Finite Group Databasebeta

← #26113821

Subgroups — #26113821

Order 12032.

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

normal subgroups only

This group has 422 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#20246569C211yesmaximal, characteristic
2#20242114C211yesmaximal, characteristic
3#20241939C212yesmaximal
4#20241253C212yesmaximal
5C47×#3139C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689030—22no—
8#15689009—24no—
9#15688991C2212yes—
10#15688872—22no—
11#15688860—24no—
12C47×#533C2211yescharacteristic
13C2×C4⋊Q8×C47C2211yescharacteristic
14C47×#491—22no—
15C47×#487C414yescomplemented
16C47×#419—22no—
17C4×M16×C47C2211yescharacteristic
18C2×C4⋊Dic47—22no—
19C2×C4×Dic47—22no—
20C4⋊(C47⋊C8)—22no—
21C22⋊(C47⋊C8)—22no—
22#13093935—22no—
23C47⋊#95—24no—
24C47⋊#95—22no—
25C47⋊#95—22no—
26C47⋊#95D414yes—
27C22×Q8×C47D412yes—
28C2×M16×C47—21no—
29C4⋊Q8×C47—22no—
30C4⋊Q8×C47C2×C412yes—
31C4⋊Q8×C47C2×C412yes—
32C4⋊Q8×C47D412yes—
33#125×C47—21no—
34C22⋊Q8×C47—22no—
35C4×Q8×C47—22no—
36#117×C47—21no—
37C2×C4⋊C4×C47—22no—
38C2×C42×C47C2311yescharacteristic
39#103×C47—28no—
40#97×C47—21no—
41C4×C8×C47C2×C412yes—
42C2×Q8×C47—22no—
43C2×Q8×C47—22no—
44C2×Q8×C47—42no—
45C2×Q8×C47—42no—
46C2×Q8×C47C22⋊C414yes—
47C22×C4×C47—22no—
48C22×C4×C47C2×D411yescharacteristic
49C22×C4×C47C2×D412yes—
50C22×Dic47—24no—
51M16×C47—42no—
52C2×C8×C47—22no—
53C4⋊C4×C47—24no—
54C4⋊C4×C47—24no—
55C4⋊C4×C47—22no—
56C4⋊C4×C47—22no—
57C4⋊C4×C47—42no—
58C22⋊C4×C47—41no—
59C42×C47—41no—
60C42×C47C22×C411yescharacteristic
61C42×C47C2×D411yescharacteristic
62C42×C47#4112yes—
63C4⋊Dic47—82no—
64C4×Dic47—82no—
65C2×C47⋊C8—42no—
66C23×C47#12011yescharacteristic
67Q8×C47—44no—
68Q8×C47—82no—
69Q8×C47—82no—
70C2×C4×C47—22no—
71C2×C4×C47—24no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47—42no—
76C2×C4×C47C2×C22⋊C411yescharacteristic
77C2×C4×C47C4×D412yes—
78C2×C4×C47#12112yes—
79C2×C4×C47#12311yescharacteristic
80C2×Dic47—48no—
81C2×Dic47—44no—
82C8×C47—42no—
83C47⋊C8—82no—
84#9426—471nomaximal, Sylow
85C22×C47—22no—
86C22×C47#38611yescharacteristic
87C22×C47#45512yes—
88C4×C47—21no—
89C4×C47—21no—
90C4×C47—22no—
91C4×C47—22no—
92C4×C47—44no—
93C4×C47—82no—
94Dic47—84no—
95#3139D4711yescharacteristic
96#2381—471no—
97#1558—471no—
98#1515—472no—
99#1288—472no—
100C2×C47—22no—
101C2×C47#188511yescharacteristic
102C2×C47#190512yes—
103#533C2×D4711yescharacteristic
104C2×C4⋊Q8C2×D4711yescharacteristic
105#491—22no—
106#487Dic4714yescomplemented
107#432—472no—
108#419—22no—
109C4×M16C2×D4711yescharacteristic
110#391—942no—
111#387—944no—
112#383—472no—
113#333—942no—
114#328—944no—
115C47#942611yescharacteristic, complemented, Sylow
116C22×Q8C22⋊D4712yes—
117C2×M16—21no—
118C4⋊Q8—22no—
119C4⋊Q8C2×Dic4712yes—
120C4⋊Q8C2×Dic4712yes—
121C4⋊Q8C22⋊D4712yes—
122#125—21no—
123C22⋊Q8—22no—
124C4×Q8—22no—
125#117—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—942no—
128C2×C42—942no—
129C2×C42C22×D4711yescharacteristic
130C4⋊C8—942no—
131#103—28no—
132C22⋊C8—942no—
133#97—21no—
134#97—942no—
135C4×C8C2×Dic4712yes—
136#95—474no—
137#95—944no—
138#95—942no—
139#95—942no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C22⋊Dic4714yes—
145C22×C4—22no—
146C22×C4—944no—
147C22×C4D4×D4712yes—
148C22×C4C2×C22⋊D4711yescharacteristic
149M16—42no—
150C2×C8—22no—
151C2×C8—1882no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—3762no—
158C22⋊C4—41no—
159C42—41no—
160C42—3762no—
161C42#1062535212yes—
162C42C22×Dic4711yescharacteristic
163C42C2×C22⋊D4711yescharacteristic
164C23#1309399211yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—82no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—1888no—
175C2×C4—1884no—
176C2×C4C2×C22⋊Dic4711yescharacteristic
177C2×C4D4×Dic4712yes—
178C2×C4#1309399612yes—
179C2×C4#1309400511yescharacteristic
180C8—42no—
181C8—3762no—
182C22—22no—
183C22#1568900311yescharacteristic
184C22#1568930012yes—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—82no—
191C4—3764no—
192C2—22no—
193C2#2024377711yescharacteristic
194C2#2024388412yes—
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3139C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#3139D47
hypocenterC47#9426
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
15. C47×#487C4semidirect
106. #487Dic47semidirect
115. C47#9426semidirect

Maximal subgroups

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

subgroupindex
#202465692
#202421142
#202419392
#202412532
C47×#31392
#942647