814yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4C2×C47⋊C812yes—
194C22×C4#1062532912yes—
195C22×C4C22⋊Dic4712yes—
196C22×C4C22⋊Dic4711yescharacteristic
197C22×C4C22×Dic4711yescharacteristic
198C22×C4C2×C22⋊D4712yes—
199C2×C8—1884no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C42—22no—
208C42—22no—
209C42—42no—
210C23—21no—
211C23—21no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—44no—
216C23—44no—
217C23—48no—
218C23—42no—
219C23—42no—
220C23—88no—
221C23—88no—
222C23—84no—
223C23—84no—
224C23C22⋊(C47⋊C8)12yes—
225C23C2×C22⋊Dic4711yescharacteristic
226D4—44no—
227D4—44no—
228D4—48no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4C22⋊(C47⋊C8)12yes—
241C2×C4#1309393912yes—
242C2×C4#1309394112yes—
243C2×C4C22×C47⋊C811yescharacteristic
244C2×C4C2×#1062532911yescharacteristic
245C8—3764no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—48no—
252C22—41no—
253C22—41no—
254C22—44no—
255C22—84no—
256C22—816no—
257C22—82no—
258C22—82no—
259C22C2×C22⋊(C47⋊C8)11yescharacteristic
260C22C2×#1309393911yescharacteristic
261C22C2×#1309394111yescharacteristic
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—82no—
267C4#1568884714yes—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—84no—
272C2C2×#1568884711yescharacteristic
273C2#2024506011yescharacteristic
274C2#2024508011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC2×C4×C47C22×C8
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2309C2
socleC22×C47C2×#99
prime radicalC23×D4×C47C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2309D47
hypocenterC47#11433
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
1. #20241702C2semidirect
2. #20241320C2semidirect
5. #15688849C22semidirect
22. C22×D4×C47C8semidirect
151. C47#11433semidirect
157. C22×D4C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202417022
#202413202
C47×#23092
#1143347