Finite Group Databasebeta

← #26125685

Subgroups — #26125685

Order 12032.

434 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 77 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#20246009C211yesmaximal, characteristic
2#20241716C212yesmaximal
3#20241334C211yesmaximal, characteristic, complemented
4#20241331C212yesmaximal, complemented
5C47×#1325C211yesmaximal, characteristic, complemented
6C2×#13093982C411yescharacteristic
7C2×#13093981C411yescharacteristic
8C2×C4⋊C4×D47C2211yescharacteristic
9#15689085—22no—
10#15689018—22no—
11#15688994—22no—
12#15688992C411yescharacteristic
13#15688989C411yescharacteristic
14#15688988C2211yescharacteristic
15#15688849C2212yes—
16#15688848—22no—
17C47×#384C2211yescharacteristic
18C47×#324C2212yes—
19C22×C4×D47C2×C411yescharacteristic
20C22×C4×D47D412yes—
21C2×#10625324—42no—
22#13093982—42no—
23#13093981—22no—
24#13093981—22no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4D412yes—
30C4⋊C4×D47—42no—
31C2×(C2×D47)⋊C4—21no—
32C2×(C2×D47)⋊C4—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47C2×C411yescharacteristic
35C22⋊(C47⋊C8)—22no—
36(C2×D47)⋊C8—42no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C42×C47C2×C411yescharacteristic
43C22⋊C8×C47—22no—
44C23×D47C22⋊C411yescharacteristic
45C22×C4×C47—21no—
46C22×C4×C47—21no—
47C22×C4×C47C22×C411yescharacteristic
48C22×C4×C47C2×D412yes—
49C22×Dic47—21no—
50C22×Dic47—21no—
51C22×Dic47C22⋊C411yescharacteristic
52C22×Dic47C22⋊C412yes—
53C2×C4×D47—22no—
54C2×C4×D47—21no—
55C2×C4×D47—21no—
56C2×C4×D47—22no—
57C2×C4×D47—22no—
58C2×C4×D47—22no—
59C2×C4×D47—82no—
60C2×C8×C47—42no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—41no—
63C4⋊C4×C47—82no—
64C42×C47—21no—
65C42×C47M1612yescomplemented
66(C2×D47)⋊C4—24no—
67(C2×D47)⋊C4—42no—
68C4⋊Dic47—41no—
69C4⋊Dic47—41no—
70Dic47⋊C4—24no—
71Dic47⋊C4—42no—
72Dic47⋊C4—82no—
73C4×Dic47—21no—
74C4×Dic47—42no—
75C4×Dic47M1612yescomplemented
76C2×C47⋊C8—42no—
77#10625324—84no—
78C23×C47C2×C22⋊C411yescharacteristic
79C22×D47—21no—
80C22×D47—21no—
81C22×D47—41no—
82C22×D47—41no—
83C22×D47#9912yes—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—41no—
93C2×C4×C47C2×M1611yescharacteristic
94C2×C4×C47C2×M1611yescharacteristic
95C2×Dic47—21no—
96C2×Dic47—21no—
97C2×Dic47—22no—
98C2×Dic47—22no—
99C2×Dic47—41no—
100C2×Dic47—41no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—41no—
104C2×Dic47#10112yes—
105C4×D47—24no—
106C4×D47—24no—
107C4×D47—44no—
108C4×D47—82no—
109C8×C47—82no—
110C47⋊C8—82no—
111#11514—471nomaximal, Sylow
112C22×C47—21no—
113C22×C47—21no—
114C22×C47#40711yescharacteristic
115C22×C47C2×#9911yescharacteristic
116C22×C47C2×#10111yescharacteristic
117C2×D47—22no—
118C2×D47—22no—
119C2×D47—22no—
120C2×D47—42no—
121C2×D47—42no—
122C4×C47—21no—
123C4×C47—21no—
124C4×C47—42no—
125C4×C47—42no—
126C4×C47—81no—
127Dic47—22no—
128Dic47—42no—
129Dic47—42no—
130Dic47—81no—
131#2317—471no—
132#1463—472no—
133#1325—471no—
134#1325D4711yescharacteristic, complemented
135#1324—472no—
136C2×C47—21no—
137C2×C47—21no—
138C2×C47#146311yescharacteristic
139C2×C47#212711yescharacteristic
140C2×C47#213311yescharacteristic
141D47—42no—
142C2×#117—471no—
143C2×#117—471no—
144C22×C4⋊C4—471no—
145#407—942no—
146#389—942no—
147#385—942no—
148#384—471no—
149#384C2×D4711yescharacteristic
150#382—471no—
151#382—471no—
152#324—472no—
153#324—942no—
154#324C2×D4712yes—
155C47#1151411yescharacteristic, complemented, Sylow
156C23×C4—471no—
157C23×C4—472no—
158C2×M16—1882no—
159#117—942no—
160#117—942no—
161#117—1882no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—471no—
164C2×C4⋊C4—472no—
165C2×C4⋊C4—471no—
166C2×C4⋊C4—941no—
167C2×C4⋊C4—942no—
168C2×C4⋊C4—1882no—
169C2×C4⋊C4C4×D4711yescharacteristic
170C2×C4⋊C4C22×D4711yescharacteristic
171C2×C22⋊C4—941no—
172C2×C22⋊C4—941no—
173C2×C42—471no—
174C2×C42—941no—
175C2×C42C4×D4711yescharacteristic
176C22⋊C8—22no—
177C22⋊C8—942no—
178C22⋊C8—1882no—
179#95—942no—
180#95—942no—
181C24—471no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—471no—
185C22×C4—472no—
186C22×C4—942no—
187C22×C4—942no—
188C22×C4—942no—
189C22×C4—942no—
190C22×C4—941no—
191C22×C4—941no—
192C22×C4—941no—
193C22×C4—941no—
194C22×C4—3762no—
195C22×C4C2×C4×D4711yescharacteristic
196C22×C4D4×D4712yes—
197M16—3764no—
198C2×C8—42no—
199C2×C8—1882no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—82no—
203C4⋊C4—944no—
204C4⋊C4—3762no—
205C4⋊C4—1882no—
206C4⋊C4—1881no—
207C4⋊C4—1881no—
208C22⋊C4—944no—
209C22⋊C4—1882no—
210C42—21no—
211C42—472no—
212C42—941no—
213C42—1882no—
214C42#1062532412yescomplemented
215C23—472no—
216C23—941no—
217C23—941no—
218C23—1881no—
219C23—1881no—
220C23C22⋊C4×D4711yescharacteristic
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—472no—
231C2×C4—944no—
232C2×C4—944no—
233C2×C4—941no—
234C2×C4—941no—
235C2×C4—942no—
236C2×C4—942no—
237C2×C4—3762no—
238C2×C4—1884no—
239C2×C4—1882no—
240C2×C4—1881no—
241C2×C4—1881no—
242C2×C4—1881no—
243C2×C4—1882no—
244C2×C4M16×D4711yescharacteristic
245C2×C4C2×#1062532411yescharacteristic
246C8—82no—
247C8—3762no—
248C22—21no—
249C22—21no—
250C22—942no—
251C22—942no—
252C22—942no—
253C22—1882no—
254C22—1882no—
255C22#1568908511yescharacteristic
256C22#99×D4711yescharacteristic
257C22#101×D4711yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—942no—
264C4—3761no—
265C4—1882no—
266C4—1882no—
267C2—21no—
268C2—21no—
269C2—1882no—
270C2#2024171611yescharacteristic
271C2#2024505411yescharacteristic
272C2#2024507611yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1325C2
socleC22×C47C2×#99
prime radicalC23×D47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1325D47
hypocenterC47#11514
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #20241334C2semidirect
4. #20241331C2semidirect
5. C47×#1325C2semidirect
65. C42×C47M16semidirect
75. C4×Dic47M16semidirect
134. #1325D47semidirect
155. C47#11514semidirect
214. C42#10625324semidirect

Maximal subgroups

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

subgroupindex
#202460092
#202417162
#202413342
#202413312
C47×#13252
#1151447