Finite Group Databasebeta

← #36325269

Subgroups — #36325269

Order 21376.

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 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#24923831C211yesmaximal, characteristic
2C2×#19128815C212yesmaximal, complemented
3C2×#19128746C212yesmaximal, complemented
4#24923036C211yesmaximal, characteristic, complemented
5#381×C167C211yesmaximal, characteristic, complemented
6C22×C22⋊D167C2212yes—
7#19128816—22no—
8#19128815—24no—
9#19128815—24no—
10#19128806—22no—
11#19128802—22no—
12C2×Dic167⋊C4—22no—
13C2×C22⋊Dic167C2212yes—
14C2×(C2×D167)⋊C4—22no—
15#19128746—24no—
16C23×C4×C167C2211yescharacteristic
17C2×C22⋊C4×C167C2211yescharacteristic
18C2×C22⋊C4×C167C2211yescharacteristic
19#95×C167—22no—
20C23×D167D412yes—
21C2×D668—22no—
22C2×C22⋊D167—22no—
23C2×C22⋊D167—22no—
24C2×C22⋊D167—22no—
25C2×C22⋊D167—22no—
26C2×C22⋊D167—22no—
27C2×C22⋊D167—22no—
28C22×Dic167—22no—
29C22×Dic167D412yes—
30C4⋊Dic167—22no—
31Dic167⋊C4—28no—
32C22⋊Dic167—22no—
33C22⋊Dic167D414yescomplemented
34(C2×D167)⋊C4—28no—
35(C2×D167)⋊C4—24no—
36C24×C167C2311yescharacteristic
37C22×C4×C167—21no—
38C22×C4×C167—22no—
39C22×C4×C167—22no—
40C22×C4×C167—21no—
41C22×C4×C167—21no—
42C22×C4×C167D412yes—
43C22×C4×C167D412yescomplemented
44C22⋊C4×C167—22no—
45C22⋊C4×C167—22no—
46C22×D167—22no—
47C22×D167—22no—
48C22×D167—22no—
49C22×D167—44no—
50D668—82no—
51C22⋊D167—42no—
52C22⋊D167—42no—
53C22⋊D167—42no—
54C22⋊D167—42no—
55C2×Dic167—24no—
56C2×Dic167—22no—
57C2×Dic167—22no—
58C2×Dic167—22no—
59C2×Dic167—44no—
60C23×C167—42no—
61C23×C167C2×D412yes—
62C23×C167C2×D412yes—
63C23×C167C2×D411yescharacteristic
64C23×C167C2×D411yescharacteristic
65C23×C167#4111yescharacteristic
66C2×C4×C167—22no—
67C2×C4×C167—24no—
68C2×C4×C167—22no—
69C2×C4×C167—41no—
70C2×C4×C167—42no—
71C2×C4×C167—41no—
72C2×C4×C167—42no—
73C2×C4×C167—42no—
74C2×C4×C167—41no—
75C22×C167—22no—
76C22×C167—24no—
77C22×C167—42no—
78C22×C167—41no—
79C22×C167—41no—
80C22×C167#12011yescharacteristic
81C22×C167#12012yes—
82C22×C167#12112yes—
83C22×C167#12111yescharacteristic
84C22×C167#12711yescharacteristic
85C2×D167—44no—
86C2×D167—42no—
87C2×D167—44no—
88C2×D167—82no—
89C4×C167—42no—
90C4×C167—42no—
91C4×C167—81no—
92Dic167—44no—
93Dic167—42no—
94C2×C167—22no—
95C2×C167—21no—
96C2×C167—22no—
97C2×C167—21no—
98C2×C167#39211yescharacteristic
99C2×C167#45812yes—
100D167—82no—
101C167#202211yescharacteristic, complemented, Sylow
102#2022—1671nomaximal, Sylow
103#534—1671no—
104C2×#123—1672no—
105C2×#99—1672no—
106#381D16711yescharacteristic, complemented
107#342—1671no—
108C22×D4—1672no—
109C23×C4C2×D16711yescharacteristic
110#123—3342no—
111#123—3344no—
112#123—3344no—
113#121—3342no—
114#120—3342no—
115C2×C4⋊C4—3342no—
116C2×C22⋊C4—3342no—
117C2×C22⋊C4—1672no—
118C2×C22⋊C4C2×D16711yescharacteristic
119C2×C22⋊C4C2×D16711yescharacteristic
120#99—3344no—
121#95—22no—
122C24—1672no—
123C24C22×D16711yescharacteristic
124C2×D4—3342no—
125C2×D4—3342no—
126C2×D4—3342no—
127C2×D4—3342no—
128C2×D4—3342no—
129C2×D4—3342no—
130C2×D4—3342no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—3342no—
137C22×C4—1672no—
138C22×C4C22⋊D16712yes—
139C22×C4D66812yescomplemented
140C4⋊C4—3348no—
141C4⋊C4—3342no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3348no—
145C22⋊C4—3344no—
146C22⋊C4—3342no—
147C22⋊C4—1674no—
148C23—42no—
149C23—3342no—
150C23—3342no—
151C23—3342no—
152C23—6684no—
153C23D4×D16712yes—
154C23D4×D16712yes—
155C23C2×C22⋊D16711yescharacteristic
156C23C2×D66811yescharacteristic
157C23#1458195011yescharacteristic
158D4—13362no—
159D4—6682no—
160D4—6682no—
161D4—6682no—
162D4—6682no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—3342no—
173C2×C4—3344no—
174C2×C4—3342no—
175C2×C4—3342no—
176C2×C4—6684no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—13362no—
183C22—6684no—
184C22—6682no—
185C22—6684no—
186C22#1912880011yescharacteristic
187C22#1912880212yes—
188C22#1912880512yes—
189C22#1912880611yescharacteristic
190C22#1912882811yescharacteristic
191C4—42no—
192C4—42no—
193C4—81no—
194C4—6684no—
195C4—6682no—
196C2—21no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—13362no—
201C2#2492317811yescharacteristic
202C2#2492345812yes—
subgroupG/H
centerC22#19128800
commutator, cosocleC24×C167C23
FrattiniC24C22×D167
Fitting#381×C167C2
socleC22×C167#120
prime radical#24923831C2
hypercenter#381D167
hypocenterC167#2022
LCSG ▷ C24×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#19128815C2semidirect
3. C2×#19128746C2semidirect
4. #24923036C2semidirect
5. #381×C167C2semidirect
33. C22⋊Dic167D4semidirect
43. C22×C4×C167D4semidirect
101. C167#2022semidirect
106. #381D167semidirect
139. C22×C4D668semidirect

Maximal subgroups

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

subgroupindex
#249238312
C2×#191288152
C2×#191287462
#249230362
#381×C1672
#2022167