Finite Group Databasebeta

← #26113837

Subgroups — #26113837

Order 12032.

508 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 99 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#20253845C211yesmaximal, characteristic
2#20242114C211yesmaximal, characteristic, complemented
3#20241934C212yesmaximal
4#20241252C212yesmaximal, complemented
5C47×#2381C211yesmaximal, characteristic, complemented
6C2×#13094055—22no—
7#15689673C2211yescharacteristic
8#15689515—22no—
9#15689502C412yescomplemented
10#15689497C412yescomplemented
11#15689180C2212yes—
12C4×#10625329C2211yescharacteristic
13#15688872—22no—
14#15688856—24no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C47×#391—22no—
17C47×#387—24no—
18C47×#383C2212yes—
19C2×#10625350—21no—
20#13094055—44no—
21#13094055—44no—
22C2×#10625337—24no—
23C2×#10625336—24no—
24C2×#10625329—21no—
25#13094024C2×C411yescharacteristic
26#13094021C2×C411yescharacteristic
27#13094012—21no—
28#13093998—22no—
29C4×D188—22no—
30C4⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32#13093935—21no—
33#13093935—22no—
34C4×C47⋊C8C2×C412yes—
35C22×Q8×C47D412yes—
36C4⋊Q8×C47—22no—
37C4⋊Q8×C47C2×C412yes—
38C4⋊Q8×C47D412yes—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C42×C47—22no—
42C2×C42×C47C2311yescharacteristic
43#95×C47—24no—
44#95×C47—22no—
45#95×C47—22no—
46#95×C47D414yes—
47C2×Q8×C47—22no—
48C2×Q8×C47—42no—
49C2×Q8×C47—42no—
50C2×Q8×C47C22⋊C414yes—
51C22×C4×C47—22no—
52C22×C4×C47—24no—
53C22×C4×C47C2×D411yescharacteristic
54C22×C4×C47C2×D412yes—
55C2×C22⋊D47—41no—
56#10625350—42no—
57#10625350—42no—
58C2×D188—21no—
59C2×D188—22no—
60C2×C4×D47—41no—
61C2×Dic94—21no—
62C4⋊C4×C47—24no—
63C4⋊C4×C47—24no—
64C4⋊C4×C47—82no—
65C42×C47—82no—
66C42×C47C22×C411yescharacteristic
67C42×C47C2×D411yescharacteristic
68C42×C47#4112yes—
69#10625337—48no—
70#10625336—48no—
71(C2×D47)⋊C4—42no—
72C4⋊Dic47—22no—
73#10625329—24no—
74C2×C47⋊C8—22no—
75C2×C47⋊C8—42no—
76C23×C47#12011yescharacteristic
77C22×D47—41no—
78Q8×C47—44no—
79Q8×C47—82no—
80C2×C4×C47—22no—
81C2×C4×C47—24no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—48no—
86C2×C4×C47—44no—
87C2×C4×C47C2×C22⋊C411yescharacteristic
88C2×C4×C47C4×D412yes—
89C2×C4×C47#12112yes—
90C2×C4×C47#12311yescharacteristic
91C22⋊D47—82no—
92C2×Dic47—41no—
93D188—42no—
94D188—82no—
95C4×D47—81no—
96C4×D47—81no—
97Dic94—42no—
98C47⋊C8—24no—
99C47⋊C8—82no—
100#9429—471nomaximal, Sylow
101C22×C47—22no—
102C22×C47#38611yescharacteristic
103C22×C47#45312yes—
104C2×D47—81no—
105C2×D47—82no—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—22no—
109C4×C47—22no—
110C4×C47—44no—
111C4×C47—84no—
112Dic47—81no—
113#3152—471no—
114#2381D4711yescharacteristic, complemented
115#1558—471no—
116#1514—472no—
117#1288—472no—
118C2×C47—22no—
119C2×C47#188711yescharacteristic
120C2×C47#190512yes—
121D47—161no—
122C2×#137—942no—
123#532—471no—
124C2×C4⋊Q8C2×D4711yescharacteristic
125#497—942no—
126#494—472no—
127#492—472no—
128#432—472no—
129C4×M16—471no—
130#391—22no—
131#387—24no—
132#383C2×D4712yes—
133#333—942no—
134#327—944no—
135C47#942911yescharacteristic, complemented, Sylow
136C2×#41—941no—
137C22×Q8C22⋊D4712yes—
138#137—1884no—
139#137—1884no—
140C2×Q16—944no—
141C2×SD16—944no—
142C2×M16—941no—
143C4⋊Q8—22no—
144C4⋊Q8—471no—
145C4⋊Q8C4×D4712yes—
146C4⋊Q8D18812yes—
147#127—471no—
148#124—941no—
149#121—942no—
150C4×D4—942no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C22×D4711yescharacteristic
155C4⋊C8—942no—
156C22⋊C8—942no—
157#97—941no—
158#97—942no—
159C4×C8—472no—
160#95—24no—
161#95—22no—
162#95—22no—
163#95C22⋊D4714yes—
164#41—1882no—
165#41—1882no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—941no—
170C2×Q8(C2×D47)⋊C414yes—
171C2×D4—941no—
172C2×D4—942no—
173C2×D4—1881no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—1881no—
177C22×C4C2×C22⋊D4711yescharacteristic
178C22×C4C2×C22⋊D4712yes—
179Q16—1888no—
180SD16—1888no—
181M16—944no—
182C2×C8—942no—
183C2×C8—1882no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—82no—
187C4⋊C4—942no—
188C22⋊C4—1882no—
189C42—82no—
190C42C2×C4×D4711yescharacteristic
191C42C2×D18811yescharacteristic
192C42#1062535012yes—
193C23—1881no—
194C23#1309399311yescharacteristic
195Q8—44no—
196Q8—82no—
197Q8—1882no—
198D4—3762no—
199D4—3762no—
200D4—1882no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—48no—
207C2×C4—44no—
208C2×C4—3761no—
209C2×C4—3761no—
210C2×C4—1881no—
211C2×C4C2×(C2×D47)⋊C411yescharacteristic
212C2×C4C4×C22⋊D4712yes—
213C2×C4#1309399812yes—
214C2×C4#1309400711yescharacteristic
215C8—944no—
216C8—3762no—
217C22—22no—
218C22—3761no—
219C22—3762no—
220C22#1568900211yescharacteristic
221C22#1568929112yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—84no—
228C4—3761no—
229C2—22no—
230C2—7521no—
231C2#2024378611yescharacteristic
232C2#2024388212yes—
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2381C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2381D47
hypocenterC47#9429
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. #20242114C2semidirect
4. #20241252C2semidirect
5. C47×#2381C2semidirect
9. #15689502C4semidirect
10. #15689497C4semidirect
114. #2381D47semidirect
135. C47#9429semidirect

Maximal subgroups

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

subgroupindex
#202538452
#202421142
#202419342
#202412522
C47×#23812
#942947