Finite Group Databasebeta

← #36324552

Subgroups — #36324552

Order 21376.

386 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 97 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#24923744C211yesmaximal, characteristic
2C2×#19128747C211yesmaximal, characteristic
3#24923132C212yesmaximal, complemented
4#24923035C212yesmaximal, complemented
5C2×#123×C167C211yesmaximal, characteristic
6D4×Dic167—24no—
7#19128791—21no—
8C2×C22⋊Dic167—21no—
9C2×C22⋊Dic167C2212yes—
10C2×C22⋊Dic167C2212yes—
11#19128747—22no—
12C167⋊#95—24no—
13C22×D4×C167C2211yescharacteristic
14C23×C4×C167C2211yescharacteristic
15#123×C167—22no—
16#123×C167C414yescomplemented
17C2×C4⋊C4×C167—21no—
18C2×C22⋊C4×C167—21no—
19C2×C22⋊C4×C167C2211yescharacteristic
20C22×Dic167—22no—
21C22×Dic167—22no—
22C22×Dic167D414yes—
23C4⋊Dic167—22no—
24C22⋊Dic167—22no—
25C22⋊Dic167—22no—
26C22⋊Dic167—22no—
27C22⋊Dic167—22no—
28C22⋊Dic167—22no—
29C4×Dic167—22no—
30C24×C167D411yescharacteristic
31C24×C167C2311yescharacteristic
32C2×D4×C167—22no—
33C2×D4×C167—22no—
34C2×D4×C167C2×C412yes—
35C2×D4×C167D412yes—
36C22×C4×C167—21no—
37C22×C4×C167—21no—
38C22×C4×C167—21no—
39C22×C4×C167—22no—
40C22×C4×C167—22no—
41C22×C4×C167C2×C412yescomplemented
42C22×C4×C167D411yescharacteristic
43C4⋊C4×C167—24no—
44C22⋊C4×C167—24no—
45C22⋊C4×C167—21no—
46C22⋊C4×C167C2×C412yescomplemented
47C2×Dic167—24no—
48C2×Dic167—44no—
49C2×Dic167—44no—
50C2×Dic167—42no—
51C2×Dic167—82no—
52C23×C167—21no—
53C23×C167—21no—
54C23×C167—42no—
55C23×C167—42no—
56C23×C167C22⋊C412yes—
57C23×C167C22×C411yescharacteristic
58C23×C167C2×D412yes—
59C23×C167C2×D411yescharacteristic
60C23×C167C2×D411yescharacteristic
61C23×C167#4112yes—
62D4×C167—24no—
63D4×C167—24no—
64C2×C4×C167—21no—
65C2×C4×C167—21no—
66C2×C4×C167—22no—
67C2×C4×C167—22no—
68C2×C4×C167—24no—
69C2×C4×C167—42no—
70C2×C4×C167—41no—
71C2×C4×C167—42no—
72C2×C4×C167—41no—
73C2×C4×C167C22⋊C412yes—
74C22×C167—22no—
75C22×C167—24no—
76C22×C167—22no—
77C22×C167—41no—
78C22×C167—41no—
79C22×C167—42no—
80C22×C167—42no—
81C22×C167—42no—
82C22×C167—81no—
83C22×C167C2×C22⋊C411yescharacteristic
84C22×C167C4×D412yes—
85C22×C167#12011yescharacteristic
86C22×C167#12112yes—
87C22×C167#12311yescharacteristic
88C4×C167—22no—
89C4×C167—42no—
90C4×C167—42no—
91Dic167—44no—
92Dic167—82no—
93C2×C167—22no—
94C2×C167—21no—
95C2×C167—21no—
96C2×C167—22no—
97C2×C167—42no—
98C2×C167#38611yescharacteristic
99C2×C167#45812yes—
100C167#189011yescharacteristic, complemented, Sylow
101#1890—1671nomaximal, Sylow
102C2×#123D16711yescharacteristic
103#518—1671no—
104C2×#99—1671no—
105#381—1672no—
106#342—1672no—
107C22×D4C2×D16711yescharacteristic
108C23×C4C2×D16711yescharacteristic
109#123—22no—
110#123Dic16714yescomplemented
111C4×D4—3344no—
112#117—3341no—
113C2×C4⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—3341no—
116C2×C22⋊C4—1672no—
117C2×C22⋊C4—1672no—
118C2×C22⋊C4C2×D16711yescharacteristic
119#99—3342no—
120#95—3344no—
121C24C22⋊D16711yescharacteristic
122C24C22×D16711yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C2×Dic16712yes—
126C2×D4C22⋊D16712yes—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—3342no—
133C22×C4—3342no—
134C22×C4—1674no—
135C22×C4C2×Dic16712yescomplemented
136C22×C4C22⋊D16711yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—3342no—
139C22⋊C4—24no—
140C22⋊C4—21no—
141C22⋊C4—3342no—
142C22⋊C4—3342no—
143C22⋊C4—3342no—
144C22⋊C4—3342no—
145C22⋊C4—3342no—
146C22⋊C4C2×Dic16712yescomplemented
147C42—3342no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23C22⋊Dic16712yes—
153C23C22×Dic16711yescharacteristic
154C23D4×D16712yes—
155C23C2×C22⋊D16711yescharacteristic
156C23C2×C22⋊D16711yescharacteristic
157C23#1458194912yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—13362no—
170C2×C4—3344no—
171C2×C4—6684no—
172C2×C4—6682no—
173C2×C4—6684no—
174C2×C4C22⋊Dic16712yes—
175C22—22no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×C22⋊Dic16711yescharacteristic
185C22D4×Dic16712yes—
186C22#1912880011yescharacteristic
187C22#1912880412yes—
188C22#1912881311yescharacteristic
189C4—22no—
190C4—42no—
191C4—42no—
192C4—13362no—
193C4—6684no—
194C2—21no—
195C2—22no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#2492314911yescharacteristic
200C2#2492345812yes—
subgroupG/H
centerC22#19128800
commutatorC23×C167C22×C4
FrattiniC24C22×D167
FittingC2×#123×C167C2
socleC22×C167#120
prime radicalC22×D4×C167C22
cosocleC24×C167C23
hypercenterC2×#123D167
hypocenterC167#1890
LCSG ▷ C23×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #24923132C2semidirect
4. #24923035C2semidirect
16. #123×C167C4semidirect
41. C22×C4×C167C2×C4semidirect
46. C22⋊C4×C167C2×C4semidirect
100. C167#1890semidirect
110. #123Dic167semidirect
135. C22×C4C2×Dic167semidirect
146. C22⋊C4C2×Dic167semidirect

Maximal subgroups

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

subgroupindex
#249237442
C2×#191287472
#249231322
#249230352
C2×#123×C1672
#1890167