Finite Group Databasebeta

← #36334604

Subgroups — #36334604

Order 21376.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19128865C212yesmaximal, complemented
2C2×#19128860C212yesmaximal, complemented
3#24923819C211yesmaximal, characteristic, complemented
4C2×#19128820C211yesmaximal, characteristic, complemented
5#24923663C212yesmaximal, complemented
6#24923658C212yesmaximal, complemented
7#24923653C214yesmaximal, complemented
8C4×M16×C167C211yesmaximal, characteristic, complemented
9C2×#14581950C2212yes—
10C22×Dic334C2211yescharacteristic
11C22×D668C2211yescharacteristic
12#19128865—28no—
13#19128860—28no—
14C2×Dic668C2214yescomplemented
15C2×#14581939C2214yes—
16C2×#14581939C2214yescomplemented
17C2×D1336C2214yes—
18#19128832C2211yescharacteristic, complemented
19#19128829C2211yescharacteristic, complemented
20#19128820—22no—
21#19128820C2212yes—
22#19128820C2212yes—
23#19128820C2212yescomplemented
24#19128806—22no—
25C4×D668—22no—
26C2×(C2×D167)⋊C4—22no—
27C2×M16×C167C2212yes—
28C2×C42×C167C2211yescharacteristic
29#97×C167C2212yescomplemented
30C4×C8×C167C2212yescomplemented
31C23×D167—21no—
32#14581950—24no—
33#14581950—24no—
34C2×Dic334—21no—
35C2×Dic334—24no—
36C2×Dic334C2312yes—
37C2×Dic334C2312yes—
38C2×D668—24no—
39C2×D668—21no—
40C2×D668—22no—
41C2×D668C2312yes—
42C2×D668C2312yes—
43C2×C22⋊D167—22no—
44C22×Dic167—21no—
45C2×C4×D167—22no—
46Dic668—28no—
47#14581939—28no—
48#14581939—28no—
49D1336—28no—
50C4⋊Dic167—22no—
51(C2×D167)⋊C4—24no—
52(C2×D167)⋊C4—24no—
53(C2×D167)⋊C4—24no—
54C22×C4×C167D412yes—
55C22×C4×C167C2311yescharacteristic
56M16×C167D418yescomplemented
57C2×C8×C167C2314yes—
58C42×C167D412yes—
59C42×C167C2311yescharacteristic
60C42×C167C2311yescharacteristic
61C22×D167—22no—
62C22×D167—22no—
63C22×D167—44no—
64C22×D167—42no—
65Dic334—22no—
66Dic334—24no—
67Dic334—22no—
68Dic334—42no—
69D668—22no—
70D668—24no—
71D668—22no—
72D668—44no—
73D668—42no—
74C22⋊D167—44no—
75C2×Dic167—22no—
76C2×Dic167—22no—
77C2×Dic167—42no—
78C4×D167—44no—
79C23×C167C2×D411yescharacteristic
80C2×C4×C167—42no—
81C2×C4×C167C2×D412yes—
82C2×C4×C167C2×D412yes—
83C2×C4×C167C2×D411yescharacteristic
84C2×C4×C167C2×D414yes—
85C2×C4×C167C2411yescharacteristic
86C8×C167C2×D418yes—
87C22×C167—22no—
88C22×C167#12712yes—
89C22×C167C22×D411yescharacteristic
90C2×D167—42no—
91C2×D167—42no—
92C2×D167—44no—
93C2×D167—44no—
94C2×D167—84no—
95C4×C167—22no—
96C4×C167—22no—
97C4×C167#12712yes—
98C4×C167C22×D412yes—
99Dic167—42no—
100Dic167—42no—
101C2×C167—22no—
102C2×C167C2×#12711yescharacteristic
103C2×C167#57512yes—
104D167—82no—
105D167—82no—
106C167#315311yescharacteristic, complemented, Sylow
107#3153—1671nomaximal, Sylow
108C2×#137—1672no—
109C2×AGL(1,ℤ/8)—1672no—
110#532—1671no—
111C2×#124—1671no—
112#497—1672no—
113#496—1672no—
114#495—1674no—
115C4×M16D16711yescharacteristic, complemented
116C2×#41—1672no—
117C22×Q8—1671no—
118C22×D4—1671no—
119#137—3348no—
120AGL(1,ℤ/8)—3348no—
121C2×Q16—1674no—
122C2×SD16—1674no—
123C2×SD16—1674no—
124C2×D8—1674no—
125C2×M16C2×D16712yes—
126C4⋊Q8—1671no—
127#127—1671no—
128#124—3342no—
129#124—1672no—
130#124—1672no—
131#124—1672no—
132#121—3342no—
133C4×D4—3342no—
134C2×C22⋊C4—3342no—
135C2×C42C2×D16711yescharacteristic
136#97C2×D16712yescomplemented
137C4×C8C2×D16712yescomplemented
138C24—3341no—
139#41—3344no—
140#41—3344no—
141C2×Q8—3341no—
142C2×Q8—3344no—
143C2×Q8—1672no—
144C2×Q8—1672no—
145C2×D4—3342no—
146C2×D4—3344no—
147C2×D4—3342no—
148C2×D4—3341no—
149C2×D4—1672no—
150C2×D4—1672no—
151C22×C4—3341no—
152C22×C4—3342no—
153C22×C4D66812yes—
154C22×C4C22×D16711yescharacteristic
155Q16—3348no—
156SD16—3348no—
157SD16—3348no—
158D8—3348no—
159M16D66818yescomplemented
160C2×C8C22×D16714yes—
161C4⋊C4—3342no—
162C22⋊C4—3344no—
163C22⋊C4—3344no—
164C22⋊C4—3344no—
165C42D66812yes—
166C42C22×D16711yescharacteristic
167C42C22×D16711yescharacteristic
168C23—3342no—
169C23—3342no—
170C23—6684no—
171C23—6682no—
172C23C2×D66811yescharacteristic
173Q8—3342no—
174Q8—3344no—
175Q8—3342no—
176Q8—6682no—
177D4—3342no—
178D4—3344no—
179D4—3342no—
180D4—6684no—
181D4—6684no—
182D4—6682no—
183C2×C4—42no—
184C2×C4—3342no—
185C2×C4—3342no—
186C2×C4—6682no—
187C2×C4—6684no—
188C2×C4C2×D66812yes—
189C2×C4C2×D66812yes—
190C2×C4C2×D66814yes—
191C2×C4C2×D66811yescharacteristic
192C2×C4C23×D16711yescharacteristic
193C8C2×D66818yes—
194C22—22no—
195C22—13364no—
196C22—6682no—
197C22—6682no—
198C22—6684no—
199C22—6684no—
200C22#1912882912yes—
201C22C22×D66811yescharacteristic
202C4—22no—
203C4—22no—
204C4—6682no—
205C4—6682no—
206C4#1912882912yes—
207C4C22×D66812yes—
208C2—22no—
209C2—13362no—
210C2—13362no—
211C2C2×#1912882911yescharacteristic
212C2#2492406912yes—
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC4×M16×C167C2
socleC22×C167C22×D4
hypercenterC4×M16D167
hypocenterC167#3153
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#19128865C2semidirect
2. C2×#19128860C2semidirect
3. #24923819C2semidirect
4. C2×#19128820C2semidirect
5. #24923663C2semidirect
6. #24923658C2semidirect
7. #24923653C2semidirect
8. C4×M16×C167C2semidirect
14. C2×Dic668C22semidirect
16. C2×#14581939C22semidirect
18. #19128832C22semidirect
19. #19128829C22semidirect
23. #19128820C22semidirect
29. #97×C167C22semidirect
30. C4×C8×C167C22semidirect
56. M16×C167D4semidirect
106. C167#3153semidirect
115. C4×M16D167semidirect
136. #97C2×D167semidirect
137. C4×C8C2×D167semidirect
159. M16D668semidirect

Maximal subgroups

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

subgroupindex
C2×#191288652
C2×#191288602
#249238192
C2×#191288202
#249236632
#249236582
#249236532
C4×M16×C1672
#3153167