Finite Group Databasebeta

← #26103595

Subgroups — #26103595

Order 12032.

482 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 71 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#20241715C212yesmaximal
2C47×#1440C211yesmaximal, characteristic
3#15689087—22no—
4#15688847—22no—
5#15688847C414yescomplemented
6C22×C22⋊C4×C47C412yes—
7C22×C22⋊C4×C47C2211yescharacteristic
8C2×#95×C47—22no—
9C2×#10625329—42no—
10C22⋊(C47⋊C8)—24no—
11C22⋊(C47⋊C8)—42no—
12C25×C47C2×C411yescharacteristic
13C23×C4×C47—22no—
14C23×C4×C47D412yes—
15C2×C22⋊C4×C47—42no—
16C2×C22⋊C4×C47—42no—
17C2×C22⋊C4×C47—44no—
18C2×C22⋊C4×C47C2×C412yes—
19C2×C22⋊C4×C47D411yescharacteristic
20C2×C22⋊C4×C47Q811yescharacteristic
21#95×C47—44no—
22C24×C47—22no—
23C24×C47—42no—
24C24×C47—42no—
25C24×C47—42no—
26C24×C47C4211yescharacteristic
27C24×C47C22⋊C411yescharacteristic
28C24×C47C4⋊C411yescharacteristic
29C22×C4×C47—22no—
30C22×C4×C47—22no—
31C22×C4×C47—24no—
32C22×C4×C47—42no—
33C22×C4×C47—42no—
34C22×C4×C47—42no—
35C22×C4×C47—42no—
36C22×C4×C47—42no—
37C22×C4×C47—42no—
38C22×C4×C47—42no—
39C22×C4×C47—42no—
40C22×C4×C47C22⋊C412yes—
41C22×C4×C47C4⋊C412yes—
42C22⋊C4×C47—42no—
43C22⋊C4×C47—41no—
44C22⋊C4×C47—41no—
45C22⋊C4×C47—44no—
46C22⋊C4×C47—44no—
47#10625329—82no—
48#10625329—82no—
49C2×C47⋊C8—44no—
50C23×C47—22no—
51C23×C47—22no—
52C23×C47—22no—
53C23×C47—21no—
54C23×C47—21no—
55C23×C47—22no—
56C23×C47—22no—
57C23×C47—44no—
58C23×C47—44no—
59C23×C47—42no—
60C23×C47—42no—
61C23×C47—42no—
62C23×C47—42no—
63C23×C47—82no—
64C23×C47—82no—
65C23×C47—82no—
66C23×C47—82no—
67C23×C47#9511yescharacteristic
68C23×C47#9912yes—
69C2×C4×C47—22no—
70C2×C4×C47—22no—
71C2×C4×C47—42no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47—44no—
76C2×C4×C47—44no—
77C2×C4×C47—44no—
78C2×C4×C47—82no—
79C2×C4×C47—82no—
80C2×C4×C47—82no—
81C2×C4×C47—82no—
82C2×C4×C47—82no—
83C47⋊C8—84no—
84#7324—471nomaximal, Sylow
85C22×C47—22no—
86C22×C47—22no—
87C22×C47—22no—
88C22×C47—22no—
89C22×C47—22no—
90C22×C47—22no—
91C22×C47—22no—
92C22×C47—41no—
93C22×C47—41no—
94C22×C47—42no—
95C22×C47—42no—
96C22×C47—82no—
97C22×C47—82no—
98C22×C47—84no—
99C22×C47—84no—
100C22×C47#34211yescharacteristic
101C22×C47#34412yes—
102C22×C47AΣL(1,𝔽16)12yes—
103C22×C47#35212yes—
104C4×C47—42no—
105C4×C47—42no—
106C4×C47—84no—
107#1462—472no—
108#1440D4711yescharacteristic
109C2×C47—22no—
110C2×C47—22no—
111C2×C47—22no—
112C2×C47—81no—
113C2×C47—81no—
114C2×C47#130611yescharacteristic
115C2×C47#139012yes—
116C22×C22⋊C4Dic4712yes—
117C22×C22⋊C4C2×D4711yescharacteristic
118#407—942no—
119C2×#95—22no—
120#323—474no—
121#323—942no—
122C47#732411yescharacteristic, complemented, Sylow
123C25C2×Dic4711yescharacteristic
124C23×C4—22no—
125C23×C4C22⋊D4712yes—
126C2×M16—1882no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4Dic9411yescharacteristic
131C2×C22⋊C4C4×D4712yes—
132C2×C22⋊C4D18811yescharacteristic
133C22⋊C8—944no—
134C22⋊C8—1882no—
135#95—44no—
136C24—22no—
137C24—42no—
138C24—42no—
139C24—42no—
140C24C4×Dic4711yescharacteristic
141C24C4⋊Dic4711yescharacteristic
142C24C22⋊Dic4711yescharacteristic
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4Dic47⋊C412yes—
155C22×C4(C2×D47)⋊C412yes—
156M16—3762no—
157M16—3762no—
158C2×C8—1884no—
159C22⋊C4—41no—
160C22⋊C4—42no—
161C22⋊C4—41no—
162C22⋊C4—44no—
163C22⋊C4—44no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—22no—
171C23—44no—
172C23—44no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—82no—
178C23—82no—
179C23—82no—
180C23—82no—
181C23C47⋊#9511yescharacteristic
182C23#1309393912yes—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C8—3764no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—42no—
206C22—42no—
207C22—41no—
208C22—41no—
209C22—82no—
210C22—82no—
211C22—84no—
212C22—84no—
213C22#1568889011yescharacteristic
214C22#1568889312yes—
215C22#1568891012yes—
216C22#1568891312yes—
217C4—42no—
218C4—42no—
219C4—84no—
220C2—22no—
221C2—22no—
222C2—22no—
223C2—81no—
224C2—81no—
225C2#2024129611yescharacteristic
226C2#2024150612yes—
subgroupG/H
centerC22#15688890
commutatorC24×C47C42
FrattiniC22×C22⋊C4C2×D47
FittingC47×#1440C2
socleC22×C47#342
prime radicalC25×C47C2×C4
cosocleC22×C22⋊C4×C47C22
hypercenter#1440D47
hypocenterC47#7324
LCSG ▷ C24×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1440

Complements

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

NH ≅ G/Ntype
5. #15688847C4semidirect
122. C47#7324semidirect

Maximal subgroups

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

subgroupindex
#202417152
C47×#14402
#732447