Finite Group Databasebeta

← #26103557

Subgroups — #26103557

Order 12032.

794 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 79 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#20243303C212yesmaximal
2C47×#1438C211yesmaximal, characteristic
3C2×#13093939—22no—
4C2×#13093939C414yescomplemented
5#15688979—22no—
6C23×D4×C47C2211yescharacteristic
7C22×C22⋊C4×C47—22no—
8C2×#95×C47C412yescomplemented
9C2×C22⋊Dic47—24no—
10C2×C22⋊Dic47—44no—
11#13093939—44no—
12#13093939—44no—
13C25×C47D412yes—
14C22×D4×C47—42no—
15C22×D4×C47—44no—
16C22×D4×C47C2×C412yes—
17C22×D4×C47D411yescharacteristic
18C22×D4×C47Q811yescharacteristic
19C23×C4×C47—22no—
20C23×C4×C47C2×C411yescharacteristic
21C2×C22⋊C4×C47—48no—
22C2×C22⋊C4×C47—44no—
23#95×C47—42no—
24C24×C47—24no—
25C24×C47—24no—
26C24×C47—42no—
27C24×C47—44no—
28C24×C47—44no—
29C24×C47C22⋊C411yescharacteristic
30C24×C47C22⋊C412yes—
31C24×C47C4⋊C412yes—
32C2×D4×C47—24no—
33C2×D4×C47—24no—
34C2×D4×C47—42no—
35C2×D4×C47—42no—
36C2×D4×C47—41no—
37C2×D4×C47—41no—
38C2×D4×C47—42no—
39C2×D4×C47—416no—
40C22×C4×C47—22no—
41C22×C4×C47—24no—
42C22×C4×C47—42no—
43C22×C4×C47—44no—
44C22×C4×C47—44no—
45C22×C4×C47—42no—
46C22×C4×C47—42no—
47C22×C4×C47C4211yescharacteristic
48C22×C4×C47C4⋊C411yescharacteristic
49C22×Dic47—44no—
50C22⋊C4×C47—416no—
51C22⋊Dic47—44no—
52C22⋊Dic47—44no—
53C22⋊Dic47—84no—
54C22⋊Dic47—84no—
55C23×C47—22no—
56C23×C47—22no—
57C23×C47—22no—
58C23×C47—22no—
59C23×C47—22no—
60C23×C47—22no—
61C23×C47—22no—
62C23×C47—22no—
63C23×C47—48no—
64C23×C47—44no—
65C23×C47—44no—
66C23×C47—44no—
67C23×C47—48no—
68C23×C47—44no—
69C23×C47—44no—
70C23×C47—44no—
71C23×C47—84no—
72C23×C47—82no—
73C23×C47—84no—
74C23×C47—82no—
75C23×C47#9511yescharacteristic
76C23×C47#9914yes—
77C23×C47#9912yes—
78D4×C47—44no—
79D4×C47—48no—
80D4×C47—44no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—22no—
84C2×C4×C47—22no—
85C2×C4×C47—44no—
86C2×C4×C47—48no—
87C2×C4×C47—44no—
88C2×C4×C47—84no—
89C2×C4×C47—82no—
90C2×C4×C47—82no—
91C2×Dic47—84no—
92C2×Dic47—88no—
93#7307—471nomaximal, Sylow
94C22×C47—22no—
95C22×C47—24no—
96C22×C47—22no—
97C22×C47—22no—
98C22×C47—22no—
99C22×C47—22no—
100C22×C47—24no—
101C22×C47—44no—
102C22×C47—44no—
103C22×C47—44no—
104C22×C47—42no—
105C22×C47—44no—
106C22×C47—82no—
107C22×C47—88no—
108C22×C47—88no—
109C22×C47—82no—
110C22×C47#34212yes—
111C22×C47#34211yescharacteristic
112C22×C47#35312yes—
113C22×C47#35412yes—
114C4×C47—41no—
115C4×C47—41no—
116C4×C47—84no—
117Dic47—164no—
118#1781—472no—
119#1438D4711yescharacteristic
120C2×C47—22no—
121C2×C47—22no—
122C2×C47—22no—
123C2×C47—44no—
124C2×C47—82no—
125C2×C47#130411yescharacteristic
126C2×C47#139312yes—
127C23×D4C2×D4711yescharacteristic
128C22×C22⋊C4—22no—
129C2×#99—474no—
130C2×#99—942no—
131#379—942no—
132C2×#95Dic4712yescomplemented
133C47#730711yescharacteristic, complemented, Sylow
134C25C22⋊D4712yes—
135C22×D4—42no—
136C22×D4—44no—
137C22×D4Dic9411yescharacteristic
138C22×D4C4×D4712yes—
139C22×D4D18811yescharacteristic
140C23×C4—22no—
141C23×C4C2×Dic4711yescharacteristic
142C2×C22⋊C4—48no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—944no—
145C2×C22⋊C4—1884no—
146#99—1884no—
147#99—1884no—
148#95—42no—
149C24—24no—
150C24—24no—
151C24—42no—
152C24—44no—
153C24—44no—
154C24Dic47⋊C412yes—
155C24(C2×D47)⋊C412yes—
156C24C22⋊Dic4711yescharacteristic
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—42no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—41no—
164C2×D4—416no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—1884no—
173C22×C4C4×Dic4711yescharacteristic
174C22×C4C4⋊Dic4711yescharacteristic
175C22⋊C4—416no—
176C22⋊C4—3764no—
177C22⋊C4—3764no—
178C22⋊C4—1884no—
179C22⋊C4—1884no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—48no—
189C23—44no—
190C23—44no—
191C23—44no—
192C23—48no—
193C23—44no—
194C23—44no—
195C23—44no—
196C23—84no—
197C23—82no—
198C23—84no—
199C23—82no—
200C23C47⋊#9511yescharacteristic
201C23#1309393814yes—
202C23#1309393912yes—
203D4—44no—
204D4—48no—
205D4—44no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—48no—
212C2×C4—44no—
213C2×C4—84no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—3764no—
217C2×C4—3768no—
218C22—22no—
219C22—24no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—44no—
226C22—44no—
227C22—44no—
228C22—44no—
229C22—42no—
230C22—82no—
231C22—88no—
232C22—88no—
233C22—82no—
234C22#1568888912yes—
235C22#1568889011yescharacteristic
236C22#1568891612yes—
237C22#1568891812yes—
238C4—41no—
239C4—41no—
240C4—84no—
241C4—7524no—
242C2—22no—
243C2—22no—
244C2—22no—
245C2—44no—
246C2—82no—
247C2C47⋊#130411yescharacteristic
248C2#2024151312yes—
subgroupG/H
centerC22#15688890
commutatorC22×C4×C47C42
FrattiniC23×D4C2×D47
FittingC47×#1438C2
socleC22×C47#342
prime radical, cosocleC23×D4×C47C22
hypercenter#1438D47
hypocenterC47#7307
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #1438

Complements

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

NH ≅ G/Ntype
4. C2×#13093939C4semidirect
8. C2×#95×C47C4semidirect
132. C2×#95Dic47semidirect
133. C47#7307semidirect

Maximal subgroups

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

subgroupindex
#202433032
C47×#14382
#730747