Finite Group Databasebeta

← #26125702

Subgroups — #26125702

Order 12032.

542 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 99 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#20241336C214yesmaximal, complemented
3C47×#2315C211yesmaximal, characteristic
4#15689087—22no—
5#15688847—22no—
6#15688847C2214yes—
7C22×C22⋊C4×C47C411yescharacteristic
8C22×C22⋊C4×C47C411yescharacteristic
9C22×C22⋊C4×C47C2211yescharacteristic
10C47×#385—22no—
11C47×#385C412yes—
12C47×#385C2212yes—
13C47×#380—22no—
14C2×#10625329—42no—
15C22⋊(C47⋊C8)—24no—
16C22⋊(C47⋊C8)—42no—
17C25×C47C2×C411yescharacteristic
18C23×C4×C47—21no—
19C23×C4×C47C2×C412yes—
20C23×C4×C47D412yes—
21C2×C4⋊C4×C47—22no—
22C2×C4⋊C4×C47—22no—
23C2×C22⋊C4×C47—21no—
24C2×C22⋊C4×C47—24no—
25C2×C22⋊C4×C47—21no—
26C2×C22⋊C4×C47—21no—
27C2×C22⋊C4×C47—41no—
28C2×C22⋊C4×C47—41no—
29C2×C22⋊C4×C47—41no—
30C2×C22⋊C4×C47—41no—
31C2×C22⋊C4×C47C2×C412yes—
32C2×C22⋊C4×C47C2×C411yescharacteristic
33C2×C22⋊C4×C47D412yes—
34C2×C22⋊C4×C47C2311yescharacteristic
35#95×C47—22no—
36#95×C47—22no—
37C24×C47—21no—
38C24×C47—21no—
39C24×C47—22no—
40C24×C47—22no—
41C24×C47—82no—
42C24×C47C22⋊C411yescharacteristic
43C24×C47C22⋊C411yescharacteristic
44C24×C47C22×C411yescharacteristic
45C22×C4×C47—22no—
46C22×C4×C47—22no—
47C22×C4×C47—24no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—42no—
51C22×C4×C47—82no—
52C22×C4×C47—82no—
53C22×C4×C47—82no—
54C22×C4×C47C22⋊C412yes—
55C22×C4×C47C2×D412yes—
56C4⋊C4×C47—82no—
57C4⋊C4×C47—82no—
58C22⋊C4×C47—24no—
59C22⋊C4×C47—22no—
60C22⋊C4×C47—41no—
61C22⋊C4×C47—42no—
62C22⋊C4×C47—41no—
63C22⋊C4×C47—42no—
64C22⋊C4×C47—41no—
65C22⋊C4×C47—41no—
66C22⋊C4×C47M1614yescomplemented
67#10625329—84no—
68C2×C47⋊C8—44no—
69C23×C47—24no—
70C23×C47—24no—
71C23×C47—21no—
72C23×C47—21no—
73C23×C47—21no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47—21no—
77C23×C47—42no—
78C23×C47—42no—
79C23×C47—42no—
80C23×C47—82no—
81C23×C47—82no—
82C23×C47—84no—
83C23×C47—84no—
84C23×C47#9911yescharacteristic
85C23×C47#9911yescharacteristic
86C23×C47#10112yes—
87C23×C47C2×C22⋊C411yescharacteristic
88C23×C47C2×M1612yes—
89C2×C4×C47—24no—
90C2×C4×C47—22no—
91C2×C4×C47—22no—
92C2×C4×C47—44no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—82no—
101C2×C4×C47—84no—
102C2×C4×C47—84no—
103C47⋊C8—84no—
104#11519—471nomaximal, Sylow
105C22×C47—22no—
106C22×C47—22no—
107C22×C47—22no—
108C22×C47—21no—
109C22×C47—21no—
110C22×C47—22no—
111C22×C47—22no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—42no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47—42no—
119C22×C47—42no—
120C22×C47—84no—
121C22×C47—84no—
122C22×C47#40711yescharacteristic
123C22×C47C2×#9911yescharacteristic
124C22×C47C2×#10111yescharacteristic
125C4×C47—44no—
126C4×C47—42no—
127C4×C47—42no—
128C4×C47—82no—
129#2315D4711yescharacteristic
130#1462—472no—
131#1326—474no—
132C2×C47—21no—
133C2×C47—21no—
134C2×C47—41no—
135C2×C47—41no—
136C2×C47—42no—
137C2×C47—42no—
138C2×C47#146311yescharacteristic
139C2×C47#212112yes—
140C22×C22⋊C4Dic4711yescharacteristic
141C22×C22⋊C4Dic4711yescharacteristic
142C22×C22⋊C4C2×D4711yescharacteristic
143#407—942no—
144#385—22no—
145#385Dic4712yes—
146#385C2×D4712yes—
147#380—22no—
148#323—474no—
149#323—942no—
150C47#1151911yescharacteristic, complemented, Sylow
151C25C2×Dic4711yescharacteristic
152C23×C4—21no—
153C23×C4C2×Dic4712yes—
154C23×C4C22⋊D4712yes—
155C2×M16—1882no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4C2×Dic4712yes—
167C2×C22⋊C4C2×Dic4711yescharacteristic
168C2×C22⋊C4C22⋊D4712yes—
169C2×C22⋊C4C22×D4711yescharacteristic
170C22⋊C8—944no—
171C22⋊C8—1882no—
172#95—22no—
173#95—22no—
174C24—21no—
175C24—21no—
176C24—22no—
177C24—22no—
178C24—82no—
179C24C22⋊Dic4711yescharacteristic
180C24C22⋊Dic4711yescharacteristic
181C24C22×Dic4711yescharacteristic
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4C22⋊Dic4712yes—
192C22×C4C2×C22⋊D4712yes—
193M16—3764no—
194C2×C8—1884no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4#1062532914yescomplemented
206C23—24no—
207C23—24no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—82no—
218C23—82no—
219C23—84no—
220C23—84no—
221C23#1309393911yescharacteristic
222C23#1309393911yescharacteristic
223C23#1309394312yes—
224C23C2×C22⋊Dic4711yescharacteristic
225C23C2×#1062532912yes—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—84no—
239C2×C4—84no—
240C8—3764no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—21no—
245C22—21no—
246C22—22no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—84no—
257C22—84no—
258C22#1568908711yescharacteristic
259C22C2×#1309393911yescharacteristic
260C22C2×#1309394311yescharacteristic
261C4—44no—
262C4—42no—
263C4—42no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—41no—
268C2—41no—
269C2—42no—
270C2—42no—
271C2#2024171811yescharacteristic
272C2#2024503412yes—
subgroupG/H
centerC22C2×#13093939
commutatorC24×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2315C2
socleC22×C47C2×#99
prime radicalC25×C47C2×C4
cosocleC2×C22⋊C4×C47C23
hypercenter#2315D47
hypocenterC47#11519
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #20241336C2semidirect
66. C22⋊C4×C47M16semidirect
150. C47#11519semidirect
205. C22⋊C4#10625329semidirect

Maximal subgroups

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

subgroupindex
#202417152
#202413362
C47×#23152
#1151947