Finite Group Databasebeta

← #26113820

Subgroups — #26113820

Order 12032.

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

normal subgroups only

This group has 422 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#20253768C211yesmaximal, characteristic
2#20242114C211yesmaximal, characteristic
3#20241938C212yesmaximal
4#20241252C212yesmaximal
5C47×#2381C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7#15689491—22no—
8#15689473C414yescomplemented
9#15689180C2212yes—
10#15689132—22no—
11C4×#10625329C2211yescharacteristic
12#15688872—22no—
13#15688859—24no—
14C2×C4⋊Q8×C47C2211yescharacteristic
15C47×#391—22no—
16C47×#387—24no—
17C47×#383C2212yes—
18C2×#10625329—21no—
19#13094024C2×C412yes—
20#13094016—21no—
21#13094003—22no—
22C4×Dic94—22no—
23#13093983—21no—
24C4⋊(C47⋊C8)—22no—
25#13093948—28no—
26C22⋊(C47⋊C8)—22no—
27#13093935—21no—
28#13093935—22no—
29C4×C47⋊C8C2×C412yes—
30C22×Q8×C47D412yes—
31C4⋊Q8×C47—22no—
32C4⋊Q8×C47C2×C412yes—
33C4⋊Q8×C47D412yes—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—22no—
36C2×C42×C47—22no—
37C2×C42×C47C2311yescharacteristic
38#95×C47—24no—
39#95×C47—22no—
40#95×C47—22no—
41#95×C47D414yes—
42C2×Q8×C47—22no—
43C2×Q8×C47—42no—
44C2×Q8×C47—42no—
45C2×Q8×C47C22⋊C414yes—
46C22×C4×C47—22no—
47C22×C4×C47—24no—
48C22×C4×C47C2×D411yescharacteristic
49C22×C4×C47C2×D412yes—
50C2×Dic94—22no—
51C4⋊C4×C47—24no—
52C4⋊C4×C47—24no—
53C4⋊C4×C47—82no—
54C42×C47—82no—
55C42×C47C22×C411yescharacteristic
56C42×C47C2×D411yescharacteristic
57C42×C47#4112yes—
58C22⋊Dic47—41no—
59C4⋊Dic47—22no—
60C4⋊Dic47—22no—
61Dic47⋊C4—42no—
62C4×Dic47—41no—
63#10625329—42no—
64C2×C47⋊C8—22no—
65C2×C47⋊C8—42no—
66C23×C47#12011yescharacteristic
67Q8×C47—44no—
68Q8×C47—82no—
69C2×C4×C47—22no—
70C2×C4×C47—24no—
71C2×C4×C47—42no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—48no—
75C2×C4×C47—44no—
76C2×C4×C47C2×C22⋊C411yescharacteristic
77C2×C4×C47C4×D412yes—
78C2×C4×C47#12112yes—
79C2×C4×C47#12311yescharacteristic
80C2×Dic47—42no—
81Dic94—82no—
82C47⋊C8—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—84no—
94Dic47—82no—
95#3139—471no—
96#2381D4711yescharacteristic
97#1558—471no—
98#1515—472no—
99#1288—472no—
100C2×C47—22no—
101C2×C47#188511yescharacteristic
102C2×C47#190512yes—
103#533—471no—
104C2×C4⋊Q8C2×D4711yescharacteristic
105#491—942no—
106#487—474no—
107#432—472no—
108#419—942no—
109C4×M16—471no—
110#391—22no—
111#387—24no—
112#383C2×D4712yes—
113#333—942no—
114#328—944no—
115C47#942611yescharacteristic, complemented, Sylow
116C22×Q8C22⋊D4712yes—
117C2×M16—941no—
118C4⋊Q8—22no—
119C4⋊Q8—472no—
120C4⋊Q8C4×D4712yes—
121C4⋊Q8D18812yes—
122#125—941no—
123C22⋊Q8—942no—
124C4×Q8—942no—
125#117—941no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C42—22no—
129C2×C42C22×D4711yescharacteristic
130C4⋊C8—942no—
131#103—948no—
132C22⋊C8—942no—
133#97—941no—
134#97—942no—
135C4×C8—472no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95C22⋊D4714yes—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—942no—
144C2×Q8(C2×D47)⋊C414yes—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4C2×C22⋊D4711yescharacteristic
148C22×C4C2×C22⋊D4712yes—
149M16—1882no—
150C2×C8—942no—
151C2×C8—1882no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—82no—
155C4⋊C4—942no—
156C4⋊C4—942no—
157C4⋊C4—1882no—
158C22⋊C4—1881no—
159C42—82no—
160C42—1881no—
161C42C2×C4×D4711yescharacteristic
162C42C2×D18811yescharacteristic
163C42#1062535012yes—
164C23#1309399311yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—3762no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—48no—
174C2×C4—44no—
175C2×C4—1882no—
176C2×C4C2×(C2×D47)⋊C411yescharacteristic
177C2×C4C4×C22⋊D4712yes—
178C2×C4#1309399812yes—
179C2×C4#1309400711yescharacteristic
180C8—3762no—
181C8—1882no—
182C22—22no—
183C22#1568900211yescharacteristic
184C22#1568930112yes—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—84no—
191C4—3762no—
192C2—22no—
193C2#2024377611yescharacteristic
194C2#2024388212yes—
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2381C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#2381D47
hypocenterC47#9426
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
8. #15689473C4semidirect
115. C47#9426semidirect

Maximal subgroups

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

subgroupindex
#202537682
#202421142
#202419382
#202412522
C47×#23812
#942647