Finite Group Databasebeta

← #26108893

Subgroups — #26108893

Order 12032.

910 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 139 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#20243462C216yesmaximal
2C22×#126×C47C211yesmaximal, characteristic
3#15688979C2214yes—
4#15688890—212no—
5C2×#126×C47—212no—
6C22×C4⋊C4×C47C413yescomplemented
7C22×C22⋊C4×C47C2213yes—
8C22×C42×C47C411yescharacteristic, complemented
9C2×C22⋊Dic47—212no—
10C25×C47C2311yescharacteristic
11C23×C4×C47C2×C413yes—
12C23×C4×C47C2×C413yes—
13#126×C47—232no—
14C2×C4⋊C4×C47—26no—
15C2×C4⋊C4×C47—26no—
16C2×C4⋊C4×C47—26no—
17C2×C22⋊C4×C47—26no—
18C2×C22⋊C4×C47—26no—
19C2×C22⋊C4×C47D4112yes—
20C2×C42×C47—26no—
21C24×C47—46no—
22C24×C47C22×C411yescharacteristic
23C24×C47C2×D416yes—
24C22×C4×C47—23no—
25C22×C4×C47—23no—
26C22×C4×C47—23no—
27C22×C4×C47—23no—
28C22×C4×C47—46no—
29C22×C4×C47—46no—
30C22×C4×C47C22⋊C416yes—
31C22×C4×C47C22⋊C416yes—
32C22×Dic47—44no—
33C4⋊C4×C47—224no—
34C22⋊C4×C47—224no—
35C42×C47—28no—
36C22⋊Dic47—424no—
37C23×C47—26no—
38C23×C47—212no—
39C23×C47—46no—
40C23×C47—46no—
41C23×C47—88no—
42C23×C47C2×C22⋊C413yes—
43C23×C47#12014yes—
44C2×C4×C47—26no—
45C2×C4×C47—26no—
46C2×C4×C47—46no—
47C2×C4×C47—46no—
48C2×C4×C47—46no—
49C2×C4×C47—46no—
50C2×C4×C47—46no—
51C2×C4×C47—46no—
52C2×Dic47—812no—
53#8545—471nomaximal, Sylow
54C22×C47—28no—
55C22×C47—23no—
56C22×C47—23no—
57C22×C47—212no—
58C22×C47—812no—
59C22×C47#37911yescharacteristic
60C22×C47#41016yes—
61C4×C47—46no—
62C4×C47—46no—
63Dic47—164no—
64C22×#126D4711yescharacteristic
65#1819—476no—
66C2×C47—26no—
67C2×C47—44no—
68C2×C47#178213yes—
69C2×#126—212no—
70C22×C4⋊C4Dic4713yescomplemented
71C22×C22⋊C4C2×D4713yes—
72C22×C42Dic4711yescharacteristic, complemented
73#379—474no—
74#342—9412no—
75C47#854511yescharacteristic, complemented, Sylow
76C25C22×D4711yescharacteristic
77C23×C4C2×Dic4713yes—
78C23×C4C2×Dic4713yes—
79#126—232no—
80C2×C4⋊C4—26no—
81C2×C4⋊C4—26no—
82C2×C4⋊C4—26no—
83C2×C22⋊C4—26no—
84C2×C22⋊C4—26no—
85C2×C22⋊C4—9412no—
86C2×C22⋊C4C22⋊D47112yes—
87C2×C42—26no—
88C24—46no—
89C24C22×Dic4711yescharacteristic
90C24C2×C22⋊D4716yes—
91C22×C4—23no—
92C22×C4—23no—
93C22×C4—23no—
94C22×C4—23no—
95C22×C4—46no—
96C22×C4—46no—
97C22×C4—1884no—
98C22×C4C22⋊Dic4716yes—
99C22×C4C22⋊Dic4716yes—
100C4⋊C4—224no—
101C22⋊C4—224no—
102C22⋊C4—18824no—
103C42—28no—
104C23—26no—
105C23—212no—
106C23—46no—
107C23—46no—
108C23—88no—
109C23C2×C22⋊Dic4713yes—
110C23#1309399314yes—
111C2×C4—26no—
112C2×C4—26no—
113C2×C4—46no—
114C2×C4—46no—
115C2×C4—46no—
116C2×C4—46no—
117C2×C4—46no—
118C2×C4—46no—
119C2×C4—37612no—
120C22—28no—
121C22—23no—
122C22—23no—
123C22—212no—
124C22—812no—
125C22#1568897911yescharacteristic
126C22#1568909716yes—
127C4—46no—
128C4—46no—
129C4—7524no—
130C2—26no—
131C2—44no—
132C2#2024330613yes—
subgroupG/H
centerC22#15688979
commutatorC24×C47C22×C4
FrattiniC25C22×D47
FittingC22×#126×C47C2
socleC22×C47#379
prime radical, cosocleC25×C47C23
hypercenterC22×#126D47
hypocenterC47#8545
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊C4×C47C4semidirect
8. C22×C42×C47C4semidirect
70. C22×C4⋊C4Dic47semidirect
72. C22×C42Dic47semidirect
75. C47#8545semidirect

Maximal subgroups

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

subgroupindex
#202434622
C22×#126×C472
#854547