Finite Group Databasebeta

← #26125767

Subgroups — #26125767

Order 12032.

440 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 75 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#20246102C211yesmaximal, characteristic
2#20241725C212yesmaximal
3#20241333C212yesmaximal, complemented
4#20241322C211yesmaximal, characteristic, complemented
5C47×#1319C211yesmaximal, characteristic, complemented
6C2×#13094013C411yescharacteristic
7C2×Q8×Dic47C411yescharacteristic
8C2×#13093986C2211yescharacteristic
9#15689085—22no—
10#15689025C411yescharacteristic
11#15689012—22no—
12#15688997—22no—
13#15688992C2211yescharacteristic
14#15688969C411yescharacteristic
15#15688849C2212yes—
16#15688846—22no—
17C47×#391C2211yescharacteristic
18C47×#324C2212yes—
19C22×D188C2×C411yescharacteristic
20C22×C4×D47D412yes—
21C2×#10625324—42no—
22#13094013—22no—
23#13094013—22no—
24Q8×Dic47—42no—
25#13093986—42no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4D412yes—
32C2×C4×Dic47—21no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35C22⋊(C47⋊C8)—22no—
36(C2×D47)⋊C8—42no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C22×Q8×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2311yescharacteristic
41C2×C42×C47C2×C411yescharacteristic
42C22⋊C8×C47—22no—
43#95×C47—22no—
44C23×D47C22⋊C412yes—
45C2×Q8×C47—21no—
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C2×Q8×C47M1612yescomplemented
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47C22×C411yescharacteristic
52C22×C4×C47C2×D412yes—
53C22×Dic47—21no—
54C22×Dic47—21no—
55C22×Dic47C22⋊C411yescharacteristic
56C22×Dic47C22⋊C411yescharacteristic
57C2×D188—21no—
58C2×D188—21no—
59C2×D188—41no—
60C2×D188—41no—
61C2×C4×D47—22no—
62C2×C4×D47—22no—
63C2×C4×D47—82no—
64C2×C8×C47—42no—
65C4⋊C4×C47—41no—
66C42×C47—41no—
67(C2×D47)⋊C4—28no—
68(C2×D47)⋊C4—42no—
69(C2×D47)⋊C4—82no—
70C4⋊Dic47—42no—
71C4⋊Dic47—41no—
72Dic47⋊C4—82no—
73C4×Dic47—21no—
74C4×Dic47—22no—
75C4×Dic47—42no—
76C4×Dic47M1612yescomplemented
77C2×C47⋊C8—42no—
78#10625324—84no—
79C23×C47C2×C22⋊C411yescharacteristic
80C22×D47—22no—
81C22×D47—22no—
82C22×D47—22no—
83C22×D47—42no—
84C22×D47—42no—
85Q8×C47—42no—
86Q8×C47—42no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—22no—
90C2×C4×C47—22no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47C2×M1611yescharacteristic
98C2×C4×C47C2×M1611yescharacteristic
99C2×Dic47—22no—
100C2×Dic47—21no—
101C2×Dic47—41no—
102C2×Dic47—41no—
103C2×Dic47—41no—
104C2×Dic47—41no—
105C2×Dic47—42no—
106C2×Dic47—41no—
107C2×Dic47#9912yes—
108D188—42no—
109D188—42no—
110C4×D47—44no—
111C4×D47—82no—
112C8×C47—82no—
113C47⋊C8—82no—
114#11535—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47#40711yescharacteristic
118C22×C47C2×#9911yescharacteristic
119C22×C47#41311yescharacteristic
120C2×D47—44no—
121C2×D47—42no—
122C2×D47—42no—
123C2×D47—42no—
124C2×D47—82no—
125C4×C47—21no—
126C4×C47—21no—
127C4×C47—42no—
128C4×C47—42no—
129C4×C47—81no—
130Dic47—22no—
131Dic47—42no—
132Dic47—41no—
133Dic47—81no—
134#2326—471no—
135#1465—472no—
136#1325—472no—
137#1319—471no—
138#1319D4711yescharacteristic, complemented
139C2×C47—21no—
140C2×C47—21no—
141C2×C47#146511yescharacteristic
142C2×C47#212711yescharacteristic
143C2×C47#213511yescharacteristic
144D47—82no—
145C2×#124—471no—
146C2×C4×Q8—471no—
147C2×C4×D4—471no—
148#407—942no—
149#391C2×D4711yescharacteristic
150#390—471no—
151#388—942no—
152#385—942no—
153#384—471no—
154#376—471no—
155#324—472no—
156#324C2×D4712yes—
157#323—942no—
158C47#1153511yescharacteristic, complemented, Sylow
159C22×Q8C4×D4711yescharacteristic
160C22×D4—471no—
161C23×C4—472no—
162C2×M16—1882no—
163#124—942no—
164#124—942no—
165C4×Q8—1882no—
166C4×D4—1882no—
167C2×C4⋊C4—471no—
168C2×C4⋊C4—942no—
169C2×C4⋊C4—941no—
170C2×C4⋊C4C22×D4711yescharacteristic
171C2×C22⋊C4—472no—
172C2×C22⋊C4—942no—
173C2×C22⋊C4—942no—
174C2×C42—471no—
175C2×C42—471no—
176C2×C42—941no—
177C2×C42C4×D4711yescharacteristic
178C22⋊C8—22no—
179C22⋊C8—942no—
180C22⋊C8—1882no—
181#95—22no—
182#95—941no—
183#95—941no—
184C24—472no—
185C2×Q8—21no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8#1062532412yescomplemented
189C2×D4—941no—
190C2×D4—941no—
191C2×D4—1881no—
192C2×D4—1881no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—471no—
196C22×C4—471no—
197C22×C4—942no—
198C22×C4—942no—
199C22×C4—941no—
200C22×C4—941no—
201C22×C4—3762no—
202C22×C4C2×C4×D4711yescharacteristic
203C22×C4D4×D4712yes—
204M16—3764no—
205C2×C8—42no—
206C2×C8—1882no—
207C4⋊C4—41no—
208C4⋊C4—3762no—
209C4⋊C4—1881no—
210C4⋊C4—1882no—
211C22⋊C4—948no—
212C22⋊C4—3762no—
213C22⋊C4—1882no—
214C42—41no—
215C42—472no—
216C42—942no—
217C42—941no—
218C42—1882no—
219C23—942no—
220C23—942no—
221C23—942no—
222C23—1882no—
223C23—1882no—
224C23C22⋊C4×D4711yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—1882no—
228D4—1882no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—472no—
240C2×C4—942no—
241C2×C4—941no—
242C2×C4—3762no—
243C2×C4—1884no—
244C2×C4—1881no—
245C2×C4—1882no—
246C2×C4—1881no—
247C2×C4—1881no—
248C2×C4—1881no—
249C2×C4—1881no—
250C2×C4M16×D4711yescharacteristic
251C2×C4C2×#1062532411yescharacteristic
252C8—82no—
253C8—3762no—
254C22—21no—
255C22—21no—
256C22—3762no—
257C22—1884no—
258C22—1882no—
259C22—1882no—
260C22—1882no—
261C22#1568908511yescharacteristic
262C22#99×D4711yescharacteristic
263C22#1568910411yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—942no—
270C4—3761no—
271C4—1882no—
272C4—1881no—
273C2—21no—
274C2—21no—
275C2—3762no—
276C2#2024172311yescharacteristic
277C2#2024505611yescharacteristic
278C2#2024508111yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1319C2
socleC22×C47C2×#99
prime radicalC22×D188C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1319D47
hypocenterC47#11535
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #20241333C2semidirect
4. #20241322C2semidirect
5. C47×#1319C2semidirect
48. C2×Q8×C47M16semidirect
76. C4×Dic47M16semidirect
138. #1319D47semidirect
158. C47#11535semidirect
188. C2×Q8#10625324semidirect

Maximal subgroups

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

subgroupindex
#202461022
#202417252
#202413332
#202413222
C47×#13192
#1153547