Finite Group Databasebeta

← #26113913

Subgroups — #26113913

Order 12032.

456 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 103 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#20246494C211yesmaximal, characteristic, complemented
2#20242015C211yesmaximal, characteristic, complemented
3#20241896C212yesmaximal
4#20241282C212yesmaximal, complemented
5C47×#3142C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689009—22no—
8#15689009—22no—
9C4⋊C4×Dic47—22no—
10#15688969C2212yes—
11#15688865—22no—
12#15688857—22no—
13#15688857—22no—
14C47×#532C2211yescharacteristic
15C2×#125×C47C2211yescharacteristic
16C47×#491C412yescomplemented
17C47×#490C412yescomplemented
18C47×#488—22no—
19C47×#417—22no—
20C4×M16×C47C2211yescharacteristic
21C2×C4⋊Dic47—22no—
22C2×C4×Dic47—22no—
23C2×C4×Dic47—22no—
24C4⋊(C47⋊C8)—22no—
25C22⋊(C47⋊C8)—22no—
26#13093935—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—22no—
30C47⋊#95D412yes—
31C47⋊#95D412yes—
32C2×#41×C47—21no—
33C2×M16×C47—21no—
34C4⋊Q8×C47C2×C411yescharacteristic
35#127×C47C2×C411yescharacteristic
36#125×C47—22no—
37#125×C47C2×C412yes—
38#125×C47D412yes—
39#124×C47—21no—
40#121×C47—22no—
41C4×D4×C47—22no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47D412yes—
44C2×C42×C47C2311yescharacteristic
45#103×C47—24no—
46#102×C47—24no—
47#97×C47C2×C412yes—
48C4×C8×C47—21no—
49#41×C47—44no—
50C2×Q8×C47—21no—
51C2×D4×C47—21no—
52C2×D4×C47—22no—
53C2×D4×C47—41no—
54C22×C4×C47—22no—
55C22×C4×C47—41no—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D412yes—
58C22×Dic47—22no—
59C22×Dic47—22no—
60M16×C47—42no—
61C2×C8×C47—22no—
62C4⋊C4×C47—24no—
63C4⋊C4×C47—24no—
64C4⋊C4×C47—22no—
65C4⋊C4×C47—22no—
66C4⋊C4×C47C22⋊C414yes—
67C22⋊C4×C47—42no—
68C42×C47C22×C411yescharacteristic
69C42×C47C2×D411yescharacteristic
70C42×C47#4112yes—
71C4⋊Dic47—44no—
72C4×Dic47—44no—
73C4×Dic47—44no—
74C2×C47⋊C8—42no—
75C23×C47—41no—
76C23×C47#12011yescharacteristic
77Q8×C47—42no—
78D4×C47—42no—
79D4×C47—82no—
80D4×C47—82no—
81C2×C4×C47—22no—
82C2×C4×C47—24no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—41no—
87C2×C4×C47—82no—
88C2×C4×C47C2×C22⋊C411yescharacteristic
89C2×C4×C47C4×D412yes—
90C2×C4×C47#12112yes—
91C2×C4×C47#12311yescharacteristic
92C2×Dic47—24no—
93C2×Dic47—42no—
94C2×Dic47—42no—
95C2×Dic47—42no—
96C2×Dic47—42no—
97C2×Dic47—42no—
98C8×C47—42no—
99C47⋊C8—82no—
100#9442—471nomaximal, Sylow
101C22×C47—22no—
102C22×C47—81no—
103C22×C47—82no—
104C22×C47#38611yescharacteristic
105C22×C47#45412yes—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—22no—
109C4×C47—22no—
110C4×C47—44no—
111C4×C47—81no—
112Dic47—44no—
113Dic47—82no—
114#3142D4711yescharacteristic
115#2371—471no—
116#1534—471no—
117#1505—472no—
118#1300—472no—
119C2×C47—22no—
120C2×C47—161no—
121C2×C47#188811yescharacteristic
122C2×C47#190612yes—
123#532C2×D4711yescharacteristic
124C2×#125C2×D4711yescharacteristic
125#491Dic4712yescomplemented
126#490Dic4712yescomplemented
127#488—22no—
128#432—472no—
129#417—22no—
130C4×M16C2×D4711yescharacteristic
131#387—942no—
132#387—942no—
133C4×C4⋊C4—942no—
134#376—472no—
135#330—942no—
136#327—942no—
137#327—942no—
138C47#944211yescharacteristic, complemented, Sylow
139C2×#41—21no—
140C2×M16—21no—
141C4⋊Q8C2×Dic4711yescharacteristic
142#127C2×Dic4711yescharacteristic
143#125—22no—
144#125C2×Dic4712yes—
145#125C22⋊D4712yes—
146#124—21no—
147#121—22no—
148C4×D4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—942no—
151C2×C4⋊C4C22⋊D4712yes—
152C2×C42—942no—
153C2×C42—942no—
154C2×C42C22×D4711yescharacteristic
155C4⋊C8—942no—
156#103—24no—
157#102—24no—
158C22⋊C8—942no—
159#97—942no—
160#97C2×Dic4712yes—
161C4×C8—21no—
162#95—472no—
163#95—472no—
164#95—942no—
165#95—942no—
166#95—942no—
167#41—44no—
168C2×Q8—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—41no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—942no—
175C22×C4—942no—
176C22×C4D4×D4712yes—
177C22×C4C2×C22⋊D4711yescharacteristic
178M16—42no—
179C2×C8—22no—
180C2×C8—1882no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—1884no—
186C4⋊C4C22⋊Dic4714yes—
187C22⋊C4—42no—
188C42—1884no—
189C42—1884no—
190C42#1062535212yes—
191C42C22×Dic4711yescharacteristic
192C42C2×C22⋊D4711yescharacteristic
193C23—41no—
194C23#1309399211yescharacteristic
195Q8—42no—
196D4—42no—
197D4—82no—
198D4—82no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—82no—
206C2×C4—944no—
207C2×C4—1882no—
208C2×C4—1882no—
209C2×C4—1882no—
210C2×C4—1882no—
211C2×C4—1882no—
212C2×C4C2×C22⋊Dic4711yescharacteristic
213C2×C4D4×Dic4712yes—
214C2×C4#1309399612yes—
215C2×C4#1309400511yescharacteristic
216C8—42no—
217C8—3762no—
218C22—22no—
219C22—81no—
220C22—82no—
221C22#1568900311yescharacteristic
222C22#1568929512yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—81no—
229C4—3762no—
230C4—1884no—
231C2—22no—
232C2—161no—
233C2#2024379211yescharacteristic
234C2#2024389112yes—
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3142C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3142D47
hypocenterC47#9442
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. #20246494C2semidirect
2. #20242015C2semidirect
4. #20241282C2semidirect
16. C47×#491C4semidirect
17. C47×#490C4semidirect
125. #491Dic47semidirect
126. #490Dic47semidirect
138. C47#9442semidirect

Maximal subgroups

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

subgroupindex
#202464942
#202420152
#202418962
#202412822
C47×#31422
#944247