Finite Group Databasebeta

← #30748446

Subgroups — #30748446

Order 17152.

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#22560283C216yesmaximal
2C22×#126×C67C211yesmaximal, characteristic
3#16863252C2214yes—
4#16863163—212no—
5C2×#126×C67—212no—
6C22×C4⋊C4×C67C413yescomplemented
7C22×C22⋊C4×C67C2213yes—
8C22×C42×C67C411yescharacteristic, complemented
9C2×C22⋊Dic67—212no—
10C25×C67C2311yescharacteristic
11C23×C4×C67C2×C413yes—
12C23×C4×C67C2×C413yes—
13#126×C67—232no—
14C2×C4⋊C4×C67—26no—
15C2×C4⋊C4×C67—26no—
16C2×C4⋊C4×C67—26no—
17C2×C22⋊C4×C67—26no—
18C2×C22⋊C4×C67—26no—
19C2×C22⋊C4×C67D4112yes—
20C2×C42×C67—26no—
21C22×Dic67—44no—
22C22⋊Dic67—424no—
23C24×C67—46no—
24C24×C67C22×C411yescharacteristic
25C24×C67C2×D416yes—
26C22×C4×C67—23no—
27C22×C4×C67—23no—
28C22×C4×C67—23no—
29C22×C4×C67—23no—
30C22×C4×C67—46no—
31C22×C4×C67—46no—
32C22×C4×C67C22⋊C416yes—
33C22×C4×C67C22⋊C416yes—
34C4⋊C4×C67—224no—
35C22⋊C4×C67—224no—
36C42×C67—28no—
37C23×C67—26no—
38C23×C67—212no—
39C23×C67—46no—
40C23×C67—46no—
41C23×C67—88no—
42C23×C67C2×C22⋊C413yes—
43C23×C67#12014yes—
44C2×C4×C67—26no—
45C2×C4×C67—26no—
46C2×C4×C67—46no—
47C2×C4×C67—46no—
48C2×C4×C67—46no—
49C2×C4×C67—46no—
50C2×C4×C67—46no—
51C2×C4×C67—46no—
52C2×Dic67—812no—
53C22×C67—28no—
54C22×C67—23no—
55C22×C67—23no—
56C22×C67—212no—
57C22×C67—812no—
58C22×C67#37911yescharacteristic
59C22×C67#41016yes—
60C4×C67—46no—
61C4×C67—46no—
62Dic67—164no—
63#8545—671nomaximal, Sylow
64C2×C67—26no—
65C2×C67—44no—
66C2×C67#178213yes—
67C22×#126D6711yescharacteristic
68#1819—676no—
69C67#854511yescharacteristic, complemented, Sylow
70C2×#126—212no—
71C22×C4⋊C4Dic6713yescomplemented
72C22×C22⋊C4C2×D6713yes—
73C22×C42Dic6711yescharacteristic, complemented
74#379—674no—
75#342—13412no—
76C25C22×D6711yescharacteristic
77C23×C4C2×Dic6713yes—
78C23×C4C2×Dic6713yes—
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—13412no—
86C2×C22⋊C4C22⋊D67112yes—
87C2×C42—26no—
88C24—46no—
89C24C22×Dic6711yescharacteristic
90C24C2×C22⋊D6716yes—
91C22×C4—23no—
92C22×C4—23no—
93C22×C4—23no—
94C22×C4—23no—
95C22×C4—46no—
96C22×C4—46no—
97C22×C4—2684no—
98C22×C4C22⋊Dic6716yes—
99C22×C4C22⋊Dic6716yes—
100C4⋊C4—224no—
101C22⋊C4—224no—
102C22⋊C4—26824no—
103C42—28no—
104C23—26no—
105C23—212no—
106C23—46no—
107C23—46no—
108C23—88no—
109C23C2×C22⋊Dic6713yes—
110C23#1452401214yes—
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—53612no—
120C22—28no—
121C22—23no—
122C22—23no—
123C22—212no—
124C22—812no—
125C22#1686325211yescharacteristic
126C22#1686337016yes—
127C4—46no—
128C4—46no—
129C4—10724no—
130C2—26no—
131C2—44no—
132C2#2256012713yes—
subgroupG/H
centerC22#16863252
commutatorC24×C67C22×C4
FrattiniC25C22×D67
FittingC22×#126×C67C2
socleC22×C67#379
prime radical, cosocleC25×C67C23
hypercenterC22×#126D67
hypocenterC67#8545
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C25 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊C4×C67C4semidirect
8. C22×C42×C67C4semidirect
69. C67#8545semidirect
71. C22×C4⋊C4Dic67semidirect
73. C22×C42Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602832
C22×#126×C672
#854567