Finite Group Databasebeta

← #26125658

Subgroups — #26125658

Order 12032.

598 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 107 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#20241718C212yesmaximal
2#20241322C214yesmaximal, complemented
3C2×C47×#391C211yesmaximal, characteristic
4#15689087—22no—
5#15688849—22no—
6#15688849C2214yes—
7C23×Q8×C47C411yescharacteristic
8C22×C4⋊C4×C47C2211yescharacteristic
9C22×C42×C47C411yescharacteristic
10C47×#391—22no—
11C47×#391C412yes—
12C47×#391C2212yes—
13C2×#95×C47—22no—
14C2×#10625329—42no—
15C22⋊(C47⋊C8)—24no—
16C22⋊(C47⋊C8)—42no—
17C22×Q8×C47—24no—
18C22×Q8×C47—21no—
19C22×Q8×C47—42no—
20C22×Q8×C47—41no—
21C22×Q8×C47—41no—
22C22×Q8×C47C2×C412yes—
23C23×C4×C47—21no—
24C23×C4×C47—21no—
25C23×C4×C47C2×C411yescharacteristic
26C23×C4×C47D412yes—
27C2×C4⋊C4×C47—41no—
28C2×C4⋊C4×C47—41no—
29C2×C4⋊C4×C47C2×C411yescharacteristic
30C2×C4⋊C4×C47D412yes—
31C2×C4⋊C4×C47C2311yescharacteristic
32C2×C42×C47—21no—
33C2×C42×C47—42no—
34C2×C42×C47C2×C412yes—
35#95×C47—24no—
36#95×C47—24no—
37C24×C47C22⋊C411yescharacteristic
38C2×Q8×C47—24no—
39C2×Q8×C47—22no—
40C2×Q8×C47—48no—
41C2×Q8×C47—48no—
42C2×Q8×C47—42no—
43C2×Q8×C47—42no—
44C2×Q8×C47—42no—
45C2×Q8×C47—42no—
46C2×Q8×C47M1614yescomplemented
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—24no—
52C22×C4×C47—24no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—42no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47—41no—
61C22×C4×C47—41no—
62C22×C4×C47—44no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D412yes—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—42no—
69C4⋊C4×C47—41no—
70C42×C47—42no—
71C42×C47—42no—
72#10625329—84no—
73C2×C47⋊C8—44no—
74C23×C47—22no—
75C23×C47—22no—
76C23×C47—21no—
77C23×C47—21no—
78C23×C47#9911yescharacteristic
79C23×C47#9911yescharacteristic
80C23×C47C2×C22⋊C411yescharacteristic
81Q8×C47—44no—
82Q8×C47—48no—
83Q8×C47—44no—
84C2×C4×C47—24no—
85C2×C4×C47—24no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—24no—
91C2×C4×C47—24no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—48no—
95C2×C4×C47—42no—
96C2×C4×C47—44no—
97C2×C4×C47—44no—
98C2×C4×C47—44no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—82no—
105C2×C4×C47#10012yes—
106C2×C4×C47C2×M1612yes—
107C47⋊C8—84no—
108#11506—471nomaximal, Sylow
109C22×C47—24no—
110C22×C47—24no—
111C22×C47—21no—
112C22×C47—21no—
113C22×C47—21no—
114C22×C47—21no—
115C22×C47—21no—
116C22×C47—21no—
117C22×C47#35412yes—
118C22×C47#35612yes—
119C22×C47#40711yescharacteristic
120C22×C47C2×#9911yescharacteristic
121C22×C47C2×#10011yescharacteristic
122C4×C47—22no—
123C4×C47—22no—
124C4×C47—44no—
125C4×C47—42no—
126C4×C47—42no—
127C4×C47—44no—
128C2×#391D4711yescharacteristic
129#1463—472no—
130#1319—474no—
131C2×C47—21no—
132C2×C47—22no—
133C2×C47—22no—
134C2×C47—21no—
135C2×C47#146211yescharacteristic
136C2×C47C2×#35411yescharacteristic
137C2×C47C2×#35611yescharacteristic
138C23×Q8Dic4711yescharacteristic
139C22×C4⋊C4C2×D4711yescharacteristic
140C22×C42Dic4711yescharacteristic
141#407—942no—
142#391—22no—
143#391Dic4712yes—
144#391C2×D4712yes—
145C2×#95—22no—
146#324—474no—
147#324—942no—
148C47#1150611yescharacteristic, complemented, Sylow
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—42no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8C2×Dic4712yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C2×Dic4711yescharacteristic
158C23×C4C22⋊D4712yes—
159C2×M16—1882no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C2×Dic4711yescharacteristic
163C2×C4⋊C4C22⋊D4712yes—
164C2×C4⋊C4C22×D4711yescharacteristic
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C2×Dic4712yes—
168C22⋊C8—944no—
169C22⋊C8—1882no—
170#95—24no—
171#95—24no—
172C24C22⋊Dic4711yescharacteristic
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8#1062532914yescomplemented
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—44no—
198C22×C4C22⋊Dic4711yescharacteristic
199C22×C4C22⋊Dic4712yes—
200C22×C4C22×Dic4711yescharacteristic
201C22×C4C2×C22⋊D4712yes—
202M16—3764no—
203C2×C8—1884no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C42—42no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23#1309393911yescharacteristic
214C23#1309393911yescharacteristic
215C23C2×C22⋊Dic4711yescharacteristic
216Q8—44no—
217Q8—48no—
218Q8—44no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—48no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4#1309394112yes—
241C2×C4C2×#1062532912yes—
242C8—3764no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#1568891812yes—
252C22#1568892212yes—
253C22#1568908711yescharacteristic
254C22C2×#1309393911yescharacteristic
255C22C2×#1309394111yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—44no—
260C4—42no—
261C4—42no—
262C2—21no—
263C2—21no—
264C2—22no—
265C2—22no—
266C2#2024171511yescharacteristic
267C2C2×#1568891811yescharacteristic
268C2C2×#1568892211yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC2×C47×#391C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenterC2×#391D47
hypocenterC47#11506
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. #20241322C2semidirect
46. C2×Q8×C47M16semidirect
148. C47#11506semidirect
181. C2×Q8#10625329semidirect

Maximal subgroups

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

subgroupindex
#202417182
#202413222
C2×C47×#3912
#1150647