Finite Group Databasebeta

← #26113816

Subgroups — #26113816

Order 12032.

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

normal subgroups only

This group has 534 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#20246565C211yesmaximal, characteristic
2#20242102C211yesmaximal, characteristic
3#20241939C212yesmaximal, complemented
4#20241253C212yesmaximal, complemented
5C47×#3138C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689003—24no—
8#15689003—22no—
9#15688991C2212yes—
10#15688867—22no—
11#15688860—24no—
12C47×#533C2211yescharacteristic
13C2×#127×C47C2211yescharacteristic
14C47×#490—22no—
15C47×#486C414yescomplemented
16C47×#418—22no—
17C4×M16×C47C2211yescharacteristic
18C23×Dic47—22no—
19C2×C22⋊Dic47—24no—
20C2×C22⋊Dic47—22no—
21C4⋊(C47⋊C8)—22no—
22C22⋊(C47⋊C8)—22no—
23#13093935—22no—
24C47⋊#95—22no—
25C47⋊#95—22no—
26C47⋊#95D414yes—
27C22×D4×C47—22no—
28C22×D4×C47D412yes—
29C2×M16×C47—21no—
30C4⋊Q8×C47C2×C412yescomplemented
31#127×C47—22no—
32#127×C47C2×C412yes—
33#127×C47D412yes—
34#125×C47—21no—
35C22⋊Q8×C47—22no—
36C4×Q8×C47—22no—
37#117×C47—21no—
38C2×C42×C47C2311yescharacteristic
39#102×C47—28no—
40#97×C47—21no—
41C4×C8×C47C2×C412yescomplemented
42C24×C47—22no—
43C2×Q8×C47—22no—
44C2×D4×C47—24no—
45C2×D4×C47—24no—
46C2×D4×C47—22no—
47C2×D4×C47—42no—
48C2×D4×C47—42no—
49C2×D4×C47—82no—
50C2×D4×C47—82no—
51C2×D4×C47C22⋊C414yes—
52C22×C4×C47C2×D411yescharacteristic
53C22×C4×C47C2×D412yes—
54C22×Dic47—24no—
55C22×Dic47—84no—
56C22×Dic47—82no—
57M16×C47—42no—
58C2×C8×C47—22no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—22no—
61C4⋊C4×C47—42no—
62C22⋊C4×C47—41no—
63C42×C47—41no—
64C42×C47C22×C411yescharacteristic
65C42×C47C2×D411yescharacteristic
66C42×C47#4112yes—
67C22⋊Dic47—84no—
68C22⋊Dic47—82no—
69C2×C47⋊C8—42no—
70C23×C47—24no—
71C23×C47—44no—
72C23×C47—44no—
73C23×C47—42no—
74C23×C47—42no—
75C23×C47#12011yescharacteristic
76Q8×C47—82no—
77D4×C47—44no—
78D4×C47—82no—
79D4×C47—84no—
80D4×C47—84no—
81C2×C4×C47—22no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47C2×C22⋊C411yescharacteristic
85C2×C4×C47C4×D412yes—
86C2×C4×C47#12112yes—
87C2×C4×C47#12311yescharacteristic
88C2×Dic47—48no—
89C2×Dic47—44no—
90C2×Dic47—88no—
91C8×C47—42no—
92C47⋊C8—82no—
93#9425—471nomaximal, Sylow
94C22×C47—22no—
95C22×C47—44no—
96C22×C47—48no—
97C22×C47—82no—
98C22×C47—82no—
99C22×C47—84no—
100C22×C47#38611yescharacteristic
101C22×C47#45512yes—
102C4×C47—21no—
103C4×C47—21no—
104C4×C47—22no—
105C4×C47—22no—
106C4×C47—82no—
107Dic47—84no—
108#3138D4711yescharacteristic
109#2380—471no—
110#1555—471no—
111#1515—472no—
112#1288—472no—
113C2×C47—22no—
114C2×C47—84no—
115C2×C47#188511yescharacteristic
116C2×C47#190412yes—
117#533C2×D4711yescharacteristic
118C2×#127C2×D4711yescharacteristic
119#490—22no—
120#486Dic4714yescomplemented
121#432—472no—
122#418—22no—
123C4×M16C2×D4711yescharacteristic
124#386—944no—
125#386—942no—
126#383—472no—
127#331—942no—
128#328—944no—
129C47#942511yescharacteristic, complemented, Sylow
130C22×D4—22no—
131C22×D4C22⋊D4712yes—
132C23×C4—942no—
133C2×M16—21no—
134C4⋊Q8C2×Dic4712yescomplemented
135#127—22no—
136#127C2×Dic4712yes—
137#127C22⋊D4712yes—
138#125—21no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117—21no—
142C2×C22⋊C4—944no—
143C2×C22⋊C4—942no—
144C2×C42C22×D4711yescharacteristic
145C4⋊C8—942no—
146#102—28no—
147C22⋊C8—942no—
148#97—21no—
149#97—942no—
150C4×C8C2×Dic4712yescomplemented
151#95—474no—
152#95—942no—
153#95—942no—
154C24—22no—
155C2×Q8—22no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4C22⋊Dic4714yes—
164C22×C4—944no—
165C22×C4—3764no—
166C22×C4—3762no—
167C22×C4D4×D4712yes—
168C22×C4C2×C22⋊D4711yescharacteristic
169M16—42no—
170C2×C8—22no—
171C2×C8—1882no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C22⋊C4—41no—
176C22⋊C4—3764no—
177C22⋊C4—3762no—
178C42—41no—
179C42#1062535212yes—
180C42C22×Dic4711yescharacteristic
181C42C2×C22⋊D4711yescharacteristic
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1309399211yescharacteristic
188Q8—82no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—3768no—
197C2×C4—1888no—
198C2×C4—1884no—
199C2×C4C2×C22⋊Dic4711yescharacteristic
200C2×C4D4×Dic4712yes—
201C2×C4#1309399612yes—
202C2×C4#1309400511yescharacteristic
203C8—42no—
204C8—3762no—
205C22—22no—
206C22—44no—
207C22—48no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1568900311yescharacteristic
212C22#1568930012yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—82no—
218C4—3764no—
219C2—22no—
220C2—84no—
221C2#2024377711yescharacteristic
222C2#2024387712yes—
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3138C2
socleC22×C47#386
prime radicalC2×#127×C47C22
cosocleC2×C42×C47C23
hypercenter#3138D47
hypocenterC47#9425
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
3. #20241939C2semidirect
4. #20241253C2semidirect
15. C47×#486C4semidirect
30. C4⋊Q8×C47C2×C4semidirect
41. C4×C8×C47C2×C4semidirect
120. #486Dic47semidirect
129. C47#9425semidirect
134. C4⋊Q8C2×Dic47semidirect
150. C4×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465652
#202421022
#202419392
#202412532
C47×#31382
#942547