Finite Group Databasebeta

← #36325202

Subgroups — #36325202

Order 21376.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#24923776C212yesmaximal, complemented
2#24923286C212yesmaximal, complemented
3#24923232C211yesmaximal, characteristic, complemented
4#24923026C211yesmaximal, characteristic, complemented
5#575×C167C211yesmaximal, characteristic, complemented
6C22×C4×D167C2211yescharacteristic
7C2×#14581934C2211yescharacteristic
8#19128813—22no—
9#19128808—22no—
10#19128803—22no—
11#19128800—22no—
12C4×C22⋊D167—24no—
13#19128791C2212yes—
14#19128759—24no—
15(C2×D167)⋊C8—22no—
16C2×#41×C167C2212yes—
17#137×C167—22no—
18AGL(1,ℤ/8)×C167—22no—
19#135×C167—22no—
20C2×M16×C167C2211yescharacteristic
21C23×D167D411yescharacteristic
22C2×C22⋊D167—24no—
23C22×Dic167D411yescharacteristic
24C2×C4×D167—21no—
25C2×C4×D167—22no—
26C2×C4×D167—22no—
27C2×C4×D167—21no—
28#14581934—22no—
29C2×C167⋊C8—21no—
30C4⋊Dic167—22no—
31Dic167⋊C4—24no—
32C22⋊Dic167—24no—
33C22⋊Dic167—22no—
34(C2×D167)⋊C4—24no—
35C4×Dic167D414yescomplemented
36#41×C167—22no—
37#41×C167—22no—
38#41×C167—22no—
39#41×C167—22no—
40C2×Q8×C167D412yescomplemented
41C2×D4×C167—22no—
42C2×D4×C167D412yescomplemented
43C22×C4×C167—22no—
44C22×C4×C167C2311yescharacteristic
45Q16×C167—22no—
46SD16×C167—24no—
47D8×C167—22no—
48M16×C167—22no—
49C2×C8×C167D412yescomplemented
50C22×D167—22no—
51C22×D167—21no—
52C22×D167—42no—
53C22⋊D167—48no—
54C2×Dic167—24no—
55C2×Dic167—22no—
56C2×Dic167—21no—
57C4×D167—41no—
58C4×D167—42no—
59C4×D167—41no—
60C167⋊C8—41no—
61C23×C167—22no—
62C23×C167C2×D411yescharacteristic
63Q8×C167—22no—
64Q8×C167—22no—
65D4×C167—22no—
66D4×C167—22no—
67D4×C167—42no—
68D4×C167—42no—
69C2×C4×C167—22no—
70C2×C4×C167—42no—
71C2×C4×C167—42no—
72C2×C4×C167C2×D412yes—
73C2×C4×C167C2×D412yes—
74C2×C4×C167C2×D411yescharacteristic
75C2×C4×C167#4111yescharacteristic
76C8×C167—22no—
77C22×C167—41no—
78C22×C167—44no—
79C22×C167—42no—
80C22×C167#12012yes—
81C22×C167#12111yescharacteristic
82C2×D167—24no—
83C2×D167—22no—
84C2×D167—42no—
85C2×D167—41no—
86C2×D167—41no—
87C4×C167—42no—
88C4×C167#12011yescharacteristic
89C4×C167#12112yes—
90C4×C167#12711yescharacteristic
91Dic167—44no—
92Dic167—41no—
93C2×C167—22no—
94C2×C167—21no—
95C2×C167—82no—
96C2×C167#39211yescharacteristic
97D167—42no—
98C167#201011yescharacteristic, complemented, Sylow
99#2010—1671nomaximal, Sylow
100#575D16711yescharacteristic, complemented
101#525—1672no—
102#421—1672no—
103#407—1671no—
104H(3,ℤ/4)—1671no—
105C2×#41C2×D16712yes—
106C23×C4—1671no—
107#137—22no—
108AGL(1,ℤ/8)—22no—
109#135—22no—
110C2×M16—1671no—
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—3342no—
120C24—1671no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8C22⋊D16712yescomplemented
126C2×D4—22no—
127C2×D4—3344no—
128C2×D4C22⋊D16712yescomplemented
129C22×C4—22no—
130C22×C4—3342no—
131C22×C4—3341no—
132C22×C4—3342no—
133C22×C4—3341no—
134C22×C4—1671no—
135C22×C4C22×D16711yescharacteristic
136Q16—22no—
137SD16—24no—
138D8—22no—
139M16—22no—
140M16—3342no—
141C2×C8—3341no—
142C2×C8C22⋊D16712yescomplemented
143C4⋊C4—3344no—
144C4⋊C4—3342no—
145C22⋊C4—3344no—
146C22⋊C4—3344no—
147C22⋊C4—3342no—
148C42—1674no—
149C23—22no—
150C23—3342no—
151C23—3341no—
152C23—6682no—
153C23D4×D16711yescharacteristic
154Q8—22no—
155Q8—22no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160D4—6688no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—3344no—
165C2×C4—3342no—
166C2×C4—3341no—
167C2×C4—6681no—
168C2×C4—6682no—
169C2×C4—6681no—
170C2×C4D4×D16712yes—
171C2×C4C2×C22⋊D16712yes—
172C2×C4C2×C22⋊D16711yescharacteristic
173C2×C4#1458194911yescharacteristic
174C8—22no—
175C8—6681no—
176C22—41no—
177C22—44no—
178C22—42no—
179C22—3344no—
180C22—3342no—
181C22—6682no—
182C22—6681no—
183C22—6681no—
184C22#1912880012yes—
185C22#1912880311yescharacteristic
186C4—42no—
187C4—6684no—
188C4—6681no—
189C4#1912880111yescharacteristic
190C4#1912880412yes—
191C4#1912882811yescharacteristic
192C2—22no—
193C2—21no—
194C2—82no—
195C2—6682no—
196C2#2492317711yescharacteristic
subgroupG/H
centerC4#19128828
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#575×C167C2
socleC2×C167#392
hypercenter#575D167
hypocenterC167#2010
LCSG ▷ C22×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #24923776C2semidirect
2. #24923286C2semidirect
3. #24923232C2semidirect
4. #24923026C2semidirect
5. #575×C167C2semidirect
35. C4×Dic167D4semidirect
40. C2×Q8×C167D4semidirect
42. C2×D4×C167D4semidirect
49. C2×C8×C167D4semidirect
98. C167#2010semidirect
100. #575D167semidirect
125. C2×Q8C22⋊D167semidirect
128. C2×D4C22⋊D167semidirect
142. C2×C8C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
#249237762
#249232862
#249232322
#249230262
#575×C1672
#2010167