Finite Group Databasebeta

← #14544125

Subgroups — #14544125

Order 2176.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763830C211yesmaximal, characteristic, complemented
2C4×#63247C211yesmaximal, characteristic, complemented
3#11763657C214yesmaximal, complemented
4C2×#10587579C211yesmaximal, characteristic
5C22×D68C411yescharacteristic
6C22×C4×D17—21no—
7C2×#63247—21no—
8C2×C4⋊C4×C17C411yescharacteristic
9C2×#63227—21no—
10C2×C4×Dic17C2211yescharacteristic
11#10587579—21no—
12#10587579—21no—
13#10587579C412yes—
14#10587579C2212yes—
15C22⋊#3624—21no—
16#10587542—21no—
17#10587542—21no—
18C4⋊#3624C2212yescomplemented
19C4×#3624C2212yescomplemented
20C23×D17—21no—
21C22×C4×C17—21no—
22C22×C4×C17C2×C411yescharacteristic
23C22×Dic17D411yescharacteristic
24C22×Dic17D411yescharacteristic
25C2×D68—21no—
26C2×D68—24no—
27C2×D68C2×C412yes—
28C2×C4×D17—22no—
29C2×C4×D17—22no—
30C2×C4×D17—44no—
31#63247—24no—
32C2×#3624—22no—
33C2×#3624—22no—
34C4⋊C4×C17—21no—
35C4⋊C4×C17C2×C412yes—
36#63227—22no—
37#63227—22no—
38C4×Dic17C2×C411yescharacteristic
39C4×Dic17D411yescharacteristic
40C4×Dic17D411yescharacteristic
41C4×Dic17C2311yescharacteristic
42C23×C17C22⋊C411yescharacteristic
43C22×D17—24no—
44C22×D17—22no—
45C22×D17—42no—
46C22×D17—42no—
47C2×C4×C17—22no—
48C2×C4×C17—22no—
49C2×C4×C17—42no—
50C2×C4×C17C22⋊C411yescharacteristic
51C2×C4×C17C22×C411yescharacteristic
52C2×Dic17—22no—
53C2×Dic17—22no—
54C2×Dic17C22⋊C411yescharacteristic
55C2×Dic17C22⋊C411yescharacteristic
56C2×Dic17C2×D411yescharacteristic
57C2×Dic17C2×D411yescharacteristic
58D68—24no—
59D68—22no—
60D68M1614yescomplemented
61C4×D17—44no—
62C4×D17—44no—
63#3624—24no—
64#3624—42no—
65#1490—171nomaximal, Sylow
66C22×C17—21no—
67C22×C17—21no—
68C22×C17#10412yes—
69C22×C17C2×C22⋊C411yescharacteristic
70C2×D17—24no—
71C2×D17—48no—
72C2×D17—42no—
73C2×D17—42no—
74C4×C17—21no—
75C4×C17—42no—
76C4×C17C2×M1612yes—
77Dic17—21no—
78Dic17—21no—
79Dic17—21no—
80Dic17AGL(1,ℤ/8)12yes—
81C2×C4×D4—171no—
82#432—171no—
83C4×M16—171no—
84D4⋊C8—174no—
85C2×C17—22no—
86C2×C17#40711yescharacteristic
87C2×C17#41811yescharacteristic
88C2×C17C2×#10411yescharacteristic
89D17—44no—
90C22×D4—171no—
91C23×C4—341no—
92C2×M16—341no—
93C4×D4—172no—
94C4×D4—172no—
95C4×D4—341no—
96C4×D4—341no—
97C2×C4⋊C4#59511yescharacteristic
98C2×C22⋊C4—341no—
99C2×C42—171no—
100C4⋊C8—172no—
101C22⋊C8—341no—
102#97—341no—
103#97—341no—
104C4×C8—172no—
105C17#149011yescharacteristic, complemented, Sylow
106C24—341no—
107C2×D4—172no—
108C2×D4—344no—
109C2×D4—341no—
110C22×C4—21no—
111C22×C4—171no—
112C22×C4—171no—
113C22×C4—342no—
114C22×C4—342no—
115C22×C4—684no—
116C22×C4C2×#59511yescharacteristic
117M16—344no—
118C2×C8—342no—
119C2×C8—342no—
120C4⋊C4—21no—
121C4⋊C4C2×#59512yes—
122C22⋊C4—342no—
123C22⋊C4—342no—
124C42—171no—
125C42—171no—
126C42—171no—
127C42—171no—
128C23—344no—
129C23—342no—
130C23—682no—
131C23—682no—
132C23C22⋊#59511yescharacteristic
133D4—174no—
134D4—344no—
135D4—342no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—171no—
140C2×C4—171no—
141C2×C4—171no—
142C2×C4—171no—
143C2×C4—342no—
144C2×C4—342no—
145C2×C4—684no—
146C2×C4—684no—
147C2×C4C22⋊#59511yescharacteristic
148C2×C4C22×#59511yescharacteristic
149C8—344no—
150C8—682no—
151C22—21no—
152C22—21no—
153C22—344no—
154C22—688no—
155C22—682no—
156C22—682no—
157C22#1058755012yes—
158C22C2×C22⋊#59511yescharacteristic
159C4—21no—
160C4—42no—
161C4—172no—
162C4—341no—
163C4—341no—
164C4—341no—
165C4D17⋊M1612yes—
166C2—22no—
167C2—684no—
168C2#1176378111yescharacteristic
169C2#1176380111yescharacteristic
170C2C2×#1058755011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D68C4
cosocleC4×Dic17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1490
LCSG ▷ C2×C4×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
1. #11763830C2semidirect
2. C4×#63247C2semidirect
3. #11763657C2semidirect
18. C4⋊#3624C22semidirect
19. C4×#3624C22semidirect
60. D68M16semidirect
105. C17#1490semidirect

Maximal subgroups

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

subgroupindex
#117638302
C4×#632472
#117636572
C2×#105875792
#149017