Finite Group Databasebeta

← #26111227

Subgroups — #26111227

Order 12032.

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

normal subgroups only

This group has 416 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#20253733C211yesmaximal, characteristic
2#20242050C211yesmaximal, characteristic
3#20241860C212yesmaximal, complemented
4#20241286C212yesmaximal, complemented
5C47×#2980C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7#15689491—21no—
8#15689487—21no—
9#15689482C414yes—
10#15689176C2212yes—
11#15689119—22no—
12C2×#13093935C2211yescharacteristic
13#15688872—21no—
14#15688867—21no—
15#15688853—24no—
16C47×#532C2211yescharacteristic
17C47×#435—24no—
18C47×#432C2212yes—
19C47×#408—22no—
20C22×C47⋊C8—21no—
21#13094024C2×C412yes—
22#13094016—21no—
23#13094003—22no—
24C4×Dic94—22no—
25#13093983—21no—
26C4⋊(C47⋊C8)—22no—
27#13093948—24no—
28#13093945—24no—
29C22⋊(C47⋊C8)—22no—
30#13093935—21no—
31#13093935C2×C412yes—
32C4×C47⋊C8—22no—
33C2×#41×C47D412yes—
34C2×M16×C47—22no—
35C22×C8×C47—22no—
36C4⋊Q8×C47D411yescharacteristic
37#127×C47D411yescharacteristic
38#124×C47C2×C412yes—
39#121×C47—22no—
40C4×D4×C47—22no—
41C2×C42×C47C2311yescharacteristic
42C4⋊C8×C47D414yes—
43C22⋊C8×C47—24no—
44C22⋊C8×C47—22no—
45#97×C47—22no—
46#41×C47—42no—
47#41×C47—42no—
48C2×Q8×C47C22⋊C412yes—
49C2×D4×C47—22no—
50C2×D4×C47—22no—
51C2×D4×C47C22⋊C412yes—
52C22×C4×C47—22no—
53C22×C4×C47C2×D411yescharacteristic
54C22×C4×C47#4112yes—
55C2×Dic94—22no—
56M16×C47—82no—
57C2×C8×C47—24no—
58C2×C8×C47—82no—
59C4⋊C4×C47—22no—
60C22⋊C4×C47—24no—
61C42×C47C22×C411yescharacteristic
62C42×C47C2×D411yescharacteristic
63C42×C47C2×D412yes—
64C22⋊Dic47—41no—
65C4⋊Dic47—22no—
66C4⋊Dic47—22no—
67Dic47⋊C4—42no—
68C4×Dic47—41no—
69C2×C47⋊C8—22no—
70C2×C47⋊C8—42no—
71C2×C47⋊C8—42no—
72C23×C47—22no—
73C23×C47#12311yescharacteristic
74Q8×C47—42no—
75D4×C47—42no—
76D4×C47—82no—
77D4×C47—82no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—42no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47C2×C22⋊C411yescharacteristic
84C2×C4×C47C4×D412yes—
85C2×C4×C47#12011yescharacteristic
86C2×C4×C47#12112yes—
87C2×Dic47—42no—
88Dic94—44no—
89C8×C47—44no—
90C47⋊C8—42no—
91C47⋊C8—82no—
92#9005—471nomaximal, Sylow
93C22×C47—22no—
94C22×C47—21no—
95C22×C47—44no—
96C22×C47—42no—
97C22×C47#38611yescharacteristic
98C4×C47—22no—
99C4×C47—22no—
100C4×C47—21no—
101C4×C47—42no—
102C4×C47#45512yes—
103Dic47—82no—
104#3135—471no—
105#2980D4711yescharacteristic
106#1543—471no—
107#1497—472no—
108#1302—472no—
109C2×C47—41no—
110C2×C47—82no—
111C2×C47#188112yes—
112C2×C47#188511yescharacteristic
113#533—471no—
114#532C2×D4711yescharacteristic
115#491—941no—
116#490—941no—
117#489—474no—
118#435—24no—
119#432C2×D4712yes—
120#431—472no—
121#416—942no—
122#408—22no—
123C2×#97—471no—
124#333—941no—
125#331—941no—
126Q8⋊C8—944no—
127C47#900511yescharacteristic, complemented, Sylow
128C2×#41C22⋊D4712yes—
129C2×M16—22no—
130C22×C8—22no—
131C22×C8—941no—
132C4⋊Q8—472no—
133C4⋊Q8D18811yescharacteristic
134#127D18811yescharacteristic
135#125—941no—
136#124C4×D4712yes—
137C22⋊Q8—942no—
138#121—22no—
139C4×Q8—942no—
140C4×D4—22no—
141#117—941no—
142C2×C42C22×D4711yescharacteristic
143C4⋊C8—942no—
144C4⋊C8C22⋊D4714yes—
145#103—944no—
146#102—944no—
147C22⋊C8—24no—
148C22⋊C8—22no—
149C22⋊C8—942no—
150#97—22no—
151#97—472no—
152#97—941no—
153C4×C8—942no—
154#41—42no—
155#41—42no—
156C2×Q8—942no—
157C2×Q8(C2×D47)⋊C412yes—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4(C2×D47)⋊C412yes—
161C22×C4—22no—
162C22×C4#1062535012yes—
163C22×C4C2×C22⋊D4711yescharacteristic
164M16—82no—
165C2×C8—24no—
166C2×C8—82no—
167C2×C8—942no—
168C2×C8—1882no—
169C2×C8—1882no—
170C4⋊C4—22no—
171C4⋊C4—942no—
172C4⋊C4—942no—
173C4⋊C4—1882no—
174C22⋊C4—24no—
175C22⋊C4—1881no—
176C42—1881no—
177C42C2×C4×D4711yescharacteristic
178C42C2×D18811yescharacteristic
179C42C2×C22⋊D4712yes—
180C23—22no—
181C23#1309400711yescharacteristic
182Q8—42no—
183Q8—1884no—
184D4—42no—
185D4—82no—
186D4—82no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—1882no—
193C2×C4C2×(C2×D47)⋊C411yescharacteristic
194C2×C4C4×C22⋊D4712yes—
195C2×C4#1309399311yescharacteristic
196C2×C4#1309399812yes—
197C8—44no—
198C8—3762no—
199C8—1882no—
200C22—22no—
201C22—21no—
202C22—44no—
203C22—42no—
204C22#1568900211yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—3762no—
210C4#1568930112yes—
211C2—41no—
212C2—82no—
213C2#2024375012yes—
214C2#2024377611yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2980C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#2980D47
hypocenterC47#9005
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
3. #20241860C2semidirect
4. #20241286C2semidirect
127. C47#9005semidirect

Maximal subgroups

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

subgroupindex
#202537332
#202420502
#202418602
#202412862
C47×#29802
#900547