Finite Group Databasebeta

← #26113838

Subgroups — #26113838

Order 12032.

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

normal subgroups only

This group has 508 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#20246569C211yesmaximal, characteristic, complemented
2#20242114C211yesmaximal, characteristic, complemented
3#20241935C212yesmaximal
4#20241253C212yesmaximal, complemented
5C47×#3152C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689030—22no—
8#15689009—24no—
9#15688991C2212yes—
10#15688872—22no—
11#15688857—24no—
12C2×#137×C47—22no—
13C47×#532C2211yescharacteristic
14C2×C4⋊Q8×C47C2211yescharacteristic
15C47×#497—22no—
16C47×#494C412yescomplemented
17C47×#492C412yescomplemented
18C4×M16×C47C2211yescharacteristic
19C2×C4⋊Dic47—22no—
20C2×C4×Dic47—22no—
21C4⋊(C47⋊C8)—22no—
22C22⋊(C47⋊C8)—22no—
23#13093935—22no—
24C47⋊#95—24no—
25C47⋊#95—22no—
26C47⋊#95—22no—
27C47⋊#95D414yes—
28C2×#41×C47—21no—
29C22×Q8×C47D412yes—
30#137×C47—44no—
31#137×C47—44no—
32C2×Q16×C47—24no—
33C2×SD16×C47—24no—
34C2×M16×C47—21no—
35C4⋊Q8×C47—22no—
36C4⋊Q8×C47C2×C411yescharacteristic
37C4⋊Q8×C47C2×C412yes—
38C4⋊Q8×C47D412yes—
39#127×C47C2×C411yescharacteristic
40#124×C47—21no—
41#121×C47—22no—
42C4×D4×C47—22no—
43C2×C4⋊C4×C47—22no—
44C2×C42×C47C2311yescharacteristic
45#97×C47—21no—
46C4×C8×C47C2×C412yes—
47#41×C47—42no—
48#41×C47—42no—
49C2×Q8×C47—22no—
50C2×Q8×C47—21no—
51C2×Q8×C47—42no—
52C2×Q8×C47—42no—
53C2×Q8×C47C22⋊C414yes—
54C2×D4×C47—21no—
55C2×D4×C47—22no—
56C2×D4×C47—41no—
57C22×C4×C47—22no—
58C22×C4×C47—41no—
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D412yes—
61C22×Dic47—24no—
62Q16×C47—48no—
63SD16×C47—48no—
64M16×C47—24no—
65C2×C8×C47—22no—
66C4⋊C4×C47—24no—
67C4⋊C4×C47—24no—
68C4⋊C4×C47—22no—
69C22⋊C4×C47—42no—
70C42×C47C22×C411yescharacteristic
71C42×C47C2×D411yescharacteristic
72C42×C47#4112yes—
73C4⋊Dic47—82no—
74C4×Dic47—82no—
75C2×C47⋊C8—42no—
76C23×C47—41no—
77C23×C47#12011yescharacteristic
78Q8×C47—44no—
79Q8×C47—42no—
80Q8×C47—82no—
81D4×C47—42no—
82D4×C47—82no—
83D4×C47—82no—
84C2×C4×C47—22no—
85C2×C4×C47—24no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47—41no—
90C2×C4×C47—81no—
91C2×C4×C47—81no—
92C2×C4×C47C2×C22⋊C411yescharacteristic
93C2×C4×C47C4×D412yes—
94C2×C4×C47#12112yes—
95C2×C4×C47#12311yescharacteristic
96C2×Dic47—48no—
97C2×Dic47—44no—
98C8×C47—24no—
99C47⋊C8—82no—
100#9429—471nomaximal, Sylow
101C22×C47—22no—
102C22×C47—81no—
103C22×C47—82no—
104C22×C47#38611yescharacteristic
105C22×C47#45312yes—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—22no—
109C4×C47—22no—
110C4×C47—44no—
111C4×C47—81no—
112Dic47—84no—
113#3152D4711yescharacteristic
114#2381—471no—
115#1558—471no—
116#1514—472no—
117#1288—472no—
118C2×C47—22no—
119C2×C47—161no—
120C2×C47#188711yescharacteristic
121C2×C47#190512yes—
122C2×#137—22no—
123#532C2×D4711yescharacteristic
124C2×C4⋊Q8C2×D4711yescharacteristic
125#497—22no—
126#494Dic4712yescomplemented
127#492Dic4712yescomplemented
128#432—472no—
129C4×M16C2×D4711yescharacteristic
130#391—942no—
131#387—944no—
132#383—472no—
133#333—942no—
134#327—944no—
135C47#942911yescharacteristic, complemented, Sylow
136C2×#41—21no—
137C22×Q8C22⋊D4712yes—
138#137—44no—
139#137—44no—
140C2×Q16—24no—
141C2×SD16—24no—
142C2×M16—21no—
143C4⋊Q8—22no—
144C4⋊Q8C2×Dic4711yescharacteristic
145C4⋊Q8C2×Dic4712yes—
146C4⋊Q8C22⋊D4712yes—
147#127C2×Dic4711yescharacteristic
148#124—21no—
149#121—22no—
150C4×D4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—942no—
153C2×C42—942no—
154C2×C42C22×D4711yescharacteristic
155C4⋊C8—942no—
156C22⋊C8—942no—
157#97—21no—
158#97—942no—
159C4×C8C2×Dic4712yes—
160#95—474no—
161#95—944no—
162#95—942no—
163#95—942no—
164#41—42no—
165#41—42no—
166C2×Q8—22no—
167C2×Q8—21no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8C22⋊Dic4714yes—
171C2×D4—21no—
172C2×D4—22no—
173C2×D4—41no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—944no—
177C22×C4D4×D4712yes—
178C22×C4C2×C22⋊D4711yescharacteristic
179Q16—48no—
180SD16—48no—
181M16—24no—
182C2×C8—22no—
183C2×C8—1882no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—3762no—
188C22⋊C4—42no—
189C42—3762no—
190C42#1062535212yes—
191C42C22×Dic4711yescharacteristic
192C42C2×C22⋊D4711yescharacteristic
193C23—41no—
194C23#1309399211yescharacteristic
195Q8—44no—
196Q8—42no—
197Q8—82no—
198D4—42no—
199D4—82no—
200D4—82no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—1888no—
210C2×C4—1884no—
211C2×C4C2×C22⋊Dic4711yescharacteristic
212C2×C4D4×Dic4712yes—
213C2×C4#1309399612yes—
214C2×C4#1309400511yescharacteristic
215C8—24no—
216C8—3762no—
217C22—22no—
218C22—81no—
219C22—82no—
220C22#1568900311yescharacteristic
221C22#1568929012yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—81no—
228C4—3764no—
229C2—22no—
230C2—161no—
231C2#2024378711yescharacteristic
232C2#2024388412yes—
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3152C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3152D47
hypocenterC47#9429
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #20246569C2semidirect
2. #20242114C2semidirect
4. #20241253C2semidirect
16. C47×#494C4semidirect
17. C47×#492C4semidirect
126. #494Dic47semidirect
127. #492Dic47semidirect
135. C47#9429semidirect

Maximal subgroups

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

subgroupindex
#202465692
#202421142
#202419352
#202412532
C47×#31522
#942947