Finite Group Databasebeta

← #14545068

Subgroups — #14545068

Order 2176.

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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#11763788C212yesmaximal
2#11763716C214yesmaximal, complemented
3#11763385C211yesmaximal, characteristic
4C2×#63252C2211yescharacteristic
5C22×Dic34C411yescharacteristic, complemented
6#10587677—22no—
7C17×#117C411yescharacteristic, complemented
8#10587583—22no—
9#10587572—22no—
10#10587561C2212yes—
11#10587555C412yescomplemented
12#10587551—22no—
13#10587539C2214yes—
14C22×C4×C17C2×C411yescharacteristic
15C2×C22⋊D17D412yes—
16C22×Dic17—21no—
17#63252—42no—
18C2×D68C2311yescharacteristic
19C2×C4×D17D412yes—
20C2×Dic34—21no—
21C2×Dic34—42no—
22C2×Dic34C2×C412yes—
23C2×Dic34C2×C411yescharacteristic
24C22⋊#595—24no—
25C4⋊#595—42no—
26C4×#595—42no—
27C4⋊C4×C17—21no—
28C22⋊C4×C17—21no—
29C42×C17C2×C412yes—
30#63227—24no—
31C4⋊Dic17—24no—
32C23×C17C22⋊C411yescharacteristic
33C22×D17C2×D412yes—
34C2×#595—44no—
35C2×C4×C17—22no—
36C2×C4×C17—22no—
37C2×C4×C17C22⋊C411yescharacteristic
38C2×C4×C17C22×C411yescharacteristic
39C22⋊D17—42no—
40C2×Dic17—22no—
41C2×Dic17—42no—
42C2×Dic17C22⋊C412yes—
43D68—41no—
44C4×D17—22no—
45C4×D17—22no—
46Dic34—41no—
47Dic34—42no—
48Dic34—42no—
49#2129—171nomaximal, Sylow
50C22×C17—21no—
51C22×C17—41no—
52C22×C17C2×C22⋊C411yescharacteristic
53C2×D17—22no—
54C2×D17—42no—
55#595—84no—
56C4×C17—21no—
57C4×C17—42no—
58C4×C17#9912yes—
59Dic17—22no—
60Dic17—42no—
61#536—171no—
62#410—172no—
63#354—174no—
64C2×C17—21no—
65C2×C17—41no—
66C2×C17C2×#9911yescharacteristic
67D17—42no—
68C2×#41—171no—
69C22×Q8—171no—
70C4⋊Q8—172no—
71#124—172no—
72#123—342no—
73C22⋊Q8—342no—
74#117—342no—
75#117#59511yescharacteristic, complemented
76#99—174no—
77#99—342no—
78C17#212911yescharacteristic, complemented, Sylow
79#41—682no—
80C2×Q8—171no—
81C2×Q8—172no—
82C2×Q8—341no—
83C2×Q8—682no—
84C2×D4—172no—
85C2×D4—171no—
86C22×C4—172no—
87C22×C4—341no—
88C22×C4C2×#59511yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—344no—
91C4⋊C4—682no—
92C22⋊C4—21no—
93C22⋊C4—344no—
94C22⋊C4—344no—
95C42—682no—
96C42C2×#59512yes—
97C23—172no—
98C23C22⋊#59511yescharacteristic
99Q8—681no—
100Q8—682no—
101Q8—682no—
102D4—681no—
103D4—682no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—172no—
107C2×C4—342no—
108C2×C4—342no—
109C2×C4—342no—
110C2×C4—682no—
111C2×C4—684no—
112C2×C4C22⋊#59511yescharacteristic
113C2×C4C22×#59511yescharacteristic
114C22—21no—
115C22—41no—
116C22—342no—
117C22—682no—
118C22C2×C22⋊#59511yescharacteristic
119C4—21no—
120C4—42no—
121C4—342no—
122C4—682no—
123C4—1364no—
124C4#1058753912yes—
125C2—21no—
126C2—41no—
127C2—682no—
128C2C2×#1058753911yescharacteristic
subgroupG/H
centerC2C2×#10587539
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#117C4
socleC2×C17C2×#99
prime radicalC2×#63252C22
cosocleC2×D68C23
hypercenter#117#595
hypocenterC17#2129
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11763716C2semidirect
5. C22×Dic34C4semidirect
7. C17×#117C4semidirect
11. #10587555C4semidirect
75. #117#595semidirect
78. C17#2129semidirect

Maximal subgroups

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

subgroupindex
#117637882
#117637162
#117633852
#212917