Finite Group Databasebeta

← #14544684

Subgroups — #14544684

Order 2176.

316 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 70 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#11763781C212yesmaximal
2#11763689C214yesmaximal, complemented
3C2×C4⋊C4×D17C211yesmaximal, characteristic
4C22×C4×D17C412yes—
5C22×C4×D17C2211yescharacteristic
6C2×#63247C2214yes—
7C2×C4⋊C4×C17C411yescharacteristic
8C2×C4⋊Dic17C411yescharacteristic
9C2×#63225C2212yes—
10C4⋊C4×D17—22no—
11C4⋊C4×D17—22no—
12#10587543—22no—
13#10587543—22no—
14C23×D17C2×C411yescharacteristic
15C22×C4×C17C2×C412yes—
16C22×C4×C17C2×C411yescharacteristic
17C22×Dic17C2×C412yes—
18C22×Dic17C2311yescharacteristic
19C2×C4×D17—24no—
20C2×C4×D17—22no—
21C2×C4×D17—22no—
22C2×C4×D17—44no—
23C2×C4×D17D412yes—
24C2×C4×D17D412yes—
25#63247—28no—
26C2×#3624—24no—
27C4⋊C4×C17—22no—
28C4⋊Dic17—22no—
29#63225—24no—
30C23×C17C22×C411yescharacteristic
31C22×D17—22no—
32C22×D17—22no—
33C22×D17—22no—
34C22×D17C22⋊C411yescharacteristic
35C22×D17C22⋊C411yescharacteristic
36C2×C4×C17—22no—
37C2×C4×C17—22no—
38C2×C4×C17—42no—
39C2×C4×C17C22⋊C412yes—
40C2×Dic17—22no—
41C2×Dic17—42no—
42C2×Dic17C2×D412yes—
43C2×Dic17#4114yes—
44C4×D17—24no—
45C4×D17—24no—
46C4×D17—44no—
47C4×D17—44no—
48#3624—44no—
49#1821—171nomaximal, Sylow
50C22×C17—21no—
51C22×C17—21no—
52C22×C17C2×C22⋊C411yescharacteristic
53C22×C17#11712yes—
54C2×D17—28no—
55C2×D17—21no—
56C2×D17—21no—
57C2×D17—21no—
58C2×D17—21no—
59C2×D17#10012yes—
60C2×D17#10112yes—
61C4×C17—22no—
62C4×C17—42no—
63Dic17—42no—
64Dic17#12314yes—
65C22×C4⋊C4—171no—
66#407—172no—
67#343—174no—
68C2×C17—22no—
69C2×C17#38111yescharacteristic
70C2×C17C2×#10011yescharacteristic
71C2×C17C2×#10111yescharacteristic
72D17—22no—
73D17—22no—
74C23×C4—171no—
75C23×C4—172no—
76C2×M16—174no—
77C2×C4⋊C4—172no—
78C2×C4⋊C4—171no—
79C2×C4⋊C4—342no—
80C2×C4⋊C4—342no—
81C2×C4⋊C4#59511yescharacteristic
82C22⋊C8—342no—
83C22⋊C8—342no—
84C17#182111yescharacteristic, complemented, Sylow
85C24—171no—
86C22×C4—171no—
87C22×C4—172no—
88C22×C4—172no—
89C22×C4—172no—
90C22×C4—344no—
91C22×C4—342no—
92C22×C4—342no—
93C22×C4—684no—
94C22×C4C2×#59512yes—
95C22×C4C2×#59511yescharacteristic
96M16—348no—
97C2×C8—344no—
98C4⋊C4—22no—
99C4⋊C4—344no—
100C4⋊C4—342no—
101C23—171no—
102C23—171no—
103C23—342no—
104C23—342no—
105C23—342no—
106C23C22×#59511yescharacteristic
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—174no—
111C2×C4—172no—
112C2×C4—344no—
113C2×C4—344no—
114C2×C4—342no—
115C2×C4—684no—
116C2×C4—684no—
117C2×C4—682no—
118C2×C4C22⋊#59512yes—
119C8—684no—
120C22—21no—
121C22—21no—
122C22—172no—
123C22—172no—
124C22—348no—
125C22—341no—
126C22—341no—
127C22—341no—
128C22—341no—
129C22#1058767712yes—
130C22C2×C22⋊#59511yescharacteristic
131C4—22no—
132C4—42no—
133C4—174no—
134C4—682no—
135C2—22no—
136C2—342no—
137C2—342no—
138C2#1176375211yescharacteristic
139C2D17⋊#10011yescharacteristic
140C2D17⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17C2×C4
cosocleC22×Dic17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1821
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
2. #11763689C2semidirect
84. C17#1821semidirect

Maximal subgroups

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

subgroupindex
#117637812
#117636892
C2×C4⋊C4×D172
#182117