Finite Group Databasebeta

← #14544682

Subgroups — #14544682

Order 2176.

300 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 66 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#11763752C212yesmaximal
2#11763687C214yesmaximal
3C2×#10587578C211yesmaximal, characteristic
4C22×C4×D17C2211yescharacteristic
5C2×C22⋊#595C2214yes—
6C2×C4⋊C4×C17C411yescharacteristic, complemented
7C2×#63227C2212yes—
8C2×C4⋊Dic17C411yescharacteristic, complemented
9C2×C4×Dic17C412yescomplemented
10#10587578—24no—
11(C2×#595)⋊C4—24no—
12C23×D17C2311yescharacteristic
13C22×#595—24no—
14C22×C4×C17C2×C412yes—
15C22×C4×C17C2×C411yescharacteristic
16C22×Dic17C2×C411yescharacteristic
17C22×Dic17C2×C412yes—
18C2×C4×D17—24no—
19C2×C4×D17D414yes—
20C22⋊#595—28no—
21C4⋊C4×C17—22no—
22#63227—24no—
23C4⋊Dic17—22no—
24C4×Dic17—24no—
25C23×C17C22×C411yescharacteristic
26C22×D17—41no—
27C22×D17—41no—
28C22×D17C2×D412yes—
29C22×D17#4114yes—
30C2×#595—44no—
31C2×#595—44no—
32C2×#595—44no—
33C2×C4×C17—22no—
34C2×C4×C17—22no—
35C2×C4×C17—42no—
36C2×C4×C17C22⋊C412yes—
37C2×Dic17—22no—
38C2×Dic17—22no—
39C2×Dic17—42no—
40C2×Dic17C22⋊C412yes—
41C4×D17—28no—
42#1820—171nomaximal, Sylow
43C22×C17—21no—
44C22×C17—21no—
45C22×C17C2×C22⋊C411yescharacteristic
46C22×C17#11712yes—
47C2×D17—22no—
48C2×D17—22no—
49C2×D17—44no—
50C2×D17#12314yes—
51#595—84no—
52C4×C17—22no—
53C4×C17—42no—
54Dic17—22no—
55Dic17—42no—
56C2×#117—171no—
57#381—172no—
58#342—174no—
59C2×C17—22no—
60C2×C17#38111yescharacteristic
61C2×C17#41011yescharacteristic
62C2×C17#41011yescharacteristic
63D17—24no—
64C23×C4—171no—
65#117—344no—
66C2×C4⋊C4—171no—
67C2×C4⋊C4#59511yescharacteristic, complemented
68C2×C22⋊C4—172no—
69C2×C22⋊C4—174no—
70C2×C42—172no—
71#95—344no—
72C17#182011yescharacteristic, complemented, Sylow
73C24—171no—
74C22×C4—171no—
75C22×C4—174no—
76C22×C4—172no—
77C22×C4—344no—
78C22×C4—344no—
79C22×C4C2×#59512yes—
80C22×C4C2×#59511yescharacteristic
81C4⋊C4—22no—
82C4⋊C4—342no—
83C22⋊C4—344no—
84C22⋊C4—348no—
85C42—344no—
86C23—172no—
87C23—174no—
88C23—681no—
89C23—681no—
90C23C22×#59511yescharacteristic
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—42no—
94C2×C4—172no—
95C2×C4—348no—
96C2×C4—342no—
97C2×C4—342no—
98C2×C4—682no—
99C2×C4—684no—
100C2×C4—684no—
101C2×C4—684no—
102C2×C4C22⋊#59512yes—
103C22—21no—
104C22—21no—
105C22—174no—
106C22—342no—
107C22—342no—
108C22—684no—
109C22#1058767712yes—
110C22C2×C22⋊#59511yescharacteristic
111C4—22no—
112C4—42no—
113C4—342no—
114C4—682no—
115C4—1364no—
116C2—22no—
117C2—344no—
118C2#1176375211yescharacteristic
119C2#1176378911yescharacteristic
120C2#1176379011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radical, cosocleC23×D17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1820
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C2×C4⋊C4×C17C4semidirect
8. C2×C4⋊Dic17C4semidirect
9. C2×C4×Dic17C4semidirect
67. C2×C4⋊C4#595semidirect
72. C17#1820semidirect

Maximal subgroups

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

subgroupindex
#117637522
#117636872
C2×#105875782
#182017