Finite Group Databasebeta

← #16926939

Subgroups — #16926939

Order 4352.

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#14528858C216yesmaximal
2C22×C17×#126C211yesmaximal, characteristic
3#11762665C2214yes—
4#11762576—212no—
5C2×C17×#126—212no—
6C22×C4⋊C4×C17C413yescomplemented
7C22×C22⋊C4×C17C2213yes—
8C22×C42×C17C411yescharacteristic, complemented
9C25×C17C2311yescharacteristic
10C23×C4×C17C2×C413yes—
11C23×C4×C17C2×C413yes—
12C17×#126—232no—
13C2×C4⋊C4×C17—26no—
14C2×C4⋊C4×C17—26no—
15C2×C4⋊C4×C17—26no—
16C2×C22⋊C4×C17—26no—
17C2×C22⋊C4×C17—26no—
18C2×C22⋊C4×C17D4112yes—
19C2×C42×C17—26no—
20C2×C22⋊Dic17—212no—
21C24×C17—46no—
22C24×C17C22×C411yescharacteristic
23C24×C17C2×D416yes—
24C22×C4×C17—23no—
25C22×C4×C17—23no—
26C22×C4×C17—23no—
27C22×C4×C17—23no—
28C22×C4×C17—46no—
29C22×C4×C17—46no—
30C22×C4×C17C22⋊C416yes—
31C22×C4×C17C22⋊C416yes—
32C22×Dic17—44no—
33C4⋊C4×C17—224no—
34C22⋊C4×C17—224no—
35C42×C17—28no—
36C22⋊Dic17—424no—
37#8545—171nomaximal, Sylow
38C23×C17—26no—
39C23×C17—212no—
40C23×C17—46no—
41C23×C17—46no—
42C23×C17—88no—
43C23×C17C2×C22⋊C413yes—
44C23×C17#12014yes—
45C2×C4×C17—26no—
46C2×C4×C17—26no—
47C2×C4×C17—46no—
48C2×C4×C17—46no—
49C2×C4×C17—46no—
50C2×C4×C17—46no—
51C2×C4×C17—46no—
52C2×C4×C17—46no—
53C2×Dic17—812no—
54C22×#126D1711yescharacteristic
55#1819—176no—
56C22×C17—28no—
57C22×C17—23no—
58C22×C17—23no—
59C22×C17—212no—
60C22×C17—812no—
61C22×C17#37911yescharacteristic
62C22×C17#41016yes—
63C4×C17—46no—
64C4×C17—46no—
65Dic17—164no—
66C2×#126—212no—
67C22×C4⋊C4Dic1713yescomplemented
68C22×C22⋊C4C2×D1713yes—
69C22×C42Dic1711yescharacteristic, complemented
70#379—174no—
71#342—3412no—
72C2×C17—26no—
73C2×C17—44no—
74C2×C17#178213yes—
75C25C22×D1711yescharacteristic
76C23×C4C2×Dic1713yes—
77C23×C4C2×Dic1713yes—
78#126—232no—
79C2×C4⋊C4—26no—
80C2×C4⋊C4—26no—
81C2×C4⋊C4—26no—
82C2×C22⋊C4—26no—
83C2×C22⋊C4—26no—
84C2×C22⋊C4—3412no—
85C2×C22⋊C4C22⋊D17112yes—
86C2×C42—26no—
87C17#854511yescharacteristic, complemented, Sylow
88C24—46no—
89C24C22×Dic1711yescharacteristic
90C24C2×C22⋊D1716yes—
91C22×C4—23no—
92C22×C4—23no—
93C22×C4—23no—
94C22×C4—23no—
95C22×C4—46no—
96C22×C4—46no—
97C22×C4—684no—
98C22×C4C22⋊Dic1716yes—
99C22×C4C22⋊Dic1716yes—
100C4⋊C4—224no—
101C22⋊C4—224no—
102C22⋊C4—6824no—
103C42—28no—
104C23—26no—
105C23—212no—
106C23—46no—
107C23—46no—
108C23—88no—
109C23C2×C22⋊Dic1713yes—
110C23#1058763914yes—
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—13612no—
120C22—28no—
121C22—23no—
122C22—23no—
123C22—212no—
124C22—812no—
125C22#1176266511yescharacteristic
126C22#1176278316yes—
127C4—46no—
128C4—46no—
129C4—2724no—
130C2—26no—
131C2—44no—
132C2#1452870213yes—
subgroupG/H
centerC22#11762665
commutatorC24×C17C22×C4
FrattiniC25C22×D17
FittingC22×C17×#126C2
socleC22×C17#379
prime radical, cosocleC25×C17C23
hypercenterC22×#126D17
hypocenterC17#8545
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C25 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊C4×C17C4semidirect
8. C22×C42×C17C4semidirect
67. C22×C4⋊C4Dic17semidirect
69. C22×C42Dic17semidirect
87. C17#8545semidirect

Maximal subgroups

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

subgroupindex
#145288582
C22×C17×#1262
#854517