Finite Group Databasebeta

← #26119302

Subgroups — #26119302

Order 12032.

500 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 125 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#20244315C212yesmaximal
2#20244193C211yesmaximal, characteristic
3#20241271C211yesmaximal, characteristic, complemented
4#20241229C212yesmaximal, complemented
5C47×#2571C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7C2×C4⋊(C47⋊C8)C2212yes—
8C2×#13093944—22no—
9C2×#13093944—22no—
10C2×#13093944—22no—
11C2×C4×C47⋊C8C2211yescharacteristic
12C47×#400—22no—
13C47×#389—22no—
14C47×#384C2211yescharacteristic
15C47×#382—22no—
16C4×C4⋊C4×C47C2212yes—
17C22×D188—21no—
18C22×D188—22no—
19C22×C47⋊C8—21no—
20C22×C47⋊C8—22no—
21#13094021—22no—
22#13094021—21no—
23#13094021Q812yes—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26#13093944—44no—
27#13093944—44no—
28#13093944—44no—
29C4×C47⋊C8—22no—
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47—22no—
32C2×C4⋊C4×C47D412yes—
33C2×C4⋊C4×C47D412yes—
34C2×C4⋊C4×C47D412yes—
35C2×C42×C47—22no—
36C2×C42×C47C2311yescharacteristic
37#95×C47—22no—
38#95×C47—24no—
39#95×C47—22no—
40C23×D47—41no—
41C22×C4×C47—22no—
42C22×C4×C47—22no—
43C22×C4×C47—22no—
44C22×C4×C47C2×D411yescharacteristic
45C22×C4×C47C2×D412yes—
46C2×D188—22no—
47C2×D188—24no—
48C2×D188—42no—
49C2×D188—42no—
50C2×D188—42no—
51C2×D188—42no—
52C2×D188—42no—
53C2×D188—42no—
54C2×D188—41no—
55C2×D188—42no—
56C4⋊C4×C47—22no—
57C4⋊C4×C47—22no—
58C4⋊C4×C47—22no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—42no—
62C4⋊C4×C47—42no—
63C4⋊C4×C47—82no—
64C42×C47—42no—
65C42×C47—42no—
66C42×C47C2×Q811yescharacteristic
67C42×C47#4112yes—
68C42×C47#4111yescharacteristic
69C2×C47⋊C8—22no—
70C2×C47⋊C8—41no—
71C2×C47⋊C8—41no—
72C2×C47⋊C8—42no—
73C2×C47⋊C8—42no—
74C2×C47⋊C8—42no—
75C23×C47#12011yescharacteristic
76C22×D47—42no—
77C22×D47—82no—
78C22×D47—82no—
79C22×D47—81no—
80C22×D47—81no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—44no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47C22⋊Q811yescharacteristic
92C2×C4×C47C22⋊Q812yes—
93C2×C4×C47#12411yescharacteristic
94C2×C4×C47#12412yes—
95C2×C4×C47#13512yes—
96C2×C4×C47#13512yes—
97C2×C4×C47AGL(1,ℤ/8)12yes—
98C2×C4×C47AGL(1,ℤ/8)12yes—
99D188—44no—
100D188—44no—
101D188—44no—
102D188—82no—
103D188—82no—
104D188—82no—
105C47⋊C8—42no—
106C47⋊C8—82no—
107#10351—471nomaximal, Sylow
108C22×C47#39311yescharacteristic
109C22×C47#44912yes—
110C22×C47#44912yes—
111C22×C47#45311yescharacteristic
112C22×C47#45411yescharacteristic
113C2×D47—84no—
114C2×D47—82no—
115C2×D47—161no—
116C2×D47—161no—
117C2×D47—162no—
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—82no—
121C4×C47—82no—
122C4×C47—82no—
123C4×C47#47812yes—
124C4×C47#47812yes—
125C4×C47#48812yes—
126C4×C47#49012yes—
127#2571D4711yescharacteristic, complemented
128#1982—472no—
129#1957—471no—
130#1296—471no—
131#1279—472no—
132C2×C47#162011yescharacteristic
133C2×C47#162411yescharacteristic
134C2×C47#162812yes—
135C2×C47#202512yes—
136C2×C47#202711yescharacteristic
137D47—162no—
138C2×#127—471no—
139C2×C4⋊C8—472no—
140C2×#102—942no—
141C2×#102—942no—
142C2×#102—942no—
143C2×C4×C8—471no—
144#400—22no—
145#389—22no—
146#384C2×D4711yescharacteristic
147#382—22no—
148C4×C4⋊C4C2×D4712yes—
149C47#1035111yescharacteristic, complemented, Sylow
150C22×D4—941no—
151C22×D4—942no—
152C22×C8—941no—
153C22×C8—942no—
154#127—472no—
155#127—942no—
156#127—941no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22⋊D4712yes—
160C2×C4⋊C4C22⋊D4712yes—
161C2×C4⋊C4C22⋊D4712yes—
162C2×C42—22no—
163C2×C42C22×D4711yescharacteristic
164C4⋊C8—942no—
165C4⋊C8—942no—
166#102—1884no—
167#102—1884no—
168#102—1884no—
169C4×C8—942no—
170#95—22no—
171#95—24no—
172#95—22no—
173C24—1881no—
174C2×D4—942no—
175C2×D4—944no—
176C2×D4—1882no—
177C2×D4—1882no—
178C2×D4—1881no—
179C2×D4—1882no—
180C2×D4—1882no—
181C2×D4—1882no—
182C2×D4—1882no—
183C2×D4—1882no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C2×C22⋊D4711yescharacteristic
188C22×C4C2×C22⋊D4712yes—
189C2×C8—942no—
190C2×C8—1882no—
191C2×C8—1882no—
192C2×C8—1882no—
193C2×C8—1881no—
194C2×C8—1881no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C42—42no—
204C42—42no—
205C42Q8×D4711yescharacteristic
206C42#1062535412yes—
207C42#1062535411yescharacteristic
208C23—3762no—
209C23—3762no—
210C23—3761no—
211C23—3761no—
212C23—1882no—
213C23#1309399311yescharacteristic
214D4—3762no—
215D4—3762no—
216D4—3762no—
217D4—1884no—
218D4—1884no—
219D4—1884no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4#1309400011yescharacteristic
231C2×C4#1309400012yes—
232C2×C4#1309401311yescharacteristic
233C2×C4#1309401312yes—
234C2×C4#1309404412yes—
235C2×C4#1309404412yes—
236C2×C4#1309404912yes—
237C2×C4#1309404912yes—
238C8—3762no—
239C8—1882no—
240C22—3764no—
241C22—3762no—
242C22—7521no—
243C22—7521no—
244C22—7522no—
245C22#1568903411yescharacteristic
246C22#1568926812yes—
247C22#1568926812yes—
248C22#1568929111yescharacteristic
249C22#1568929611yescharacteristic
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4—82no—
255C4#1568942012yes—
256C4#1568942012yes—
257C4#1568948012yes—
258C4#1568948912yes—
259C2—7522no—
260C2#2024244711yescharacteristic
261C2#2024247111yescharacteristic
262C2#2024249812yes—
263C2#2024453112yes—
264C2#2024454111yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2571C2
socleC23×C47#120
prime radicalC2×#13094021C22
hypercenter#2571D47
hypocenterC47#10351
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #20241271C2semidirect
4. #20241229C2semidirect
5. C47×#2571C2semidirect
127. #2571D47semidirect
149. C47#10351semidirect

Maximal subgroups

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

subgroupindex
#202443152
#202441932
#202412712
#202412292
C47×#25712
#1035147