Finite Group Databasebeta

← #36325205

Subgroups — #36325205

Order 21376.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#24924069C211yesmaximal, characteristic, complemented
2#24923780C212yesmaximal, complemented
3#24923285C212yesmaximal, complemented
4#24923026C211yesmaximal, characteristic, complemented
5#407×C167C211yesmaximal, characteristic, complemented
6C2×#14581950C2212yes—
7#19128865—22no—
8#19128860—22no—
9#19128854—22no—
10#19128815—22no—
11#19128811—22no—
12#19128806—22no—
13#19128801—22no—
14C4×C22⋊D167—24no—
15#19128791C2212yes—
16#19128758—24no—
17C23×C4×C167C2211yescharacteristic
18C2×M16×C167C2211yescharacteristic
19C2×M16×C167C2211yescharacteristic
20C22⋊C8×C167—22no—
21#14581950—22no—
22#14581950—22no—
23#14581950—22no—
24#14581950—22no—
25C2×Dic334D412yescomplemented
26C2×D668D412yescomplemented
27C2×C22⋊D167—24no—
28C2×C22⋊D167—22no—
29C2×C4×D167—22no—
30Dic668—22no—
31#14581939—24no—
32D1336—22no—
33C4⋊Dic167—22no—
34Dic167⋊C4—24no—
35C22⋊Dic167—24no—
36C22⋊Dic167—22no—
37(C2×D167)⋊C4—24no—
38C4×Dic167D414yescomplemented
39C24×C167D411yescharacteristic
40C22×C4×C167—21no—
41C22×C4×C167—22no—
42C22×C4×C167—22no—
43C22×C4×C167—21no—
44C22×C4×C167D411yescharacteristic
45C22×C4×C167C2311yescharacteristic
46M16×C167—22no—
47M16×C167—22no—
48C2×C8×C167—21no—
49C2×C8×C167D412yescomplemented
50C22×D167—22no—
51Dic334—22no—
52Dic334—22no—
53D668—22no—
54D668—22no—
55C22⋊D167—48no—
56C22⋊D167—42no—
57C22⋊D167—42no—
58C2×Dic167—24no—
59C2×Dic167—22no—
60C4×D167—42no—
61C4×D167—42no—
62C23×C167—22no—
63C23×C167—21no—
64C23×C167—42no—
65C23×C167C2×D411yescharacteristic
66C2×C4×C167—22no—
67C2×C4×C167—21no—
68C2×C4×C167—41no—
69C2×C4×C167—42no—
70C2×C4×C167—41no—
71C2×C4×C167C2×D412yes—
72C2×C4×C167C2×D412yes—
73C2×C4×C167C2×D411yescharacteristic
74C2×C4×C167#4111yescharacteristic
75C8×C167—22no—
76C8×C167—41no—
77C22×C167—22no—
78C22×C167—24no—
79C22×C167—41no—
80C22×C167—42no—
81C22×C167—41no—
82C22×C167—41no—
83C22×C167#12012yes—
84C22×C167#12111yescharacteristic
85C2×D167—44no—
86C2×D167—42no—
87C4×C167—41no—
88C4×C167#12011yescharacteristic
89C4×C167#12112yes—
90C4×C167#12711yescharacteristic
91Dic167—44no—
92Dic167—42no—
93C2×C167—22no—
94C2×C167—21no—
95C2×C167—42no—
96C2×C167#39211yescharacteristic
97D167—82no—
98C167#201011yescharacteristic, complemented, Sylow
99#2010—1671nomaximal, Sylow
100#575—1671no—
101#525—1672no—
102#421—1672no—
103#407D16711yescharacteristic, complemented
104H(3,ℤ/4)—1671no—
105C2×#41—1672no—
106C23×C4C2×D16711yescharacteristic
107#137—3342no—
108AGL(1,ℤ/8)—3342no—
109#135—3342no—
110C2×M16C2×D16711yescharacteristic
111C2×M16C2×D16711yescharacteristic
112#123—3342no—
113C22⋊Q8—3342no—
114#121—3342no—
115#120—3342no—
116C4×D4—3344no—
117#117—1672no—
118#104—3344no—
119C22⋊C8—22no—
120C24D66811yescharacteristic
121#41—3342no—
122#41—3342no—
123#41—3342no—
124#41—3342no—
125C2×Q8—1672no—
126C2×D4—3344no—
127C2×D4—3342no—
128C2×D4—1672no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—3342no—
134C22×C4D66811yescharacteristic
135C22×C4C22×D16711yescharacteristic
136Q16—3342no—
137SD16—3344no—
138D8—3342no—
139M16—22no—
140M16—22no—
141C2×C8—21no—
142C2×C8C22⋊D16712yescomplemented
143C4⋊C4—3344no—
144C4⋊C4—3342no—
145C22⋊C4—3344no—
146C22⋊C4—3344no—
147C22⋊C4—3342no—
148C42—1674no—
149C23—22no—
150C23—21no—
151C23—42no—
152C23—3342no—
153C23C2×D66811yescharacteristic
154Q8—3342no—
155Q8—3342no—
156D4—3342no—
157D4—3342no—
158D4—6688no—
159D4—6682no—
160D4—6682no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—3344no—
167C2×C4—3342no—
168C2×C4—6682no—
169C2×C4—6682no—
170C2×C4D4×D16712yes—
171C2×C4D4×D16712yes—
172C2×C4C2×C22⋊D16711yescharacteristic
173C2×C4#1458195011yescharacteristic
174C8—22no—
175C8—41no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—6684no—
183C22—6682no—
184C22#1912880212yes—
185C22#1912880611yescharacteristic
186C4—41no—
187C4—6684no—
188C4—6682no—
189C4#1912880011yescharacteristic
190C4#1912880512yes—
191C4#1912882811yescharacteristic
192C2—22no—
193C2—21no—
194C2—42no—
195C2—13362no—
196C2#2492317811yescharacteristic
subgroupG/H
centerC4#19128828
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#407×C167C2
socleC2×C167#392
hypercenter#407D167
hypocenterC167#2010
LCSG ▷ C22×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #24924069C2semidirect
2. #24923780C2semidirect
3. #24923285C2semidirect
4. #24923026C2semidirect
5. #407×C167C2semidirect
25. C2×Dic334D4semidirect
26. C2×D668D4semidirect
38. C4×Dic167D4semidirect
49. C2×C8×C167D4semidirect
98. C167#2010semidirect
103. #407D167semidirect
142. C2×C8C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
#249240692
#249237802
#249232852
#249230262
#407×C1672
#2010167