Finite Group Databasebeta

← #26114120

Subgroups — #26114120

Order 12032.

474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 127 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#20252469C211yesmaximal, characteristic
2#20248750C211yesmaximal, characteristic
3#20241816C212yesmaximal
4#20241248C212yesmaximal
5C47×#1801C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7Q8×C47⋊C8—24no—
8#15689176—22no—
9#15689176C2212yes—
10#15689062C412yes—
11#15689043C412yes—
12#15689030—21no—
13#15689019—21no—
14#15689007—22no—
15#15688989C2211yescharacteristic
16#15688853—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#423C2212yes—
19C2×#101×C47—21no—
20C47×#343—21no—
21C22×Dic94C2×C411yescharacteristic
22C4×Dic94—22no—
23C4×Dic94—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—21no—
28C2×C4×Dic47—21no—
29C4⋊(C47⋊C8)—24no—
30C4⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C4×C47⋊C8—24no—
33C4×C47⋊C8D414yes—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—21no—
38C47⋊#95C2×C412yes—
39C22×Q8×C47D411yescharacteristic
40C2×M16×C47—22no—
41C4×Q8×C47—22no—
42C4×Q8×C47—22no—
43C2×C4⋊C4×C47C2×C412yes—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C42×C47D412yes—
46C2×C42×C47C2311yescharacteristic
47C4⋊C8×C47—22no—
48C4⋊C8×C47—22no—
49#101×C47—42no—
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—
64M16×C47—42no—
65M16×C47—42no—
66C2×C8×C47—42no—
67C4⋊C4×C47—24no—
68C4⋊C4×C47—22no—
69C42×C47—22no—
70C42×C47—22no—
71C42×C47C2×D412yes—
72C42×C47#4112yes—
73C4⋊Dic47—22no—
74C4⋊Dic47—42no—
75Dic47⋊C4—44no—
76Dic47⋊C4—41no—
77Dic47⋊C4—41no—
78C4×Dic47—42no—
79C2×C47⋊C8—24no—
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—82no—
108C47⋊C8—44no—
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#2982—471no—
126#2581—471no—
127#1801D4711yescharacteristic
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—944no—
138#431—472no—
139#431—942no—
140#423C2×D4712yes—
141C2×#101—21no—
142#400—472no—
143#395—472no—
144#391—941no—
145#389—941no—
146#387—942no—
147#382—471no—
148#343—21no—
149Q8⋊C8—944no—
150C47#948411yescharacteristic, complemented, Sylow
151C22×Q8—471no—
152C22×Q8C22⋊D4711yescharacteristic
153C2×M16—22no—
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⋊C4C22⋊D4711yescharacteristic
164C2×C42—941no—
165C2×C42D18812yes—
166C2×C42C22×D4711yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—944no—
170C4⋊C8—942no—
171#101—42no—
172C22⋊C8—942no—
173C4×C8—474no—
174C4×C8—944no—
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#1062533512yes—
185C2×Q8#1062533712yes—
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—42no—
195M16—42no—
196C2×C8—42no—
197C2×C8—944no—
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—
208C42D4×D4712yes—
209C42#1062535212yes—
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#1309394714yes—
230C2×C4(C22⋊D47)⋊C412yes—
231C2×C4#1309399411yescharacteristic
232C2×C4#1309399612yes—
233C2×C4#1309400811yescharacteristic
234C2×C4C2×#1062533511yescharacteristic
235C2×C4C2×#1062533711yescharacteristic
236C2×C4#1309404412yes—
237C8—82no—
238C8—1884no—
239C22—21no—
240C22—21no—
241C22#1568900011yescharacteristic
242C22C2×#1309394711yescharacteristic
243C22#1568911711yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—3762no—
250C4—3761no—
251C4—1882no—
252C4#1568935012yes—
253C4#1568936212yes—
254C2—21no—
255C2—21no—
256C2#2024380411yescharacteristic
257C2#2024380911yescharacteristic
258C2#2024392311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#386
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1801D47
hypocenterC47#9484
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

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
#202524692
#202487502
#202418162
#202412482
C47×#18012
#948447