Finite Group Databasebeta

← #26109715

Subgroups — #26109715

Order 12032.

514 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 155 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#20243368C212yesmaximal
2#20241271C212yesmaximal, complemented
3#20241271C212yesmaximal, complemented
4C47×#2285C211yesmaximal, characteristic
5C2×C4⋊(C47⋊C8)C2214yes—
6C2×C22⋊(C47⋊C8)—22no—
7#15688878—22no—
8C2×#117×C47C412yes—
9C22×C42×C47C2211yescharacteristic
10C47×#384C2211yescharacteristic
11C47×#384C2211yescharacteristic
12C47×#382C412yes—
13C47×#380—22no—
14C4×C4⋊C4×C47—22no—
15C22×C47⋊C8—24no—
16C4⋊(C47⋊C8)—28no—
17C22⋊(C47⋊C8)—44no—
18C23×C4×C47C2×C411yescharacteristic
19C23×C4×C47D412yes—
20#117×C47—44no—
21C2×C4⋊C4×C47—22no—
22C2×C4⋊C4×C47—22no—
23C2×C4⋊C4×C47C2×C412yes—
24C2×C4⋊C4×C47C2×C412yes—
25C2×C22⋊C4×C47—22no—
26C2×C42×C47—24no—
27C2×C42×C47—22no—
28C2×C42×C47C2×C411yescharacteristic
29C2×C42×C47D412yes—
30C2×C42×C47C2311yescharacteristic
31#95×C47—24no—
32C24×C47C22⋊C411yescharacteristic
33C22×C4×C47—22no—
34C22×C4×C47—22no—
35C22×C4×C47—22no—
36C22×C4×C47—24no—
37C22×C4×C47—24no—
38C22×C4×C47—22no—
39C22×C4×C47—22no—
40C22×C4×C47—22no—
41C22×C4×C47—22no—
42C22×C4×C47C22⋊C412yes—
43C22×C4×C47C22⋊C411yescharacteristic
44C22×C4×C47C22×C411yescharacteristic
45C22×C4×C47C2×D412yes—
46C4⋊C4×C47—42no—
47C4⋊C4×C47—42no—
48C4⋊C4×C47—44no—
49C4⋊C4×C47—42no—
50C4⋊C4×C47—42no—
51C22⋊C4×C47—44no—
52C42×C47—22no—
53C42×C47—24no—
54C42×C47—44no—
55C42×C47#4114yes—
56C2×C47⋊C8—44no—
57C2×C47⋊C8—44no—
58C2×C47⋊C8—44no—
59C23×C47—21no—
60C23×C47—21no—
61C23×C47—22no—
62C23×C47—22no—
63C23×C47#10112yes—
64C23×C47C2×C22⋊C411yescharacteristic
65C2×C4×C47—28no—
66C2×C4×C47—21no—
67C2×C4×C47—21no—
68C2×C4×C47—21no—
69C2×C4×C47—21no—
70C2×C4×C47—22no—
71C2×C4×C47—22no—
72C2×C4×C47—22no—
73C2×C4×C47—44no—
74C2×C4×C47—44no—
75C2×C4×C47—42no—
76C2×C4×C47—42no—
77C2×C4×C47—44no—
78C2×C4×C47—44no—
79C2×C4×C47#10012yes—
80C2×C4×C47#11712yes—
81C2×C4×C47#12314yes—
82C2×C4×C47#13514yes—
83C2×C4×C47AGL(1,ℤ/8)12yes—
84C2×C4×C47#13712yes—
85C47⋊C8—84no—
86#8728—471nomaximal, Sylow
87C22×C47—24no—
88C22×C47—22no—
89C22×C47—41no—
90C22×C47—41no—
91C22×C47—42no—
92C22×C47#38111yescharacteristic
93C22×C47C2×#10011yescharacteristic
94C22×C47C2×#10111yescharacteristic
95C22×C47#41612yes—
96C22×C47#41712yes—
97C4×C47—22no—
98C4×C47—22no—
99C4×C47—42no—
100C4×C47—82no—
101C4×C47—82no—
102C4×C47#48214yes—
103C4×C47#48514yes—
104#2285D4711yescharacteristic
105#1797—472no—
106#1296—472no—
107#1296—472no—
108C2×C47—42no—
109C2×C47#154311yescharacteristic
110C2×C47#155212yes—
111C2×C47#156011yescharacteristic
112C2×C47#182111yescharacteristic
113C2×C47#182512yes—
114C2×#117Dic4712yes—
115C22×C42C2×D4711yescharacteristic
116C2×C4⋊C8—474no—
117C2×C22⋊C8—942no—
118#384C2×D4711yescharacteristic
119#384C2×D4711yescharacteristic
120#382Dic4712yes—
121#380—22no—
122C4×C4⋊C4—22no—
123#336—942no—
124C47#872811yescharacteristic, complemented, Sylow
125C23×C4C2×Dic4711yescharacteristic
126C23×C4C22⋊D4712yes—
127C22×C8—944no—
128#117—44no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4C2×Dic4712yes—
132C2×C4⋊C4C2×Dic4712yes—
133C2×C22⋊C4—22no—
134C2×C42—24no—
135C2×C42—22no—
136C2×C42C2×Dic4711yescharacteristic
137C2×C42C22⋊D4712yes—
138C2×C42C22×D4711yescharacteristic
139C4⋊C8—948no—
140C22⋊C8—1884no—
141#95—24no—
142C24C22⋊Dic4711yescharacteristic
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4C22⋊Dic4712yes—
153C22×C4C22⋊Dic4711yescharacteristic
154C22×C4C22×Dic4711yescharacteristic
155C22×C4C2×C22⋊D4712yes—
156C2×C8—1884no—
157C2×C8—1884no—
158C2×C8—1884no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C22⋊C4—44no—
165C42—22no—
166C42—24no—
167C42—44no—
168C42#1062535014yes—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23#1309394312yes—
174C23C2×C22⋊Dic4711yescharacteristic
175C2×C4—28no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4#1309394112yes—
190C2×C4#1309398312yes—
191C2×C4#1309400714yes—
192C2×C4#1309404414yes—
193C2×C4#1309404912yes—
194C2×C4#1309405412yes—
195C8—3764no—
196C22—22no—
197C22—24no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22#1568898611yescharacteristic
202C22C2×#1309394111yescharacteristic
203C22C2×#1309394311yescharacteristic
204C22#1568911912yes—
205C22#1568912412yes—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—82no—
210C4—82no—
211C4#1568944714yes—
212C4#1568946414yes—
213C2—42no—
214C2#2024205011yescharacteristic
215C2#2024208812yes—
216C2#2024212311yescharacteristic
217C2#2024346811yescharacteristic
218C2#2024348312yes—
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2285C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2285D47
hypocenterC47#8728
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #20241271C2semidirect
3. #20241271C2semidirect
124. C47#8728semidirect

Maximal subgroups

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

subgroupindex
#202433682
#202412712
C47×#22852
#872847