Finite Group Databasebeta

← #26129034

Subgroups — #26129034

Order 12032.

404 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244544C211yesmaximal, characteristic
2#20242395C212yesmaximal, complemented
3#20241394C212yesmaximal, complemented
4#20241320C211yesmaximal, characteristic
5C47×#2724C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689453—22no—
8#15689128—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C2×#123×C47—21no—
14C47×#398C2212yes—
15C47×#397—22no—
16C47×#390C2211yescharacteristic
17C47×#389—21no—
18C47×#385—22no—
19C2×#10625329—21no—
20#13094023—41no—
21#13094015—41no—
22#13094003—22no—
23#13093983—21no—
24#13093955—42no—
25#13093947—42no—
26#13093945—41no—
27#13093945—41no—
28C22⋊(C47⋊C8)—22no—
29C22×D4×C47D411yescharacteristic
30C23×C4×C47—21no—
31#123×C47—42no—
32#123×C47—42no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47D412yes—
37C2×C4⋊C4×C47C2311yescharacteristic
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—21no—
40C2×C22⋊C4×C47—22no—
41C2×C42×C47D411yescharacteristic
42#95×C47—22no—
43#95×C47—22no—
44#95×C47D412yes—
45C24×C47—21no—
46C2×D4×C47—21no—
47C2×D4×C47—41no—
48C2×D4×C47—41no—
49C2×D4×C47SD1612yes—
50C22×C4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—42no—
54C22×C4×C47—81no—
55C22×C4×C47—81no—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D412yes—
58C2×Dic94—22no—
59C4⋊C4×C47—41no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—42no—
62C4⋊C4×C47—42no—
63C4⋊C4×C47—42no—
64C4⋊C4×C47—81no—
65C22⋊C4×C47—44no—
66C22⋊C4×C47—42no—
67C22⋊C4×C47—44no—
68C42×C47—41no—
69C22⋊Dic47—41no—
70C4⋊Dic47—21no—
71C4⋊Dic47—21no—
72C4⋊Dic47—22no—
73Dic47⋊C4—42no—
74C4×Dic47—41no—
75#10625329—41no—
76#10625329—41no—
77C2×C47⋊C8—42no—
78C2×C47⋊C8—41no—
79C23×C47—22no—
80C23×C47—42no—
81C23×C47—42no—
82C23×C47—41no—
83C23×C47—41no—
84C23×C47#12011yescharacteristic
85D4×C47—42no—
86D4×C47—81no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—22no—
90C2×C4×C47—22no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—84no—
101C2×C4×C47C2×SD1611yescharacteristic
102C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
103C2×Dic47—41no—
104C2×Dic47—41no—
105Dic94—82no—
106C47⋊C8—81no—
107C47⋊C8—82no—
108#12209—471nomaximal, Sylow
109C22×C47—21no—
110C22×C47—21no—
111C22×C47—44no—
112C22×C47—42no—
113C22×C47—82no—
114C22×C47—81no—
115C22×C47—81no—
116C22×C47#45011yescharacteristic
117C22×C47#45511yescharacteristic
118C22×C47B(4,𝔽2)11yescharacteristic
119C4×C47—21no—
120C4×C47—21no—
121C4×C47—42no—
122C4×C47—42no—
123C4×C47—82no—
124C4×C47—82no—
125Dic47—81no—
126Dic47—81no—
127#2724D4711yescharacteristic
128#2027—471no—
129#1611—472no—
130#1346—472no—
131#1318—471no—
132C2×C47—21no—
133C2×C47—21no—
134C2×C47—82no—
135C2×C47#160111yescharacteristic
136C2×C47#219811yescharacteristic
137C2×C47#220311yescharacteristic
138#544—471no—
139C2×#123—21no—
140#483—942no—
141#418—941no—
142#398C2×D4712yes—
143#397—22no—
144#390C2×D4711yescharacteristic
145#389—21no—
146#385—22no—
147#343—471no—
148#339—941no—
149#328—942no—
150#324—472no—
151C47#1220911yescharacteristic, complemented, Sylow
152C22×D4C22⋊D4711yescharacteristic
153C23×C4—21no—
154C2×M16—941no—
155C4⋊Q8—1881no—
156#125—1881no—
157#123—42no—
158#123—42no—
159C22⋊Q8—942no—
160#117—941no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22⋊D4712yes—
165C2×C4⋊C4C22×D4711yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42C22⋊D4711yescharacteristic
170#106—1882no—
171#103—1882no—
172#102—1881no—
173#102—1881no—
174C22⋊C8—942no—
175#95—22no—
176#95—22no—
177#95C22⋊D4712yes—
178C24—21no—
179C2×Q8—942no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4#1062533512yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×C22⋊D4711yescharacteristic
191C22×C4C2×C22⋊D4712yes—
192M16—1881no—
193M16—1881no—
194C2×C8—1882no—
195C2×C8—1881no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—941no—
203C4⋊C4—941no—
204C4⋊C4—942no—
205C4⋊C4—1882no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—1881no—
210C42—41no—
211C42—1881no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#1309399311yescharacteristic
218Q8—3762no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—1881no—
236C2×C4—1881no—
237C2×C4C2×#1062533511yescharacteristic
238C2×C4#1309405011yescharacteristic
239C8—3761no—
240C8—3762no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1568927411yescharacteristic
249C22#1568930111yescharacteristic
250C22#1568931011yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—3761no—
258C4—3761no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2024234011yescharacteristic
263C2#2024538411yescharacteristic
264C2#2024540911yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2724C2
socleC22×C47B(4,𝔽2)
prime radicalC22×D4×C47D4
hypercenter#2724D47
hypocenterC47#12209
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #20242395C2semidirect
3. #20241394C2semidirect
151. C47#12209semidirect

Maximal subgroups

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

subgroupindex
#202445442
#202423952
#202413942
#202413202
C47×#27242
#1220947