Finite Group Databasebeta

← #14545065

Subgroups — #14545065

Order 2176.

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

normal subgroups only

This group has 270 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, complemented
2#11763714C214yesmaximal, complemented
3#11763378C211yesmaximal, characteristic
4C2×#63252C2211yescharacteristic
5C22×D68C411yescharacteristic, complemented
6#10587677—22no—
7C17×#117C411yescharacteristic, complemented
8#10587581—22no—
9#10587568—22no—
10#10587561C412yescomplemented
11#10587560C2212yes—
12#10587551—22no—
13#10587539C2214yescomplemented
14C23×D17—21no—
15C22×C4×C17C2×C411yescharacteristic
16C2×C22⋊D17D412yes—
17#63252—42no—
18C2×D68—21no—
19C2×D68—24no—
20C2×D68—42no—
21C2×D68C2×C412yes—
22C2×D68C2311yescharacteristic
23C2×C4×D17D412yes—
24C2×Dic34C2×C411yescharacteristic
25C22⋊#595—24no—
26C4⋊#595—42no—
27C4×#595—42no—
28C4⋊C4×C17—21no—
29C22⋊C4×C17—21no—
30C42×C17C2×C412yes—
31#63227—24no—
32C23×C17C22⋊C411yescharacteristic
33C22×D17—22no—
34C22×D17—42no—
35C22×D17—82no—
36C22×D17C2×D412yes—
37C2×#595—44no—
38C2×C4×C17—22no—
39C2×C4×C17—22no—
40C2×C4×C17C22⋊C411yescharacteristic
41C2×C4×C17C22×C411yescharacteristic
42C22⋊D17—42no—
43C2×Dic17C22⋊C412yes—
44D68—41no—
45D68—42no—
46D68—42no—
47D68—84no—
48C4×D17—22no—
49C4×D17—22no—
50Dic34—41no—
51#2128—171nomaximal, Sylow
52C22×C17—21no—
53C22×C17—41no—
54C22×C17C2×C22⋊C411yescharacteristic
55C2×D17—22no—
56C2×D17—42no—
57C2×D17—44no—
58C2×D17—42no—
59C2×D17—84no—
60#595—84no—
61C4×C17—21no—
62C4×C17—42no—
63C4×C17#9912yes—
64Dic17—22no—
65#535—171no—
66#410—172no—
67#353—174no—
68C2×C17—21no—
69C2×C17—41no—
70C2×C17C2×#9911yescharacteristic
71D17—42no—
72D17—82no—
73C2×#41—171no—
74C22×D4—171no—
75#127—172no—
76#124—172no—
77#121—342no—
78#120—342no—
79#117—342no—
80#117#59511yescharacteristic, complemented
81#99—174no—
82#99—342no—
83C17#212811yescharacteristic, complemented, Sylow
84C24—341no—
85#41—682no—
86C2×Q8—171no—
87C2×D4—172no—
88C2×D4—171no—
89C2×D4—172no—
90C2×D4—344no—
91C2×D4—341no—
92C2×D4—682no—
93C22×C4—172no—
94C22×C4C2×#59511yescharacteristic
95C4⋊C4—21no—
96C4⋊C4—682no—
97C22⋊C4—21no—
98C22⋊C4—344no—
99C22⋊C4—344no—
100C42—682no—
101C42C2×#59512yes—
102C23—172no—
103C23—342no—
104C23—682no—
105C23—1362no—
106C23C22⋊#59511yescharacteristic
107Q8—681no—
108D4—681no—
109D4—682no—
110D4—682no—
111D4—682no—
112D4—1364no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—172no—
116C2×C4—342no—
117C2×C4—342no—
118C2×C4—684no—
119C2×C4C22⋊#59511yescharacteristic
120C2×C4C22×#59511yescharacteristic
121C22—21no—
122C22—41no—
123C22—342no—
124C22—682no—
125C22—684no—
126C22—682no—
127C22—1364no—
128C22C2×C22⋊#59511yescharacteristic
129C4—21no—
130C4—42no—
131C4—342no—
132C4—1364no—
133C4#1058753912yes—
134C2—21no—
135C2—41no—
136C2—682no—
137C2—1362no—
138C2C2×#1058753911yescharacteristic
subgroupG/H
centerC2C2×#10587539
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#117C4
socleC2×C17C2×#99
prime radical#11763378C2
cosocleC2×D68C23
hypercenter#117#595
hypocenterC17#2128
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
1. #11763788C2semidirect
2. #11763714C2semidirect
5. C22×D68C4semidirect
7. C17×#117C4semidirect
10. #10587561C4semidirect
13. #10587539C22semidirect
80. #117#595semidirect
83. C17#2128semidirect

Maximal subgroups

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

subgroupindex
#117637882
#117637142
#117633782
#212817