Finite Group Databasebeta

← #36334602

Subgroups — #36334602

Order 21376.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19128863C212yesmaximal, complemented
2C2×#19128858C212yesmaximal, complemented
3#24923819C211yesmaximal, characteristic, complemented
4#24923664C212yesmaximal, complemented
5#24923659C212yesmaximal, complemented
6#24923651C214yesmaximal, complemented
7C4×#14581934C211yesmaximal, characteristic, complemented
8C2×#124×C167C211yesmaximal, characteristic, complemented
9C2×#14581950C2212yes—
10#19128863—28no—
11#19128858—28no—
12C2×#14581940C2214yescomplemented
13C2×#14581938C2214yes—
14C2×#14581937C2214yescomplemented
15C2×#14581935C2214yes—
16C2×#14581934C2212yes—
17#19128832C2211yescharacteristic, complemented
18#19128829C2211yescharacteristic, complemented
19#19128820C2212yescomplemented
20#19128806—22no—
21C4×D668—22no—
22#19128743C2212yescomplemented
23C4×C167⋊C8C2212yescomplemented
24C22×Q8×C167C2211yescharacteristic
25C22×D4×C167C2211yescharacteristic
26#124×C167—22no—
27#124×C167C2212yes—
28#124×C167C2212yes—
29C2×C22⋊C4×C167—22no—
30C2×C42×C167C2211yescharacteristic
31#14581950—24no—
32#14581950—24no—
33C2×Dic334C2312yes—
34C2×D668—22no—
35C2×D668C2312yes—
36C2×C22⋊D167—22no—
37C2×C4×D167—22no—
38#14581940—28no—
39#14581938—28no—
40#14581937—28no—
41#14581935—28no—
42#14581934D418yescomplemented
43C2×C167⋊C8C2314yes—
44C4⋊Dic167—22no—
45(C2×D167)⋊C4—24no—
46C24×C167—21no—
47C2×Q8×C167—21no—
48C2×Q8×C167—24no—
49C2×Q8×C167C2312yes—
50C2×D4×C167—24no—
51C2×D4×C167—21no—
52C2×D4×C167C2312yes—
53C22×C4×C167—21no—
54C22×C4×C167D412yes—
55C22×C4×C167C2311yescharacteristic
56C22⋊C4×C167—24no—
57C22⋊C4×C167—24no—
58C42×C167D412yes—
59C42×C167C2311yescharacteristic
60C42×C167C2311yescharacteristic
61C22×D167—22no—
62Dic334—22no—
63Dic334—22no—
64D668—22no—
65D668—22no—
66D668—44no—
67C22⋊D167—44no—
68C2×Dic167—22no—
69C4×D167—44no—
70C167⋊C8C2×D418yes—
71C23×C167—22no—
72C23×C167—44no—
73C23×C167—42no—
74C23×C167C2×D411yescharacteristic
75Q8×C167—24no—
76Q8×C167—42no—
77D4×C167—24no—
78D4×C167—42no—
79C2×C4×C167—22no—
80C2×C4×C167—42no—
81C2×C4×C167—42no—
82C2×C4×C167C2×D414yes—
83C2×C4×C167C2×D412yes—
84C2×C4×C167C2×D412yes—
85C2×C4×C167C2×D411yescharacteristic
86C2×C4×C167C2411yescharacteristic
87C22×C167—22no—
88C22×C167—42no—
89C22×C167—44no—
90C22×C167—84no—
91C22×C167#12712yes—
92C22×C167C22×D411yescharacteristic
93C2×D167—42no—
94C2×D167—44no—
95C4×C167—22no—
96C4×C167—22no—
97C4×C167—42no—
98C4×C167#12712yes—
99C4×C167C22×D412yes—
100Dic167—42no—
101C2×C167—22no—
102C2×C167—82no—
103C2×C167C2×#12711yescharacteristic
104C2×C167#57512yes—
105D167—82no—
106C167#315311yescharacteristic, complemented, Sylow
107#3153—1671nomaximal, Sylow
108C2×#137—1672no—
109C2×AGL(1,ℤ/8)—1672no—
110#532—1671no—
111C2×#124D16711yescharacteristic, complemented
112#497—1672no—
113#496—1672no—
114#495—1674no—
115C4×M16—1671no—
116C2×#41—1672no—
117C22×Q8C2×D16711yescharacteristic
118C22×D4C2×D16711yescharacteristic
119#137—3348no—
120AGL(1,ℤ/8)—3348no—
121C2×Q16—1674no—
122C2×SD16—1674no—
123C2×SD16—1674no—
124C2×D8—1674no—
125C2×M16—1672no—
126C4⋊Q8—1671no—
127#127—1671no—
128#124—22no—
129#124—1672no—
130#124C2×D16712yes—
131#124C2×D16712yes—
132#121—3342no—
133C4×D4—3342no—
134C2×C22⋊C4—22no—
135C2×C42C2×D16711yescharacteristic
136#97—1672no—
137C4×C8—1672no—
138C24—21no—
139#41—3344no—
140#41—3344no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—1672no—
144C2×Q8C22×D16712yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3342no—
148C2×D4—3342no—
149C2×D4—1672no—
150C2×D4C22×D16712yes—
151C22×C4—21no—
152C22×C4—3342no—
153C22×C4C22⋊D16712yes—
154C22×C4C22×D16711yescharacteristic
155Q16—3348no—
156SD16—3348no—
157SD16—3348no—
158D8—3348no—
159M16—1678no—
160C2×C8—1674no—
161C4⋊C4—3342no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—3344no—
165C42C22⋊D16712yes—
166C42C22×D16711yescharacteristic
167C42C22×D16711yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—3342no—
172C23C2×C22⋊D16711yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—3342no—
176Q8—3342no—
177D4—24no—
178D4—42no—
179D4—3342no—
180D4—3342no—
181D4—6684no—
182D4—6684no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—3342no—
187C2×C4—6684no—
188C2×C4D4×D16714yes—
189C2×C4C2×C22⋊D16712yes—
190C2×C4C2×C22⋊D16712yes—
191C2×C4C2×C22⋊D16711yescharacteristic
192C2×C4C23×D16711yescharacteristic
193C8—1678no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—6682no—
199C22—6684no—
200C22#1912882812yes—
201C22C22×C22⋊D16711yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—6682no—
206C4#1912882812yes—
207C4C2×D4×D16712yes—
208C2—22no—
209C2—82no—
210C2—13362no—
211C2C2×#1912882811yescharacteristic
212C2#2492406712yes—
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×#124×C167C2
socleC22×C167C22×D4
hypercenterC2×#124D167
hypocenterC167#3153
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#19128863C2semidirect
2. C2×#19128858C2semidirect
3. #24923819C2semidirect
4. #24923664C2semidirect
5. #24923659C2semidirect
6. #24923651C2semidirect
7. C4×#14581934C2semidirect
8. C2×#124×C167C2semidirect
12. C2×#14581940C22semidirect
14. C2×#14581937C22semidirect
17. #19128832C22semidirect
18. #19128829C22semidirect
19. #19128820C22semidirect
22. #19128743C22semidirect
23. C4×C167⋊C8C22semidirect
42. #14581934D4semidirect
106. C167#3153semidirect
111. C2×#124D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288632
C2×#191288582
#249238192
#249236642
#249236592
#249236512
C4×#145819342
C2×#124×C1672
#3153167