Finite Group Databasebeta

← #26114122

Subgroups — #26114122

Order 12032.

474 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 131 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#20248750C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic
3#20241818C212yesmaximal
4#20241247C212yesmaximal
5C47×#2982C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7#15689147C2212yes—
8C2×#13093943—21no—
9#15689062C412yes—
10#15689043C412yes—
11#15689030—21no—
12#15689019—21no—
13#15689007—22no—
14#15688989C2211yescharacteristic
15#15688892—21no—
16#15688855—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C8×Q8×C47—24no—
19C47×#431—22no—
20C47×#431C2212yes—
21C22×Dic94C2×C411yescharacteristic
22C2×#10625329—22no—
23C4×Dic94—22no—
24C4×Dic94—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—21no—
29C2×C4×Dic47—21no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32#13093943—42no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—21no—
37C47⋊#95C2×C412yes—
38C22×Q8×C47D411yescharacteristic
39C4×Q8×C47—22no—
40C4×Q8×C47—22no—
41C2×C4⋊C4×C47C2×C412yes—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C42×C47D412yes—
44C2×C42×C47C2311yescharacteristic
45C4⋊C8×C47—24no—
46C4⋊C8×C47—22no—
47C22⋊C8×C47—22no—
48C4×C8×C47—24no—
49C4×C8×C47D414yes—
50C2×Q8×C47—22no—
51C2×Q8×C47—22no—
52C2×Q8×C47SD1612yes—
53C2×Q8×C47Q1612yes—
54C22×C4×C47C22⋊C412yes—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—21no—
60C22×Dic47—22no—
61C22×Dic47—21no—
62C2×Dic94—22no—
63C2×Dic94—42no—
64C2×C8×C47—24no—
65C4⋊C4×C47—24no—
66C4⋊C4×C47—22no—
67C42×C47—22no—
68C42×C47—22no—
69C42×C47C2×D412yes—
70C42×C47#4112yes—
71C4⋊Dic47—22no—
72C4⋊Dic47—42no—
73Dic47⋊C4—44no—
74Dic47⋊C4—41no—
75Dic47⋊C4—41no—
76C4×Dic47—42no—
77#10625329—42no—
78#10625329—42no—
79C2×C47⋊C8—42no—
80C23×C47C2×C22⋊C411yescharacteristic
81Q8×C47—24no—
82Q8×C47—24no—
83C2×C4×C47—22no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—42no—
89C2×C4×C47#10314yes—
90C2×C4×C47C4×D412yes—
91C2×C4×C47#12011yescharacteristic
92C2×C4×C47#12112yes—
93C2×C4×C47#12311yescharacteristic
94C2×C4×C47C2×SD1611yescharacteristic
95C2×C4×C47C2×Q1611yescharacteristic
96C2×C4×C47#13512yes—
97C2×Dic47—22no—
98C2×Dic47—41no—
99C2×Dic47—41no—
100C2×Dic47—42no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—41no—
104C2×Dic47—42no—
105Dic94—24no—
106Dic94—42no—
107C8×C47—44no—
108C47⋊C8—82no—
109#9484—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47#38611yescharacteristic
113C22×C47C2×#10311yescharacteristic
114C22×C47#41611yescharacteristic
115C4×C47—22no—
116C4×C47—24no—
117C4×C47—21no—
118C4×C47—21no—
119C4×C47—42no—
120C4×C47#46512yes—
121C4×C47#46712yes—
122Dic47—42no—
123Dic47—82no—
124Dic47—81no—
125#2982D4711yescharacteristic
126#2581—471no—
127#1801—471no—
128#1487—472no—
129#1286—472no—
130C2×C47—21no—
131C2×C47—21no—
132C2×C47#189111yescharacteristic
133C2×C47#189211yescharacteristic
134C2×C47#191211yescharacteristic
135C2×C4×Q8—471no—
136C2×C4×Q8C2×D4711yescharacteristic
137C8×Q8—24no—
138#431—22no—
139#431C2×D4712yes—
140#423—472no—
141C2×#101—941no—
142#400—472no—
143#395—472no—
144#391—941no—
145#389—941no—
146#387—942no—
147#382—471no—
148#343—941no—
149Q8⋊C8—944no—
150C47#948411yescharacteristic, complemented, Sylow
151C22×Q8—471no—
152C22×Q8D18811yescharacteristic
153C2×M16—942no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—942no—
157C4×Q8—942no—
158C2×C4⋊C4—471no—
159C2×C4⋊C4—942no—
160C2×C4⋊C4—941no—
161C2×C4⋊C4—941no—
162C2×C4⋊C4C4×D4712yes—
163C2×C4⋊C4D18811yescharacteristic
164C2×C42—941no—
165C2×C42C22⋊D4712yes—
166C2×C42C22×D4711yescharacteristic
167C4⋊C8—24no—
168C4⋊C8—22no—
169C4⋊C8—942no—
170C4⋊C8—942no—
171#101—1882no—
172C22⋊C8—22no—
173C4×C8—24no—
174C4×C8C22⋊D4714yes—
175#95—472no—
176#95—942no—
177#95—942no—
178#95—942no—
179#95—941no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—942no—
183C2×Q8—1882no—
184C2×Q8#1062532512yes—
185C2×Q8Dic18812yes—
186C22×C4—941no—
187C22×C4—942no—
188C22×C4—941no—
189C22×C4(C2×D47)⋊C412yes—
190C22×C4(C2×D47)⋊C412yes—
191C22×C4C2×C4×D4711yescharacteristic
192C22×C4C2×D18811yescharacteristic
193C22×C4C2×C22⋊D4711yescharacteristic
194M16—1882no—
195M16—1882no—
196C2×C8—24no—
197C2×C8—1882no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—942no—
201C4⋊C4—1884no—
202C4⋊C4—1881no—
203C4⋊C4—1881no—
204C4⋊C4—1882no—
205C42—22no—
206C42—22no—
207C42—1882no—
208C42#1062535012yes—
209C42C2×C22⋊D4712yes—
210C23C2×(C2×D47)⋊C411yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—944no—
214Q8—1882no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—942no—
222C2×C4—1882no—
223C2×C4—1882no—
224C2×C4—1882no—
225C2×C4—1881no—
226C2×C4—1882no—
227C2×C4—1881no—
228C2×C4—1881no—
229C2×C4#1309394914yes—
230C2×C4C4×C22⋊D4712yes—
231C2×C4#1309399311yescharacteristic
232C2×C4#1309399812yes—
233C2×C4#1309400711yescharacteristic
234C2×C4C2×#1062532511yescharacteristic
235C2×C4C2×Dic18811yescharacteristic
236C2×C4#1309404612yes—
237C8—44no—
238C8—3762no—
239C22—21no—
240C22—21no—
241C22#1568900211yescharacteristic
242C22C2×#1309394911yescharacteristic
243C22#1568912011yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—3762no—
250C4—3761no—
251C4—1882no—
252C4#1568935212yes—
253C4#1568936412yes—
254C2—21no—
255C2—21no—
256C2#2024380611yescharacteristic
257C2#2024381111yescharacteristic
258C2#2024392511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2982C2
socleC22×C47#386
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2982D47
hypocenterC47#9484
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
150. C47#9484semidirect

Maximal subgroups

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

subgroupindex
#202487502
#202433842
#202418182
#202412472
C47×#29822
#948447