Finite Group Databasebeta

← #26113912

Subgroups — #26113912

Order 12032.

456 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 99 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#20253791C211yesmaximal, characteristic
2#20242015C211yesmaximal, characteristic, complemented
3#20241895C212yesmaximal
4#20241281C212yesmaximal, complemented
5C47×#2371C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689493C412yescomplemented
8#15689489C412yescomplemented
9#15689480—22no—
10#15689180C2212yes—
11#15689122—22no—
12C4×#10625329C2211yescharacteristic
13#15688865—22no—
14#15688856—22no—
15#15688856—22no—
16C2×#125×C47C2211yescharacteristic
17C47×#387—22no—
18C47×#387—22no—
19C4×C4⋊C4×C47—22no—
20C47×#376C2212yes—
21C2×#10625350—21no—
22C2×#10625329—21no—
23#13094024C2×C411yescharacteristic
24#13094021C2×C411yescharacteristic
25#13094012—21no—
26#13093998—22no—
27C4×D188—22no—
28C4⋊(C47⋊C8)—22no—
29#13093947—24no—
30#13093944—24no—
31C22⋊(C47⋊C8)—22no—
32#13093935—22no—
33#13093935C2×C412yes—
34C4×C47⋊C8—21no—
35#125×C47—22no—
36#125×C47C2×C412yes—
37#125×C47D412yes—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47D412yes—
41C2×C42×C47—22no—
42C2×C42×C47—22no—
43C2×C42×C47C2311yescharacteristic
44#95×C47—22no—
45#95×C47—22no—
46#95×C47—22no—
47#95×C47D412yes—
48#95×C47D412yes—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47C2×D411yescharacteristic
53C22×C4×C47C2×D412yes—
54C2×C22⋊D47—41no—
55#10625350—44no—
56C2×D188—21no—
57C2×D188—22no—
58C2×C4×D47—41no—
59C2×Dic94—21no—
60C4⋊C4×C47—24no—
61C4⋊C4×C47—24no—
62C4⋊C4×C47—22no—
63C4⋊C4×C47—44no—
64C4⋊C4×C47C22⋊C414yes—
65C42×C47—44no—
66C42×C47—44no—
67C42×C47C22×C411yescharacteristic
68C42×C47C2×D411yescharacteristic
69C42×C47#4112yes—
70(C2×D47)⋊C4—42no—
71C4⋊Dic47—22no—
72#10625329—42no—
73C2×C47⋊C8—22no—
74C2×C47⋊C8—42no—
75C23×C47#12011yescharacteristic
76C22×D47—41no—
77C2×C4×C47—22no—
78C2×C4×C47—24no—
79C2×C4×C47—24no—
80C2×C4×C47—42no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47C2×C22⋊C411yescharacteristic
89C2×C4×C47C4×D412yes—
90C2×C4×C47#12112yes—
91C2×C4×C47#12311yescharacteristic
92C22⋊D47—82no—
93C2×Dic47—41no—
94D188—42no—
95D188—82no—
96C4×D47—82no—
97Dic94—42no—
98C47⋊C8—42no—
99C47⋊C8—82no—
100#9442—471nomaximal, Sylow
101C22×C47—22no—
102C22×C47#38611yescharacteristic
103C22×C47#45412yes—
104C2×D47—81no—
105C2×D47—82no—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—22no—
109C4×C47—22no—
110C4×C47—44no—
111C4×C47—44no—
112C4×C47—82no—
113Dic47—81no—
114#3142—471no—
115#2371D4711yescharacteristic, complemented
116#1534—471no—
117#1505—472no—
118#1300—472no—
119C2×C47—22no—
120C2×C47#188811yescharacteristic
121C2×C47#190612yes—
122D47—161no—
123#532—471no—
124C2×#125C2×D4711yescharacteristic
125#491—472no—
126#490—472no—
127#488—942no—
128#432—472no—
129#417—942no—
130C4×M16—471no—
131#387—22no—
132#387—22no—
133C4×C4⋊C4—22no—
134#376C2×D4712yes—
135#330—942no—
136#327—942no—
137#327—942no—
138C47#944211yescharacteristic, complemented, Sylow
139C2×#41—941no—
140C2×M16—941no—
141C4⋊Q8—471no—
142#127—471no—
143#125—22no—
144#125C4×D4712yes—
145#125D18812yes—
146#124—941no—
147#121—942no—
148C4×D4—942no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊D4712yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C22×D4711yescharacteristic
155C4⋊C8—942no—
156#103—944no—
157#102—944no—
158C22⋊C8—942no—
159#97—472no—
160#97—942no—
161C4×C8—941no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95C22⋊D4712yes—
166#95C22⋊D4712yes—
167#41—1884no—
168C2×Q8—941no—
169C2×D4—941no—
170C2×D4—942no—
171C2×D4—1881no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—1881no—
176C22×C4C2×C22⋊D4711yescharacteristic
177C22×C4C2×C22⋊D4712yes—
178M16—1882no—
179C2×C8—942no—
180C2×C8—1882no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4—942no—
186C4⋊C4(C2×D47)⋊C414yes—
187C22⋊C4—1882no—
188C42—44no—
189C42—44no—
190C42C2×C4×D4711yescharacteristic
191C42C2×D18811yescharacteristic
192C42#1062535012yes—
193C23—1881no—
194C23#1309399311yescharacteristic
195Q8—1882no—
196D4—3762no—
197D4—3762no—
198D4—1882no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—3762no—
211C2×C4—1881no—
212C2×C4C2×(C2×D47)⋊C411yescharacteristic
213C2×C4C4×C22⋊D4712yes—
214C2×C4#1309399812yes—
215C2×C4#1309400711yescharacteristic
216C8—3762no—
217C8—1882no—
218C22—22no—
219C22—3761no—
220C22—3762no—
221C22#1568900211yescharacteristic
222C22#1568929612yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—3761no—
231C2—22no—
232C2—7521no—
233C2#2024379111yescharacteristic
234C2#2024388912yes—
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2371C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2371D47
hypocenterC47#9442
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
2. #20242015C2semidirect
4. #20241281C2semidirect
5. C47×#2371C2semidirect
7. #15689493C4semidirect
8. #15689489C4semidirect
115. #2371D47semidirect
138. C47#9442semidirect

Maximal subgroups

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

subgroupindex
#202537912
#202420152
#202418952
#202412812
C47×#23712
#944247