Finite Group Databasebeta

← #36322676

Subgroups — #36322676

Order 21376.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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#24923326C211yesmaximal, characteristic, complemented
2#24923135C211yesmaximal, characteristic, complemented
3#24923006C212yesmaximal, complemented
4#24923003C212yesmaximal
5#529×C167C211yesmaximal, characteristic
6C2×C4⋊Dic167—21no—
7C4⋊(C167⋊C8)—21no—
8C22⋊(C167⋊C8)C2212yes—
9#19128743—21no—
10C167⋊#95—21no—
11C167⋊#95C2212yes—
12#126×C167—21no—
13#125×C167C411yescharacteristic, complemented
14#124×C167C411yescharacteristic, complemented
15#123×C167—21no—
16C22⋊Q8×C167C2211yescharacteristic
17#121×C167C2211yescharacteristic
18C4×Q8×C167C411yescharacteristic, complemented
19C4×D4×C167—21no—
20C4×D4×C167C411yescharacteristic, complemented
21#117×C167—21no—
22C2×C42×C167C2211yescharacteristic
23C22×Dic167—22no—
24C2×C167⋊C8—22no—
25C4⋊Dic167—42no—
26C2×Q8×C167C2×C411yescharacteristic
27C2×D4×C167—21no—
28C2×D4×C167C2×C411yescharacteristic
29C22×C4×C167—21no—
30C22×C4×C167D411yescharacteristic
31C22×C4×C167D411yescharacteristic
32C22×C4×C167C2311yescharacteristic
33C4⋊C4×C167—21no—
34C4⋊C4×C167—21no—
35C4⋊C4×C167—21no—
36C4⋊C4×C167—21no—
37C4⋊C4×C167C2×C411yescharacteristic
38C4⋊C4×C167C2×C411yescharacteristic
39C22⋊C4×C167—21no—
40C22⋊C4×C167—21no—
41C22⋊C4×C167—21no—
42C22⋊C4×C167—21no—
43C22⋊C4×C167—21no—
44C42×C167—21no—
45C42×C167C2×C411yescharacteristic
46C42×C167C2×C411yescharacteristic
47C42×C167D411yescharacteristic
48C42×C167D411yescharacteristic
49C2×Dic167—42no—
50C2×Dic167—42no—
51C2×Dic167—42no—
52C167⋊C8—42no—
53C23×C167—21no—
54C23×C167C2×D411yescharacteristic
55Q8×C167—41no—
56D4×C167—41no—
57D4×C167—41no—
58D4×C167—41no—
59C2×C4×C167—21no—
60C2×C4×C167—21no—
61C2×C4×C167—21no—
62C2×C4×C167—21no—
63C2×C4×C167—21no—
64C2×C4×C167—41no—
65C2×C4×C167—41no—
66C2×C4×C167—42no—
67C2×C4×C167C22⋊C411yescharacteristic
68C2×C4×C167C22⋊C411yescharacteristic
69C2×C4×C167C22⋊C411yescharacteristic
70C2×C4×C167C22⋊C411yescharacteristic
71C2×C4×C167C22×C411yescharacteristic
72C2×C4×C167C2×D411yescharacteristic
73C22×C167—21no—
74C22×C167—21no—
75C22×C167—41no—
76C22×C167—41no—
77C22×C167—41no—
78C22×C167C2×C22⋊C411yescharacteristic
79C22×C167AGL(1,ℤ/8)11yescharacteristic
80C22×C167#13711yescharacteristic
81C4×C167—21no—
82C4×C167—21no—
83C4×C167—21no—
84C4×C167—21no—
85C4×C167—21no—
86C4×C167—21no—
87C4×C167—41no—
88C4×C167—41no—
89C4×C167—41no—
90Dic167—82no—
91C2×C167—21no—
92C2×C167—21no—
93C2×C167—81no—
94C2×C167#41011yescharacteristic
95C2×C167#41711yescharacteristic
96C2×C167#42111yescharacteristic
97C167#151211yescharacteristic, complemented, Sylow
98#1512—1671nomaximal, Sylow
99#529D16711yescharacteristic
100#432—1671no—
101#382—1671no—
102#328—1672no—
103#327—1672no—
104#126—21no—
105#125Dic16711yescharacteristic, complemented
106#124Dic16711yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D16711yescharacteristic
109#121C2×D16711yescharacteristic
110C4×Q8Dic16711yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic16711yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—3341no—
115C2×C42C2×D16711yescharacteristic
116C4⋊C8—3341no—
117C22⋊C8—1672no—
118#97—3341no—
119#95—3341no—
120#95—1672no—
121C2×Q8C2×Dic16711yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic16711yescharacteristic
124C22×C4—21no—
125C22×C4—3342no—
126C22×C4C22⋊D16711yescharacteristic
127C22×C4C22⋊D16711yescharacteristic
128C22×C4C22×D16711yescharacteristic
129C2×C8—3342no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—6682no—
135C4⋊C4C2×Dic16711yescharacteristic
136C4⋊C4C2×Dic16711yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic16711yescharacteristic
144C42C2×Dic16711yescharacteristic
145C42C22⋊D16711yescharacteristic
146C42C22⋊D16711yescharacteristic
147C23—21no—
148C23C2×C22⋊D16711yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—6682no—
162C2×C4—6682no—
163C2×C4—6682no—
164C2×C4C22⋊Dic16711yescharacteristic
165C2×C4C22⋊Dic16711yescharacteristic
166C2×C4C22⋊Dic16711yescharacteristic
167C2×C4C22⋊Dic16711yescharacteristic
168C2×C4C22×Dic16711yescharacteristic
169C2×C4C2×C22⋊D16711yescharacteristic
170C8—6682no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic16711yescharacteristic
177C22#1912885711yescharacteristic
178C22#1912886211yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—13362no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#2492324311yescharacteristic
193C2#2492327011yescharacteristic
194C2#2492328611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic167
commutatorC2×C4×C167C22×C4
FrattiniC22×C4C22×D167
Fitting#529×C167C2
socleC22×C167C2×C22⋊C4
prime radical#121×C167C22
cosocleC22×C4×C167C23
hypercenter#529D167
hypocenterC167#1512
LCSG ▷ C2×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #24923326C2semidirect
2. #24923135C2semidirect
3. #24923006C2semidirect
13. #125×C167C4semidirect
14. #124×C167C4semidirect
18. C4×Q8×C167C4semidirect
20. C4×D4×C167C4semidirect
97. C167#1512semidirect
105. #125Dic167semidirect
106. #124Dic167semidirect
110. C4×Q8Dic167semidirect
112. C4×D4Dic167semidirect

Maximal subgroups

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

subgroupindex
#249233262
#249231352
#249230062
#249230032
#529×C1672
#1512167