Finite Group Databasebeta

← #26225735

Subgroups — #26225735

Order 12032.

990 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 315 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#20245991C212yesmaximal
2#20244374C218yesmaximal
3#20243465C214yesmaximal
4C2×C47×#544C211yesmaximal, characteristic
5#15689009—28no—
6#15688986C2214yes—
7#15688983C2214yes—
8C4×C22⋊Dic47C2214yes—
9#15688890C22116yes—
10C47×#544—28no—
11C2×C4⋊Q8×C47C414yescomplemented
12C2×#125×C47C414yescomplemented
13C2×C22⋊Q8×C47C2214yes—
14C2×#117×C47C2212yes—
15C22×C4⋊C4×C47C2211yescharacteristic
16C2×C4⋊Dic47—24no—
17C2×C22⋊Dic47C2318yes—
18C2×C4×Dic47—24no—
19C47⋊#95—24no—
20C47⋊#95—24no—
21C47⋊#95—28no—
22C22×Q8×C47C2×C414yes—
23C23×C4×C47C2311yescharacteristic
24C23×C4×C47C2312yes—
25C4⋊Q8×C47—216no—
26#125×C47—216no—
27C22⋊Q8×C47—28no—
28C22⋊Q8×C47—28no—
29#117×C47—28no—
30C2×C4⋊C4×C47C2×C412yes—
31C2×C4⋊C4×C47C2×C412yes—
32C2×C4⋊C4×C47C2×C414yes—
33C2×C4⋊C4×C47C2×C418yes—
34C2×C4⋊C4×C47C2×C414yes—
35C2×C4⋊C4×C47D418yes—
36C2×C22⋊C4×C47C2314yes—
37C2×C42×C47C2×C414yes—
38C24×C47C2411yescharacteristic
39C2×Q8×C47—28no—
40C2×Q8×C47—84no—
41C22×C4×C47—24no—
42C22×C4×C47—82no—
43C22×C4×C47C22×C412yes—
44C22×C4×C47C22×C414yes—
45C22×C4×C47C22×C414yes—
46C22×C4×C47C22×C414yes—
47C22×C4×C47C2×D414yes—
48C22×C4×C47C2×D414yes—
49C22×C4×C47C2×D414yes—
50C22×Dic47—28no—
51C4⋊C4×C47—24no—
52C4⋊C4×C47—24no—
53C4⋊C4×C47—28no—
54C4⋊C4×C47—216no—
55C4⋊C4×C47C22⋊C4116yes—
56C22⋊C4×C47—28no—
57C42×C47—28no—
58C22⋊Dic47—216no—
59C4⋊Dic47—84no—
60C4×Dic47—84no—
61C23×C47—41no—
62C23×C47—41no—
63C23×C47C23×C411yescharacteristic
64C23×C47C22×D412yes—
65C23×C472−1+414yes—
66Q8×C47—88no—
67C2×C4×C47—24no—
68C2×C4×C47—28no—
69C2×C4×C47—24no—
70C2×C4×C47—24no—
71C2×C4×C47—44no—
72C2×C4×C47—44no—
73C2×C4×C47—44no—
74C2×C4×C47—44no—
75C2×C4×C47—44no—
76C2×C4×C47C2×C22⋊C414yes—
77C2×C4×C47C2×C22⋊C418yes—
78C2×Dic47—48no—
79C2×Dic47—48no—
80C2×Dic47—48no—
81#21828—471nomaximal, Sylow
82C22×C47—22no—
83C22×C47—21no—
84C22×C47—22no—
85C22×C47—21no—
86C22×C47—44no—
87C22×C47C22×C22⋊C411yescharacteristic
88C22×C47#51912yes—
89C22×C47#53614yes—
90C4×C47—24no—
91C4×C47—28no—
92C4×C47—44no—
93C4×C47—44no—
94Dic47—88no—
95C2×#544D4711yescharacteristic
96#2313—472no—
97#1996—478no—
98#1820—474no—
99C2×C47—24no—
100C2×C47—22no—
101C2×C47#231011yescharacteristic
102C2×C47#288311yescharacteristic
103C2×C47#288411yescharacteristic
104#544—28no—
105C2×C4⋊Q8Dic4714yescomplemented
106C2×#125Dic4714yescomplemented
107C2×C22⋊Q8C2×D4714yes—
108C2×#117C2×D4712yes—
109C22×C4⋊C4C2×D4711yescharacteristic
110#387—948no—
111#381—474no—
112#380—474no—
113C4×C22⋊C4—474no—
114#342—4716no—
115C47#2182811yescharacteristic, complemented, Sylow
116C22×Q8C2×Dic4714yes—
117C23×C4C22×D4711yescharacteristic
118C23×C4C22×D4712yes—
119C4⋊Q8—216no—
120#125—216no—
121C22⋊Q8—28no—
122C22⋊Q8—28no—
123#117—28no—
124C2×C4⋊C4—944no—
125C2×C4⋊C4C2×Dic4712yes—
126C2×C4⋊C4C2×Dic4712yes—
127C2×C4⋊C4C2×Dic4714yes—
128C2×C4⋊C4C2×Dic4718yes—
129C2×C4⋊C4C2×Dic4714yes—
130C2×C4⋊C4C22⋊D4718yes—
131C2×C22⋊C4—478no—
132C2×C22⋊C4C22×D4714yes—
133C2×C42—944no—
134C2×C42C2×Dic4714yes—
135#95—944no—
136#95—944no—
137#95—948no—
138C24C23×D4711yescharacteristic
139C2×Q8—28no—
140C2×Q8—84no—
141C22×C4—24no—
142C22×C4—82no—
143C22×C4—948no—
144C22×C4C22×Dic4712yes—
145C22×C4C22×Dic4714yes—
146C22×C4C22×Dic4714yes—
147C22×C4C22×Dic4714yes—
148C22×C4C2×C22⋊D4714yes—
149C22×C4C2×C22⋊D4714yes—
150C22×C4C2×C22⋊D4714yes—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—28no—
154C4⋊C4—216no—
155C4⋊C4—3764no—
156C4⋊C4C22⋊Dic47116yes—
157C22⋊C4—28no—
158C22⋊C4—9416no—
159C42—28no—
160C42—3764no—
161C23—41no—
162C23—41no—
163C23C23×Dic4711yescharacteristic
164C23C22×C22⋊D4712yes—
165C23#1309407214yes—
166Q8—88no—
167C2×C4—24no—
168C2×C4—28no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—1888no—
177C2×C4—1888no—
178C2×C4—1888no—
179C2×C4C2×C22⋊Dic4714yes—
180C2×C4C2×C22⋊Dic4718yes—
181C22—21no—
182C22—22no—
183C22—21no—
184C22—22no—
185C22—44no—
186C22C22×C22⋊Dic4711yescharacteristic
187C22#1568960112yes—
188C22#1568969714yes—
189C4—28no—
190C4—24no—
191C4—44no—
192C4—44no—
193C4—3768no—
194C2—24no—
195C2—22no—
196C2#2024597411yescharacteristic
197C2#2025184911yescharacteristic
198C2#2025185211yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#544C2
socleC22×C47C22×C22⋊C4
prime radical, cosocleC24×C47C24
hypercenterC2×#544D47
hypocenterC47#21828
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C47C4semidirect
12. C2×#125×C47C4semidirect
105. C2×C4⋊Q8Dic47semidirect
106. C2×#125Dic47semidirect
115. C47#21828semidirect

Maximal subgroups

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

subgroupindex
#202459912
#202443742
#202434652
C2×C47×#5442
#2182847