Finite Group Databasebeta

← #26113903

Subgroups — #26113903

Order 12032.

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

normal subgroups only

This group has 434 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
2#20242015C211yesmaximal, characteristic
3#20241900C212yesmaximal
4#20241282C212yesmaximal
5C47×#3166C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689009—22no—
8#15689009—22no—
9C4⋊C4×Dic47—22no—
10#15688969C2212yes—
11#15688865—22no—
12#15688860—22no—
13#15688860—22no—
14C47×#533C2211yescharacteristic
15C2×#125×C47C2211yescharacteristic
16C47×#501C414yescomplemented
17C47×#499—22no—
18C47×#428—22no—
19C4×M16×C47C2211yescharacteristic
20C2×C4⋊Dic47—22no—
21C2×C4×Dic47—22no—
22C2×C4×Dic47—22no—
23C4⋊(C47⋊C8)—22no—
24C22⋊(C47⋊C8)—22no—
25#13093935—22no—
26C47⋊#95—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95D412yes—
30C47⋊#95D412yes—
31C2×M16×C47—21no—
32C4⋊Q8×C47C2×C412yes—
33#125×C47—21no—
34#125×C47—22no—
35#125×C47C2×C412yes—
36#125×C47D412yes—
37C22⋊Q8×C47—22no—
38C4×Q8×C47—22no—
39#117×C47—21no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47D412yes—
42C2×C42×C47C2311yescharacteristic
43#107×C47—24no—
44#106×C47—24no—
45#97×C47C2×C412yes—
46C4×C8×C47—21no—
47C2×Q8×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47C2×D412yes—
51C22×Dic47—22no—
52C22×Dic47—22no—
53M16×C47—24no—
54C2×C8×C47—22no—
55C4⋊C4×C47—24no—
56C4⋊C4×C47—24no—
57C4⋊C4×C47—22no—
58C4⋊C4×C47—22no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47C22⋊C414yes—
62C22⋊C4×C47—41no—
63C42×C47—41no—
64C42×C47C22×C411yescharacteristic
65C42×C47C2×D411yescharacteristic
66C42×C47#4112yes—
67C4⋊Dic47—44no—
68C4×Dic47—44no—
69C4×Dic47—44no—
70C2×C47⋊C8—42no—
71C23×C47#12011yescharacteristic
72Q8×C47—82no—
73C2×C4×C47—22no—
74C2×C4×C47—24no—
75C2×C4×C47—42no—
76C2×C4×C47—42no—
77C2×C4×C47—42no—
78C2×C4×C47—42no—
79C2×C4×C47C2×C22⋊C411yescharacteristic
80C2×C4×C47C4×D412yes—
81C2×C4×C47#12112yes—
82C2×C4×C47#12311yescharacteristic
83C2×Dic47—24no—
84C2×Dic47—42no—
85C2×Dic47—42no—
86C2×Dic47—42no—
87C2×Dic47—42no—
88C2×Dic47—42no—
89C8×C47—24no—
90C47⋊C8—82no—
91#9440—471nomaximal, Sylow
92C22×C47—22no—
93C22×C47#38611yescharacteristic
94C22×C47#45612yes—
95C4×C47—21no—
96C4×C47—21no—
97C4×C47—22no—
98C4×C47—22no—
99C4×C47—44no—
100C4×C47—82no—
101Dic47—44no—
102Dic47—82no—
103#3166D4711yescharacteristic
104#2371—471no—
105#1534—471no—
106#1506—472no—
107#1300—472no—
108C2×C47—22no—
109C2×C47#188611yescharacteristic
110C2×C47#190612yes—
111#533C2×D4711yescharacteristic
112C2×#125C2×D4711yescharacteristic
113#501Dic4714yescomplemented
114#499—22no—
115#432—472no—
116#428—22no—
117C4×M16C2×D4711yescharacteristic
118#387—942no—
119#387—942no—
120C4×C4⋊C4—942no—
121#376—472no—
122#330—942no—
123#328—942no—
124#328—942no—
125C47#944011yescharacteristic, complemented, Sylow
126C2×M16—21no—
127C4⋊Q8C2×Dic4712yes—
128#125—22no—
129#125—21no—
130#125C2×Dic4712yes—
131#125C22⋊D4712yes—
132C22⋊Q8—22no—
133C4×Q8—22no—
134#117—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—942no—
137C2×C4⋊C4C22⋊D4712yes—
138C2×C42—942no—
139C2×C42—942no—
140C2×C42C22×D4711yescharacteristic
141#107—24no—
142#106—24no—
143C4⋊C8—942no—
144C22⋊C8—942no—
145#97—942no—
146#97C2×Dic4712yes—
147C4×C8—21no—
148#95—472no—
149#95—472no—
150#95—942no—
151#95—942no—
152#95—942no—
153C2×Q8—22no—
154C22×C4—22no—
155C22×C4—942no—
156C22×C4—942no—
157C22×C4D4×D4712yes—
158C22×C4C2×C22⋊D4711yescharacteristic
159M16—24no—
160C2×C8—22no—
161C2×C8—1882no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—1884no—
169C4⋊C4C22⋊Dic4714yes—
170C22⋊C4—41no—
171C42—41no—
172C42—1884no—
173C42—1884no—
174C42#1062535212yes—
175C42C22×Dic4711yescharacteristic
176C42C2×C22⋊D4711yescharacteristic
177C23#1309399211yescharacteristic
178Q8—82no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—944no—
186C2×C4—1882no—
187C2×C4—1882no—
188C2×C4—1882no—
189C2×C4—1882no—
190C2×C4—1882no—
191C2×C4C2×C22⋊Dic4711yescharacteristic
192C2×C4D4×Dic4712yes—
193C2×C4#1309399612yes—
194C2×C4#1309400511yescharacteristic
195C8—24no—
196C8—3762no—
197C22—22no—
198C22#1568900311yescharacteristic
199C22#1568930512yes—
200C4—21no—
201C4—21no—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—82no—
206C4—3762no—
207C4—1884no—
208C2—22no—
209C2#2024378211yescharacteristic
210C2#2024389112yes—
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3166C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#3166D47
hypocenterC47#9440
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
16. C47×#501C4semidirect
113. #501Dic47semidirect
125. C47#9440semidirect

Maximal subgroups

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

subgroupindex
#202464942
#202420152
#202419002
#202412822
C47×#31662
#944047