Finite Group Databasebeta

← #26125690

Subgroups — #26125690

Order 12032.

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

normal subgroups only

This group has 418 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#20241721C212yesmaximal
3#20241334C211yesmaximal, characteristic, complemented
4#20241333C212yesmaximal, complemented
5C47×#1325C211yesmaximal, characteristic, complemented
6C2×#13093982C2211yescharacteristic
7C2×#13093981C411yescharacteristic
8C2×C4⋊C4×D47C411yescharacteristic
9#15689177—22no—
10#15689018—22no—
11#15688994—22no—
12#15688992C2211yescharacteristic
13#15688989C411yescharacteristic
14#15688988C411yescharacteristic
15#15688849C2212yes—
16#15688846—22no—
17C47×#384C2211yescharacteristic
18C47×#324C2212yes—
19C22×C4×D47—21no—
20C22×C4×D47C2×C411yescharacteristic
21#13093982—42no—
22#13093981—22no—
23#13093981—22no—
24C2×C4⋊Dic47C2×C411yescharacteristic
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×Dic47⋊C4—21no—
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C4⋊C4×D47—42no—
30C2×(C2×D47)⋊C4—21no—
31C2×(C2×D47)⋊C4D412yes—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34Dic47⋊C8—42no—
35C22⋊(C47⋊C8)—22no—
36#13093934—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—42no—
58C2×C4×D47—82no—
59C2×C8×C47—42no—
60C4⋊C4×C47—41no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—82no—
63C42×C47—21no—
64C42×C47M1612yes—
65(C2×D47)⋊C4—24no—
66(C2×D47)⋊C4—42no—
67C4⋊Dic47—41no—
68C4⋊Dic47—41no—
69Dic47⋊C4—24no—
70Dic47⋊C4—42no—
71Dic47⋊C4—82no—
72C4×Dic47—21no—
73C4×Dic47—22no—
74C4×Dic47—22no—
75C4×Dic47M1612yes—
76C2×C47⋊C8—42no—
77C23×C47C2×C22⋊C411yescharacteristic
78C22×D47—21no—
79C22×D47—21no—
80C22×D47—41no—
81C22×D47—41no—
82C22×D47#10112yes—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—41no—
92C2×C4×C47C2×M1611yescharacteristic
93C2×C4×C47C2×M1611yescharacteristic
94C2×Dic47—21no—
95C2×Dic47—21no—
96C2×Dic47—22no—
97C2×Dic47—22no—
98C2×Dic47—41no—
99C2×Dic47—41no—
100C2×Dic47—42no—
101C2×Dic47—42no—
102C2×Dic47—41no—
103C2×Dic47#9912yes—
104C4×D47—24no—
105C4×D47—24no—
106C4×D47—84no—
107C8×C47—82no—
108C47⋊C8—82no—
109#11516—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47#40711yescharacteristic
113C22×C47C2×#9911yescharacteristic
114C22×C47C2×#10111yescharacteristic
115C2×D47—22no—
116C2×D47—42no—
117C2×D47—41no—
118C2×D47—41no—
119C2×D47—42no—
120C4×C47—21no—
121C4×C47—21no—
122C4×C47—42no—
123C4×C47—42no—
124C4×C47—81no—
125Dic47—22no—
126Dic47—42no—
127Dic47—42no—
128Dic47—81no—
129#2317—471no—
130#1464—472no—
131#1325—471no—
132#1325—472no—
133#1325D4711yescharacteristic, complemented
134C2×C47—21no—
135C2×C47—21no—
136C2×C47#146311yescharacteristic
137C2×C47#212811yescharacteristic
138C2×C47#213511yescharacteristic
139D47—42no—
140C2×#117—471no—
141C2×#117—471no—
142C22×C4⋊C4—471no—
143#432—942no—
144#389—942no—
145#385—942no—
146#384—471no—
147#384C2×D4711yescharacteristic
148#382—471no—
149#382—471no—
150#324—472no—
151#324C2×D4712yes—
152#323—942no—
153C47#1151611yescharacteristic, complemented, Sylow
154C23×C4—471no—
155C23×C4—941no—
156#117—942no—
157#117—942no—
158#117—1882no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—471no—
161C2×C4⋊C4—471no—
162C2×C4⋊C4—941no—
163C2×C4⋊C4—942no—
164C2×C4⋊C4—941no—
165C2×C4⋊C4—1882no—
166C2×C4⋊C4C4×D4711yescharacteristic
167C2×C4⋊C4C22×D4711yescharacteristic
168C2×C22⋊C4—472no—
169C2×C22⋊C4—941no—
170C2×C42—471no—
171C2×C42—472no—
172C2×C42C4×D4711yescharacteristic
173C4⋊C8—1882no—
174C22⋊C8—22no—
175C22⋊C8—942no—
176#97—1882no—
177#95—942no—
178#95—942no—
179C24—471no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—471no—
183C22×C4—472no—
184C22×C4—942no—
185C22×C4—942no—
186C22×C4—941no—
187C22×C4—941no—
188C22×C4—941no—
189C22×C4—941no—
190C22×C4—3762no—
191C22×C4—1882no—
192C22×C4C2×C4×D4711yescharacteristic
193C22×C4D4×D4712yes—
194C2×C8—42no—
195C2×C8—1882no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—82no—
199C4⋊C4—944no—
200C4⋊C4—3762no—
201C4⋊C4—1881no—
202C4⋊C4—1881no—
203C4⋊C4—1882no—
204C22⋊C4—944no—
205C22⋊C4—1882no—
206C42—21no—
207C42—472no—
208C42—942no—
209C42—942no—
210C42—941no—
211C42#1062532412yes—
212C23—472no—
213C23—941no—
214C23—941no—
215C23—1881no—
216C23—1881no—
217C23C22⋊C4×D4711yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—472no—
228C2×C4—944no—
229C2×C4—944no—
230C2×C4—941no—
231C2×C4—941no—
232C2×C4—942no—
233C2×C4—942no—
234C2×C4—3764no—
235C2×C4—1882no—
236C2×C4—1881no—
237C2×C4—1881no—
238C2×C4—1881no—
239C2×C4—1882no—
240C2×C4M16×D4711yescharacteristic
241C2×C4C2×#1062532411yescharacteristic
242C8—82no—
243C8—3762no—
244C22—21no—
245C22—21no—
246C22—942no—
247C22—1882no—
248C22—1882no—
249C22—1881no—
250C22—1881no—
251C22#1568908511yescharacteristic
252C22#99×D4711yescharacteristic
253C22#101×D4711yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—942no—
260C4—3761no—
261C4—1882no—
262C4—1882no—
263C2—21no—
264C2—21no—
265C2—1882no—
266C2#2024171611yescharacteristic
267C2#2024505811yescharacteristic
268C2#2024508111yescharacteristic
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#11516
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. #20241333C2semidirect
5. C47×#1325C2semidirect
133. #1325D47semidirect
153. C47#11516semidirect

Maximal subgroups

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

subgroupindex
#202460092
#202417212
#202413342
#202413332
C47×#13252
#1151647