Finite Group Databasebeta

← #26128714

Subgroups — #26128714

Order 12032.

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

normal subgroups only

This group has 372 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#20244617C211yesmaximal, characteristic
2#20242390C212yesmaximal
3#20241394C212yesmaximal, complemented
4#20241332C211yesmaximal, characteristic, complemented
5C47×#2822C211yesmaximal, characteristic, complemented
6#15689704C2211yescharacteristic
7#15689436—22no—
8C2×#13093952—21no—
9#15689122—21no—
10#15688892C2211yescharacteristic
11#15688856—22no—
12#15688849C2212yes—
13C2×#125×C47—21no—
14C47×#400—22no—
15C47×#398—22no—
16C47×#398—21no—
17C47×#398C2212yes—
18C47×#382C2211yescharacteristic
19C2×#10625350—21no—
20C2×#10625329—21no—
21#13094004—41no—
22#13093999—41no—
23#13093998—22no—
24#13093957—42no—
25#13093952—42no—
26#13093947—41no—
27#13093944—41no—
28#13093944—42no—
29C22⋊(C47⋊C8)—22no—
30#125×C47—42no—
31#125×C47—42no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C4⋊C4×C47D412yes—
39C2×C4⋊C4×C47C2311yescharacteristic
40C2×C42×C47D411yescharacteristic
41#95×C47—22no—
42#95×C47—21no—
43#95×C47—22no—
44#95×C47—22no—
45#95×C47D412yes—
46C22×C4×C47—21no—
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47—21no—
50C22×C4×C47C2×D411yescharacteristic
51C22×C4×C47C2×D412yes—
52C2×C22⋊D47—41no—
53#10625350—41no—
54#10625350—41no—
55#10625350—41no—
56#10625350—41no—
57C2×D188—21no—
58C2×D188—22no—
59C2×C4×D47—41no—
60C2×Dic94—21no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—42no—
63C4⋊C4×C47—44no—
64C4⋊C4×C47—42no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—42no—
67C4⋊C4×C47—42no—
68C4⋊C4×C47—42no—
69C4⋊C4×C47—81no—
70C42×C47—21no—
71C42×C47D812yes—
72(C2×D47)⋊C4—42no—
73C4⋊Dic47—22no—
74#10625329—41no—
75#10625329—41no—
76C2×C47⋊C8—42no—
77C2×C47⋊C8—41no—
78C23×C47#12011yescharacteristic
79C22×D47—41no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—22no—
83C2×C4×C47—22no—
84C2×C4×C47—41no—
85C2×C4×C47—41no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—41no—
95C2×C4×C47C2×D811yescharacteristic
96C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
97C22⋊D47—81no—
98C22⋊D47—81no—
99C2×Dic47—41no—
100D188—41no—
101D188—41no—
102D188—82no—
103C4×D47—81no—
104C4×D47—81no—
105Dic94—41no—
106Dic94—41no—
107C47⋊C8—81no—
108C47⋊C8—82no—
109#12157—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47#44711yescharacteristic
113C22×C47#45411yescharacteristic
114C22×C47B(4,𝔽2)11yescharacteristic
115C2×D47—81no—
116C2×D47—81no—
117C2×D47—81no—
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—42no—
121C4×C47—42no—
122C4×C47—82no—
123C4×C47—82no—
124C4×C47—81no—
125Dic47—81no—
126#2822D4711yescharacteristic, complemented
127#2040—471no—
128#1610—472no—
129#1346—472no—
130#1324—471no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47#159811yescharacteristic
134C2×C47#220111yescharacteristic
135C2×C47#220311yescharacteristic
136D47—161no—
137#537—471no—
138C2×#125—21no—
139#480—942no—
140C2×#104—941no—
141#417—941no—
142#400—22no—
143#398—21no—
144#398—22no—
145#398C2×D4712yes—
146#382C2×D4711yescharacteristic
147#343—471no—
148#327—942no—
149#324—472no—
150C47#1215711yescharacteristic, complemented, Sylow
151C2×#41—941no—
152C2×M16—941no—
153#125—42no—
154#125—42no—
155C22⋊Q8—1881no—
156#121—942no—
157#121—1881no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D4711yescharacteristic
164C2×C4⋊C4C22⋊D4712yes—
165C2×C4⋊C4C22×D4711yescharacteristic
166C2×C42C22⋊D4711yescharacteristic
167#107—1882no—
168#104—1882no—
169#103—1881no—
170#102—1881no—
171#102—1882no—
172C22⋊C8—942no—
173#95—22no—
174#95—21no—
175#95—22no—
176#95—22no—
177#95C22⋊D4712yes—
178#41—1881no—
179#41—1881no—
180#41—1881no—
181#41—1881no—
182C2×Q8—941no—
183C2×D4—942no—
184C2×D4—941no—
185C2×D4—1881no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1881no—
191C22×C4C2×C22⋊D4711yescharacteristic
192C22×C4C2×C22⋊D4712yes—
193M16—1881no—
194M16—1881no—
195C2×C8—1882no—
196C2×C8—1881no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—942no—
207C22⋊C4—1882no—
208C42—21no—
209C42#1062533412yes—
210C23—1881no—
211C23#1309399311yescharacteristic
212Q8—1881no—
213Q8—1881no—
214D4—3762no—
215D4—3761no—
216D4—3761no—
217D4—1881no—
218D4—1881no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—3761no—
235C2×C4—3761no—
236C2×C4—1881no—
237C2×C4C2×#1062533411yescharacteristic
238C2×C4#1309405011yescharacteristic
239C8—3761no—
240C8—3762no—
241C22—21no—
242C22—21no—
243C22—3761no—
244C22—3761no—
245C22—3761no—
246C22#1568925711yescharacteristic
247C22#1568929611yescharacteristic
248C22#1568931011yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—3761no—
257C2—21no—
258C2—21no—
259C2—7521no—
260C2#2024232311yescharacteristic
261C2#2024540111yescharacteristic
262C2#2024540911yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2822C2
socleC22×C47B(4,𝔽2)
prime radical#15689704C22
hypercenter#2822D47
hypocenterC47#12157
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
3. #20241394C2semidirect
4. #20241332C2semidirect
5. C47×#2822C2semidirect
126. #2822D47semidirect
150. C47#12157semidirect

Maximal subgroups

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

subgroupindex
#202446172
#202423902
#202413942
#202413322
C47×#28222
#1215747