Finite Group Databasebeta

← #26109757

Subgroups — #26109757

Order 12032.

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

normal subgroups only

This group has 418 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#20243471C211yesmaximal, characteristic
2#20243465C211yesmaximal, characteristic
3#20241241C214yesmaximal
4C47×#2298C211yesmaximal, characteristic
5#15689087C2212yes—
6#15688986C2212yes—
7#15688892—22no—
8#15688890—22no—
9C2×#117×C47C2211yescharacteristic
10C47×#388—22no—
11C47×#387—22no—
12C4×C4⋊C4×C47C412yescomplemented
13C4×C22⋊C4×C47C2212yes—
14C47×#376C411yescharacteristic, complemented
15C43×C47C411yescharacteristic, complemented
16C2×#10625329—22no—
17C2×C22⋊Dic47—22no—
18C22⋊(C47⋊C8)—24no—
19C47⋊#95—24no—
20C23×C4×C47C2311yescharacteristic
21#117×C47—24no—
22C2×C4⋊C4×C47—22no—
23C2×C4⋊C4×C47D412yes—
24C2×C22⋊C4×C47—22no—
25C2×C22⋊C4×C47D412yes—
26C2×C42×C47—22no—
27C2×C42×C47C2×C411yescharacteristic
28C2×C42×C47C2×C411yescharacteristic
29C2×C42×C47C2×C412yes—
30C2×C42×C47C2×C412yes—
31#95×C47—22no—
32#95×C47—22no—
33#95×C47—22no—
34C24×C47C2×D411yescharacteristic
35C22×C4×C47—22no—
36C22×C4×C47—22no—
37C22×C4×C47—42no—
38C22×C4×C47C22⋊C412yes—
39C22×C4×C47C22⋊C412yes—
40C22×C4×C47C22×C411yescharacteristic
41C22×C4×C47C2×D411yescharacteristic
42C22×C4×C47#4114yes—
43C22×Dic47—42no—
44C4⋊C4×C47—24no—
45C4⋊C4×C47—44no—
46C22⋊C4×C47—24no—
47C22⋊C4×C47—44no—
48C42×C47—22no—
49C42×C47—22no—
50C42×C47—44no—
51C42×C47—42no—
52C42×C47—42no—
53C22⋊Dic47—44no—
54#10625329—44no—
55C2×C47⋊C8—42no—
56C23×C47—22no—
57C23×C47—42no—
58C23×C47C2×C22⋊C411yescharacteristic
59C23×C47#12312yes—
60C2×C4×C47—22no—
61C2×C4×C47—22no—
62C2×C4×C47—24no—
63C2×C4×C47—22no—
64C2×C4×C47—22no—
65C2×C4×C47—44no—
66C2×C4×C47—42no—
67C2×C4×C47—42no—
68C2×C4×C47—42no—
69C2×C4×C47—42no—
70C2×C4×C47—42no—
71C2×C4×C47—42no—
72C2×C4×C47—42no—
73C2×C4×C47#11712yes—
74C2×C4×C47#12312yes—
75C2×Dic47—82no—
76C2×Dic47—84no—
77C47⋊C8—82no—
78#8740—471nomaximal, Sylow
79C22×C47—22no—
80C22×C47—22no—
81C22×C47—24no—
82C22×C47—82no—
83C22×C47#38111yescharacteristic
84C22×C47#41012yes—
85C4×C47—22no—
86C4×C47—22no—
87C4×C47—44no—
88C4×C47—42no—
89C4×C47—42no—
90C4×C47—82no—
91Dic47—162no—
92#2298D4711yescharacteristic
93#1822—471no—
94#1820—471no—
95#1284—474no—
96C2×C47—22no—
97C2×C47—42no—
98C2×C47#182011yescharacteristic
99C2×C47#182712yes—
100C2×#117C2×D4711yescharacteristic
101#407—472no—
102#388—22no—
103#387—22no—
104#381—472no—
105C4×C4⋊C4Dic4712yescomplemented
106C4×C22⋊C4C2×D4712yes—
107#376Dic4711yescharacteristic, complemented
108C43Dic4711yescharacteristic, complemented
109#343—942no—
110#342—942no—
111C47#874011yescharacteristic, complemented, Sylow
112C23×C4C22×D4711yescharacteristic
113C2×M16—942no—
114#117—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4C22⋊D4712yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—942no—
119C2×C22⋊C4C22⋊D4712yes—
120C2×C42—22no—
121C2×C42C2×Dic4711yescharacteristic
122C2×C42C2×Dic4711yescharacteristic
123C2×C42C2×Dic4712yes—
124C2×C42C2×Dic4712yes—
125C22⋊C8—944no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—944no—
130C24C2×C22⋊D4711yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—1882no—
135C22×C4C22⋊Dic4712yes—
136C22×C4C22⋊Dic4712yes—
137C22×C4#1062535014yes—
138C22×C4C22×Dic4711yescharacteristic
139C22×C4C2×C22⋊D4711yescharacteristic
140M16—1884no—
141C2×C8—1882no—
142C4⋊C4—24no—
143C4⋊C4—44no—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C22⋊C4—1884no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42—42no—
151C42—44no—
152C23—22no—
153C23—42no—
154C23C2×C22⋊Dic4711yescharacteristic
155C23#1309400712yes—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—3762no—
170C2×C4—3764no—
171C2×C4#1309398312yes—
172C2×C4#1309400712yes—
173C8—3762no—
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#1568898611yescharacteristic
179C22#1568909712yes—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—7522no—
187C2—22no—
188C2—42no—
189C2#2024346511yescharacteristic
190C2#2024349212yes—
subgroupG/H
centerC22#15688986
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2298C2
socleC22×C47#381
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2298D47
hypocenterC47#8740
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
12. C4×C4⋊C4×C47C4semidirect
14. C47×#376C4semidirect
15. C43×C47C4semidirect
105. C4×C4⋊C4Dic47semidirect
107. #376Dic47semidirect
108. C43Dic47semidirect
111. C47#8740semidirect

Maximal subgroups

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

subgroupindex
#202434712
#202434652
#202412412
C47×#22982
#874047