Finite Group Databasebeta

← #26115659

Subgroups — #26115659

Order 12032.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#20253711C211yesmaximal, characteristic
2#20241951C212yesmaximal, complemented
3#20241947C212yesmaximal
4#20241671C211yesmaximal, characteristic, complemented
5C47×#1835C211yesmaximal, characteristic, complemented
6#15689680C2211yescharacteristic
7#15689673C2211yescharacteristic
8#15689478—21no—
9#15689478—21no—
10#15689474C412yescomplemented
11#15689470C412yescomplemented
12#15689176C2212yes—
13#15689120—22no—
14#15688859—24no—
15#15688856—24no—
16#15688844—21no—
17C8×C47⋊C8—21no—
18C2×C4×C8×C47C2211yescharacteristic
19C47×#382C2212yes—
20C47×#340—22no—
21C2×#10625350—21no—
22#13094024C2×C411yescharacteristic
23#13094024C2×C412yes—
24#13094021C2×C411yescharacteristic
25#13094016—21no—
26#13094012—21no—
27#13094003—22no—
28#13093998—22no—
29C4×Dic94—22no—
30C4×D188—22no—
31#13093983—21no—
32C4⋊(C47⋊C8)—22no—
33#13093949—22no—
34#13093949—22no—
35#13093946—22no—
36#13093946—22no—
37C22⋊(C47⋊C8)D414yes—
38C4×C47⋊C8—22no—
39C22×C8×C47D412yes—
40C2×C4⋊C4×C47—22no—
41C2×C42×C47C2311yescharacteristic
42C4×C8×C47C2×C412yes—
43C4×C8×C47D411yescharacteristic
44C4×C8×C47D411yescharacteristic
45#95×C47—22no—
46#95×C47—22no—
47C22×C4×C47—42no—
48C22×C4×C47C2×D412yes—
49C22×C4×C47C2×D411yescharacteristic
50C2×C22⋊D47—41no—
51#10625350—44no—
52C2×D188—21no—
53C2×D188—22no—
54C2×C4×D47—41no—
55C2×Dic94—21no—
56C2×Dic94—22no—
57C2×C8×C47—24no—
58C2×C8×C47C22⋊C412yes—
59C2×C8×C47C22⋊C412yes—
60C4⋊C4×C47—44no—
61C42×C47C22×C411yescharacteristic
62C42×C47C2×D411yescharacteristic
63C42×C47#4112yes—
64C22⋊Dic47—41no—
65(C2×D47)⋊C4—42no—
66C4⋊Dic47—22no—
67C4⋊Dic47—22no—
68C4⋊Dic47—22no—
69Dic47⋊C4—42no—
70C4×Dic47—41no—
71C2×C47⋊C8—24no—
72C23×C47#12711yescharacteristic
73C22×D47—41no—
74C2×C4×C47—42no—
75C2×C4×C47—84no—
76C2×C4×C47—82no—
77C2×C4×C47C2×C22⋊C411yescharacteristic
78C2×C4×C47C4×D412yes—
79C2×C4×C47#12112yes—
80C2×C4×C47#12411yescharacteristic
81C2×C4×C47#13514yes—
82C22⋊D47—82no—
83C2×Dic47—42no—
84C2×Dic47—41no—
85D188—42no—
86D188—82no—
87C4×D47—82no—
88Dic94—42no—
89Dic94—82no—
90C8×C47—22no—
91C8×C47—22no—
92C47⋊C8—44no—
93#9757—471nomaximal, Sylow
94C22×C47—22no—
95C22×C47#39011yescharacteristic
96C22×C47#49512yes—
97C2×D47—81no—
98C2×D47—82no—
99C4×C47—22no—
100C4×C47—22no—
101C4×C47—162no—
102C4×C47#41612yes—
103C4×C47#48812yes—
104Dic47—81no—
105Dic47—82no—
106#3131—471no—
107#1835D4711yescharacteristic, complemented
108#1518—472no—
109#1517—472no—
110#1450—471no—
111C2×C47—22no—
112C2×C47#195811yescharacteristic
113C2×C47#197612yes—
114D47—161no—
115#533—471no—
116#532—471no—
117#488—941no—
118#488—941no—
119#487—472no—
120#486—472no—
121#431—472no—
122#416—942no—
123C2×C4×C8C2×D4711yescharacteristic
124#382C2×D4712yes—
125#340—22no—
126#328—944no—
127#327—944no—
128#322—941no—
129C82—941no—
130C47#975711yescharacteristic, complemented, Sylow
131C2×#41—941no—
132C22×C8C22⋊D4712yes—
133C4⋊Q8—471no—
134C4⋊Q8—472no—
135#127—471no—
136#125—941no—
137#124—941no—
138C22⋊Q8—942no—
139#121—942no—
140C4×Q8—942no—
141C4×D4—942no—
142#117—941no—
143C2×C4⋊C4—22no—
144C2×C42C22×D4711yescharacteristic
145C4⋊C8—942no—
146#103—942no—
147#103—942no—
148#102—942no—
149#102—942no—
150C22⋊C8—474no—
151C4×C8—942no—
152C4×C8C4×D4712yes—
153C4×C8D18811yescharacteristic
154C4×C8D18811yescharacteristic
155#95—22no—
156#95—22no—
157#41—1884no—
158C2×Q8—941no—
159C2×Q8—942no—
160C2×D4—941no—
161C2×D4—942no—
162C2×D4—1881no—
163C22×C4—42no—
164C22×C4—1881no—
165C22×C4D4×D4712yes—
166C22×C4C2×C22⋊D4711yescharacteristic
167C2×C8—24no—
168C2×C8—944no—
169C2×C8(C2×D47)⋊C412yes—
170C2×C8(C2×D47)⋊C412yes—
171C4⋊C4—44no—
172C4⋊C4—942no—
173C4⋊C4—942no—
174C4⋊C4—942no—
175C4⋊C4—1882no—
176C22⋊C4—1881no—
177C22⋊C4—1882no—
178C42—1881no—
179C42C2×C4×D4711yescharacteristic
180C42C2×D18811yescharacteristic
181C42#1062535412yes—
182C23—1881no—
183C23#1309402011yescharacteristic
184Q8—3762no—
185Q8—1882no—
186D4—3762no—
187D4—3762no—
188D4—1882no—
189C2×C4—42no—
190C2×C4—84no—
191C2×C4—82no—
192C2×C4—3762no—
193C2×C4—1882no—
194C2×C4—1881no—
195C2×C4C2×(C2×D47)⋊C411yescharacteristic
196C2×C4#1309398612yes—
197C2×C4#1309399912yes—
198C2×C4#1309401311yescharacteristic
199C2×C4#1309404414yes—
200C8—22no—
201C8—22no—
202C8—1884no—
203C22—22no—
204C22—3762no—
205C22—3761no—
206C22#1568902411yescharacteristic
207C22#1568950512yes—
208C4—22no—
209C4—22no—
210C4—162no—
211C4—3762no—
212C4—3761no—
213C4#1568911712yes—
214C4#1568948012yes—
215C2—22no—
216C2—7521no—
217C2#2024419711yescharacteristic
218C2#2024428412yes—
subgroupG/H
centerC22#15689024
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1835C2
socleC22×C47#390
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1835D47
hypocenterC47#9757
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1835

Complements

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

NH ≅ G/Ntype
2. #20241951C2semidirect
4. #20241671C2semidirect
5. C47×#1835C2semidirect
10. #15689474C4semidirect
11. #15689470C4semidirect
107. #1835D47semidirect
130. C47#9757semidirect

Maximal subgroups

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

subgroupindex
#202537112
#202419512
#202419472
#202416712
C47×#18352
#975747