Finite Group Databasebeta

← #26118524

Subgroups — #26118524

Order 12032.

542 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 103 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#20254329C212yesmaximal
2#20253912C211yesmaximal, characteristic, complemented
3#20241951C212yesmaximal, complemented
4#20241233C211yesmaximal, characteristic, complemented
5C47×#1961C211yesmaximal, characteristic
6#15689680C2212yes—
7#15689532—22no—
8#15689532—22no—
9#15689495—22no—
10#15689469—22no—
11#15689320—24no—
12#15689274—24no—
13#15689176C2212yes—
14#15689161—22no—
15#15688859—24no—
16C2×#127×C47C2211yescharacteristic
17C2×#102×C47—22no—
18C2×#97×C47C2211yescharacteristic
19C47×#382C2211yescharacteristic
20C2×#10625335—28no—
21#13094024D412yes—
22#13094024D412yes—
23#13094016—22no—
24#13094003—22no—
25#13094003—22no—
26C4×Dic94—22no—
27C4×Dic94—22no—
28#13093983—22no—
29#13093958—24no—
30#13093956—22no—
31#13093956—22no—
32C4⋊(C47⋊C8)—22no—
33#13093945—28no—
34C22⋊(C47⋊C8)D414yes—
35C4×C47⋊C8—22no—
36C22×D4×C47—21no—
37C22×D4×C47—22no—
38C22×C8×C47D412yes—
39#127×C47—22no—
40#127×C47—21no—
41#127×C47D412yes—
42C2×C4⋊C4×C47—21no—
43C2×C42×C47C2311yescharacteristic
44#102×C47—44no—
45#97×C47—22no—
46#95×C47—22no—
47C24×C47—41no—
48C2×D4×C47—24no—
49C2×D4×C47—22no—
50C2×D4×C47—41no—
51C2×D4×C47—42no—
52C2×D4×C47—42no—
53C2×D4×C47—42no—
54C2×D4×C47—84no—
55C22×C4×C47—41no—
56C22×C4×C47C2×D412yes—
57C22×C4×C47C2×D411yescharacteristic
58C2×Dic94—22no—
59C2×Dic94—22no—
60C2×C8×C47—22no—
61C2×C8×C47—22no—
62C2×C8×C47SD1614yescomplemented
63C4⋊C4×C47—42no—
64C42×C47C2×D411yescharacteristic
65C42×C47C2×D412yes—
66C42×C47#4111yescharacteristic
67C22⋊Dic47—42no—
68#10625335—88no—
69C4⋊Dic47—22no—
70C4⋊Dic47—22no—
71C4⋊Dic47—22no—
72C4⋊Dic47—22no—
73Dic47⋊C4—42no—
74Dic47⋊C4—42no—
75C4×Dic47—42no—
76C2×C47⋊C8—24no—
77C23×C47—42no—
78C23×C47—84no—
79C23×C47—81no—
80C23×C47—81no—
81C23×C47#12711yescharacteristic
82D4×C47—48no—
83D4×C47—44no—
84D4×C47—84no—
85D4×C47—82no—
86C2×C4×C47—42no—
87C2×C4×C47—82no—
88C2×C4×C47—81no—
89C2×C4×C47#12011yescharacteristic
90C2×C4×C47#12012yes—
91C2×C4×C47#12112yes—
92C2×C4×C47#12111yescharacteristic
93C2×C4×C47C2×SD1612yes—
94C2×C4×C47#13712yes—
95C2×Dic47—42no—
96C2×Dic47—42no—
97Dic94—44no—
98Dic94—82no—
99C8×C47—24no—
100C47⋊C8—44no—
101#10220—471nomaximal, Sylow
102C22×C47—22no—
103C22×C47—84no—
104C22×C47—82no—
105C22×C47—162no—
106C22×C47—162no—
107C22×C47#39211yescharacteristic
108C22×C47#49211yescharacteristic
109C22×C47#49711yescharacteristic
110C4×C47—22no—
111C4×C47—22no—
112C4×C47—161no—
113C4×C47#44812yes—
114C4×C47#46112yes—
115Dic47—82no—
116Dic47—82no—
117#3208—472no—
118#3162—471no—
119#1961D4711yescharacteristic
120#1518—472no—
121#1281—471no—
122C2×C47—21no—
123C2×C47—21no—
124C2×C47—162no—
125C2×C47#200812yes—
126C2×C47#201311yescharacteristic
127#533—472no—
128C2×#127C2×D4711yescharacteristic
129#501—942no—
130#501—942no—
131#492—942no—
132#486—942no—
133#460—944no—
134#450—944no—
135#431—472no—
136#427—942no—
137C2×#102—22no—
138C2×#97C2×D4711yescharacteristic
139#382C2×D4711yescharacteristic
140#328—944no—
141C47#1022011yescharacteristic, complemented, Sylow
142C22×D4—22no—
143C22×D4—21no—
144C2×SD16—948no—
145C22×C8C22⋊D4712yes—
146C4⋊Q8—472no—
147C4⋊Q8—472no—
148#127—21no—
149#127—22no—
150#127D18812yes—
151#125—942no—
152C22⋊Q8—942no—
153C22⋊Q8—942no—
154C4×Q8—942no—
155C4×Q8—942no—
156#117—942no—
157C2×C4⋊C4—21no—
158C2×C42C22×D4711yescharacteristic
159#107—944no—
160#106—942no—
161#106—942no—
162C4⋊C8—942no—
163#102—44no—
164#102—948no—
165C22⋊C8—474no—
166#97—22no—
167C4×C8—942no—
168#95—22no—
169C24—41no—
170C2×Q8—942no—
171C2×Q8—942no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—84no—
179C22×C4—41no—
180C22×C4D4×D4712yes—
181C22×C4C2×C22⋊D4711yescharacteristic
182SD16—3768no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—944no—
186C2×C8#1062533514yescomplemented
187C4⋊C4—42no—
188C4⋊C4—942no—
189C4⋊C4—942no—
190C4⋊C4—942no—
191C4⋊C4—942no—
192C4⋊C4—1882no—
193C4⋊C4—1882no—
194C22⋊C4—1882no—
195C42—1882no—
196C42C2×D18811yescharacteristic
197C42#1062535011yescharacteristic
198C42D4×D4712yes—
199C23—42no—
200C23—84no—
201C23—81no—
202C23—81no—
203C23#1309402011yescharacteristic
204Q8—3762no—
205Q8—1884no—
206D4—48no—
207D4—44no—
208D4—84no—
209D4—82no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—81no—
213C2×C4—1882no—
214C2×C4—1882no—
215C2×C4#1309399211yescharacteristic
216C2×C4#1309399412yes—
217C2×C4#1309399712yes—
218C2×C4#1309399811yescharacteristic
219C2×C4C2×#1062533512yes—
220C2×C4#1309405412yes—
221C8—24no—
222C8—1884no—
223C22—22no—
224C22—84no—
225C22—82no—
226C22—162no—
227C22—162no—
228C22#1568903211yescharacteristic
229C22#1568949511yescharacteristic
230C22#1568951811yescharacteristic
231C4—22no—
232C4—22no—
233C4—161no—
234C4—3762no—
235C4—3762no—
236C4#1568926112yes—
237C4#1568932812yes—
238C2—21no—
239C2—21no—
240C2—162no—
241C2#2024444312yes—
242C2#2024447011yescharacteristic
subgroupG/H
centerC22#15689032
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1961C2
socleC22×C47#392
prime radicalC2×#127×C47C22
hypercenter#1961D47
hypocenterC47#10220
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #20253912C2semidirect
3. #20241951C2semidirect
4. #20241233C2semidirect
62. C2×C8×C47SD16semidirect
141. C47#10220semidirect
186. C2×C8#10625335semidirect

Maximal subgroups

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

subgroupindex
#202543292
#202539122
#202419512
#202412332
C47×#19612
#1022047