Finite Group Databasebeta

← #26113830

Subgroups — #26113830

Order 12032.

620 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 95 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#20253838C211yesmaximal, characteristic
2#20246565C211yesmaximal, characteristic, complemented
3#20241936C212yesmaximal, complemented
4#20241254C212yesmaximal, complemented
5C47×#1555C211yesmaximal, characteristic, complemented
6C2×#13094050—22no—
7#15689673C2211yescharacteristic
8#15689510—22no—
9#15689499C412yescomplemented
10#15689495C412yescomplemented
11C4×#10625329C2211yescharacteristic
12#15689003—24no—
13#15689003—22no—
14#15688991C2212yes—
15#15688858—24no—
16C2×#127×C47C2211yescharacteristic
17C47×#432C2212yes—
18C47×#331—22no—
19C2×#10625350—21no—
20C23×Dic47—22no—
21#13094050—44no—
22#13094050—44no—
23C2×#10625335—24no—
24C2×#10625334—24no—
25C2×#10625329—21no—
26#13094024C2×C411yescharacteristic, complemented
27#13094021C2×C411yescharacteristic, complemented
28#13094012—21no—
29#13093998—22no—
30C4×D188—22no—
31C2×C22⋊Dic47—24no—
32C2×C22⋊Dic47—22no—
33#13093935—21no—
34C4×C47⋊C8C2×C412yescomplemented
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95D414yes—
38C22×D4×C47—22no—
39C22×D4×C47D412yes—
40#127×C47—22no—
41#127×C47C2×C412yes—
42#127×C47D412yes—
43C2×C42×C47C2311yescharacteristic
44C4⋊C8×C47—22no—
45C22⋊C8×C47—22no—
46#97×C47—22no—
47C24×C47—22no—
48C2×D4×C47—24no—
49C2×D4×C47—24no—
50C2×D4×C47—22no—
51C2×D4×C47—42no—
52C2×D4×C47—42no—
53C2×D4×C47—82no—
54C2×D4×C47—82no—
55C2×D4×C47C22⋊C414yes—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D412yes—
58C2×C22⋊D47—41no—
59C22×Dic47—24no—
60C22×Dic47—84no—
61C22×Dic47—82no—
62#10625350—42no—
63#10625350—42no—
64C2×D188—21no—
65C2×D188—22no—
66C2×C4×D47—41no—
67C2×Dic94—21no—
68C2×C8×C47—42no—
69C42×C47C22×C411yescharacteristic
70C42×C47C2×D411yescharacteristic
71C42×C47#4112yes—
72C22⋊Dic47—84no—
73C22⋊Dic47—82no—
74#10625335—48no—
75#10625334—48no—
76(C2×D47)⋊C4—42no—
77C4⋊Dic47—22no—
78#10625329—24no—
79C2×C47⋊C8—22no—
80C23×C47—24no—
81C23×C47—44no—
82C23×C47—44no—
83C23×C47—42no—
84C23×C47—42no—
85C23×C47#12011yescharacteristic
86C22×D47—41no—
87D4×C47—44no—
88D4×C47—82no—
89D4×C47—84no—
90D4×C47—84no—
91C2×C4×C47—22no—
92C2×C4×C47—42no—
93C2×C4×C47C2×C22⋊C411yescharacteristic
94C2×C4×C47C4×D412yes—
95C2×C4×C47#12112yes—
96C2×C4×C47#12311yescharacteristic
97C22⋊D47—82no—
98C2×Dic47—48no—
99C2×Dic47—41no—
100C2×Dic47—44no—
101C2×Dic47—88no—
102D188—42no—
103D188—82no—
104C4×D47—81no—
105C4×D47—81no—
106Dic94—42no—
107C8×C47—82no—
108C47⋊C8—24no—
109#9428—471nomaximal, Sylow
110C22×C47—22no—
111C22×C47—44no—
112C22×C47—48no—
113C22×C47—82no—
114C22×C47—82no—
115C22×C47—84no—
116C22×C47#38611yescharacteristic
117C22×C47#45312yes—
118C2×D47—81no—
119C2×D47—82no—
120C4×C47—21no—
121C4×C47—21no—
122C4×C47—22no—
123C4×C47—22no—
124Dic47—84no—
125Dic47—81no—
126#3151—471no—
127#2380—471no—
128#1555D4711yescharacteristic, complemented
129#1514—472no—
130#1288—472no—
131C2×C47—22no—
132C2×C47—84no—
133C2×C47#188711yescharacteristic
134C2×C47#190412yes—
135D47—161no—
136C2×AGL(1,ℤ/8)—942no—
137#532—471no—
138C2×#127C2×D4711yescharacteristic
139#496—942no—
140#493—472no—
141#492—472no—
142#432C2×D4712yes—
143C4×M16—471no—
144#386—944no—
145#386—942no—
146#383—472no—
147#331—22no—
148#327—944no—
149C47#942811yescharacteristic, complemented, Sylow
150C2×#41—941no—
151C22×D4—22no—
152C22×D4D18812yes—
153C23×C4—942no—
154AGL(1,ℤ/8)—1884no—
155AGL(1,ℤ/8)—1884no—
156C2×SD16—944no—
157C2×D8—944no—
158C2×M16—941no—
159C4⋊Q8—471no—
160#127—22no—
161#127—471no—
162#127C4×D4712yes—
163#127C22⋊D4712yes—
164#124—941no—
165#121—942no—
166C4×D4—942no—
167C2×C22⋊C4—944no—
168C2×C22⋊C4—942no—
169C2×C42C22×D4711yescharacteristic
170C4⋊C8—22no—
171C22⋊C8—22no—
172#97—22no—
173#97—941no—
174C4×C8—472no—
175#95—474no—
176#95—942no—
177#95—942no—
178C24—22no—
179#41—1882no—
180#41—1882no—
181C2×Q8—941no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—82no—
188C2×D4—82no—
189C2×D4—941no—
190C2×D4—942no—
191C2×D4—1881no—
192C2×D4(C2×D47)⋊C414yes—
193C22×C4—944no—
194C22×C4—3764no—
195C22×C4—3762no—
196C22×C4—1881no—
197C22×C4C2×D18811yescharacteristic
198C22×C4D4×D4712yes—
199SD16—1888no—
200D8—1888no—
201M16—944no—
202C2×C8—42no—
203C2×C8—942no—
204C4⋊C4—942no—
205C22⋊C4—3764no—
206C22⋊C4—3762no—
207C22⋊C4—1882no—
208C42C2×C4×D4711yescharacteristic
209C42#1062535212yes—
210C42C2×C22⋊D4711yescharacteristic
211C23—24no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23—1881no—
217C23#1309399411yescharacteristic
218Q8—1882no—
219D4—44no—
220D4—82no—
221D4—84no—
222D4—84no—
223D4—3762no—
224D4—3762no—
225D4—1882no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—3761no—
229C2×C4—3761no—
230C2×C4—3768no—
231C2×C4—1888no—
232C2×C4—1881no—
233C2×C4—1884no—
234C2×C4C2×(C2×D47)⋊C411yescharacteristic
235C2×C4(C22⋊D47)⋊C412yes—
236C2×C4#1309399612yes—
237C2×C4#1309400811yescharacteristic
238C8—82no—
239C8—944no—
240C22—22no—
241C22—44no—
242C22—48no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22—3761no—
247C22—3762no—
248C22#1568900011yescharacteristic
249C22#1568929312yes—
250C4—21no—
251C4—21no—
252C4—22no—
253C4—22no—
254C4—3764no—
255C4—3761no—
256C2—22no—
257C2—84no—
258C2—7521no—
259C2#2024378411yescharacteristic
260C2#2024387312yes—
subgroupG/H
centerC22#15689000
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1555C2
socleC22×C47#386
prime radical#20253838C2
cosocleC2×C42×C47C23
hypercenter#1555D47
hypocenterC47#9428
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
2. #20246565C2semidirect
3. #20241936C2semidirect
4. #20241254C2semidirect
5. C47×#1555C2semidirect
9. #15689499C4semidirect
10. #15689495C4semidirect
26. #13094024C2×C4semidirect
27. #13094021C2×C4semidirect
34. C4×C47⋊C8C2×C4semidirect
128. #1555D47semidirect
149. C47#9428semidirect

Maximal subgroups

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

subgroupindex
#202538382
#202465652
#202419362
#202412542
C47×#15552
#942847