Finite Group Databasebeta

← #36325267

Subgroups — #36325267

Order 21376.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19128813C212yesmaximal, complemented
2C2×#19128747C212yesmaximal, complemented
3#24923130C211yesmaximal, characteristic, complemented
4#24923036C211yesmaximal, characteristic, complemented
5#534×C167C211yesmaximal, characteristic
6C23×Dic167C2211yescharacteristic
7#19128813—24no—
8#19128813—24no—
9C2×Dic167⋊C4—22no—
10C2×C22⋊Dic167—22no—
11C2×C22⋊Dic167C2211yescharacteristic
12C2×C22⋊Dic167C2212yes—
13#19128747—24no—
14C167⋊#95—22no—
15C22×D4×C167C2212yes—
16#123×C167—22no—
17#121×C167—22no—
18#120×C167—22no—
19C2×C22⋊C4×C167C2211yescharacteristic
20C22×Dic167—21no—
21C22×Dic167—22no—
22C22×Dic167—22no—
23C22×Dic167—22no—
24C22×Dic167—21no—
25C22×Dic167—21no—
26C22×Dic167D412yescomplemented
27Dic167⋊C4—28no—
28C22⋊Dic167—28no—
29C22⋊Dic167—22no—
30C22⋊Dic167—22no—
31C22⋊Dic167D414yescomplemented
32C24×C167D412yes—
33C24×C167C2311yescharacteristic
34C2×D4×C167—22no—
35C2×D4×C167—22no—
36C2×D4×C167—22no—
37C2×D4×C167—22no—
38C2×D4×C167—22no—
39C2×D4×C167—22no—
40C2×D4×C167—22no—
41C22×C4×C167D412yes—
42C22×C4×C167D412yes—
43C4⋊C4×C167—22no—
44C22⋊C4×C167—24no—
45C22⋊C4×C167—22no—
46C2×Dic167—24no—
47C2×Dic167—24no—
48C2×Dic167—22no—
49C2×Dic167—44no—
50C2×Dic167—41no—
51C2×Dic167—42no—
52C2×Dic167—41no—
53C2×Dic167—42no—
54C2×Dic167—41no—
55C23×C167—22no—
56C23×C167—22no—
57C23×C167—22no—
58C23×C167—42no—
59C23×C167—44no—
60C23×C167C2×D411yescharacteristic
61C23×C167C2×D412yes—
62C23×C167C2×D412yes—
63C23×C167C2×D411yescharacteristic
64C23×C167#4111yescharacteristic
65D4×C167—42no—
66D4×C167—42no—
67D4×C167—42no—
68D4×C167—42no—
69D4×C167—82no—
70C2×C4×C167—22no—
71C2×C4×C167—22no—
72C2×C4×C167—22no—
73C2×C4×C167—22no—
74C2×C4×C167—42no—
75C22×C167—22no—
76C22×C167—24no—
77C22×C167—41no—
78C22×C167—42no—
79C22×C167—41no—
80C22×C167—44no—
81C22×C167—42no—
82C22×C167—44no—
83C22×C167—82no—
84C22×C167#12012yes—
85C22×C167#12011yescharacteristic
86C22×C167#12111yescharacteristic
87C22×C167#12112yes—
88C22×C167#12711yescharacteristic
89C4×C167—42no—
90C4×C167—42no—
91Dic167—44no—
92Dic167—42no—
93Dic167—81no—
94C2×C167—22no—
95C2×C167—21no—
96C2×C167—22no—
97C2×C167—21no—
98C2×C167—82no—
99C2×C167#39211yescharacteristic
100C2×C167#45812yes—
101C167#202211yescharacteristic, complemented, Sylow
102#2022—1671nomaximal, Sylow
103#534D16711yescharacteristic
104C2×#123—1672no—
105C2×#99—1672no—
106#381—1671no—
107#342—1671no—
108C22×D4C2×D16712yes—
109C23×C4—1671no—
110#123—22no—
111#123—3344no—
112#123—3344no—
113#121—22no—
114#120—22no—
115C2×C4⋊C4—3342no—
116C2×C22⋊C4—3342no—
117C2×C22⋊C4—1671no—
118C2×C22⋊C4—1672no—
119C2×C22⋊C4C2×D16711yescharacteristic
120#99—3344no—
121#95—3342no—
122C24C22⋊D16712yes—
123C24C22×D16711yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C22×C4—3342no—
132C22×C4—3341no—
133C22×C4—3342no—
134C22×C4—3342no—
135C22×C4—3341no—
136C22×C4—3341no—
137C22×C4—1672no—
138C22×C4C22⋊D16712yes—
139C22×C4C22⋊D16712yes—
140C4⋊C4—22no—
141C4⋊C4—3348no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—3348no—
145C22⋊C4—3342no—
146C22⋊C4—3342no—
147C22⋊C4—1674no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—42no—
152C23—44no—
153C23D4×D16711yescharacteristic
154C23D4×D16712yes—
155C23C2×C22⋊D16712yes—
156C23C2×C22⋊D16711yescharacteristic
157C23#1458194911yescharacteristic
158D4—42no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—82no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—3344no—
169C2×C4—3344no—
170C2×C4—3342no—
171C2×C4—6682no—
172C2×C4—6681no—
173C2×C4—6684no—
174C2×C4—6681no—
175C2×C4—6682no—
176C2×C4—6681no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—44no—
183C22—42no—
184C22—44no—
185C22—82no—
186C22#1912880012yes—
187C22#1912880111yescharacteristic
188C22#1912880311yescharacteristic
189C22#1912880412yes—
190C22#1912882811yescharacteristic
191C4—42no—
192C4—42no—
193C4—13361no—
194C4—6684no—
195C4—6682no—
196C2—21no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—82no—
201C2#2492317711yescharacteristic
202C2#2492345912yes—
subgroupG/H
centerC22#19128801
commutator, cosocleC24×C167C23
FrattiniC24C22×D167
Fitting, prime radical#534×C167C2
socleC22×C167#120
hypercenter#534D167
hypocenterC167#2022
LCSG ▷ C24×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#19128813C2semidirect
2. C2×#19128747C2semidirect
3. #24923130C2semidirect
4. #24923036C2semidirect
26. C22×Dic167D4semidirect
31. C22⋊Dic167D4semidirect
101. C167#2022semidirect

Maximal subgroups

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

subgroupindex
C2×#191288132
C2×#191287472
#249231302
#249230362
#534×C1672
#2022167