Finite Group Databasebeta

← #392×D167 #36327332

Subgroups — #392×D167 #36327332

Order 21376.

1812 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 223 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
1C22×D4×D167C213yesmaximal, complemented
2C2×C22⋊C4×D167C213yesmaximal, complemented
3#24923179C211yesmaximal, characteristic, complemented
4#24923178C213yesmaximal, complemented
5#24923177C213yesmaximal, complemented
6#95×D167C211yesmaximal, characteristic, complemented
7#392×C167C211yesmaximal, characteristic, complemented
8C24×D167C2213yes—
9C22×D668C2213yescomplemented
10C22×C22⋊D167C2216yescomplemented
11C2×D4×D167—224no—
12C2×D4×D167—212no—
13C22×C4×D167C2211yescharacteristic
14C22×C4×D167C2213yes—
15C2×C22⋊Dic167C2213yescomplemented
16C2×(C2×D167)⋊C4C2213yescomplemented
17C2×(C2×D167)⋊C4C2213yescomplemented
18C22⋊C4×D167—212no—
19C167⋊#95C2213yescomplemented
20C22×D4×C167C2213yescomplemented
21C2×C22⋊C4×C167C2213yescomplemented
22#95×C167C2211yescharacteristic, complemented
23C23×D167—212no—
24C23×D167—212no—
25C23×D167—26no—
26C23×D167—26no—
27C23×D167D4112yescomplemented
28C23×D167C2313yes—
29C23×D167C2311yescharacteristic
30C2×D668—212no—
31C2×D668—26no—
32C2×C22⋊D167—212no—
33C2×C22⋊D167—212no—
34C2×C22⋊D167—212no—
35D4×D167—296no—
36C22×Dic167C2311yescharacteristic
37C22×Dic167C2313yes—
38C2×C4×D167—212no—
39C2×C4×D167—26no—
40C2×C4×D167D4112yescomplemented
41C22⋊Dic167—26no—
42(C2×D167)⋊C4—26no—
43(C2×D167)⋊C4—26no—
44C24×C167C2313yes—
45C2×D4×C167—212no—
46C2×D4×C167—26no—
47C22×C4×C167C2311yescharacteristic
48C22×C4×C167C2313yes—
49C22⋊C4×C167—26no—
50C22×D167—23no—
51C22×D167—26no—
52C22×D167—26no—
53C22×D167—23no—
54C22×D167—248no—
55C22×D167—224no—
56C22×D167—412no—
57C22×D167—424no—
58C22×D167—412no—
59C22×D167C2×D416yes—
60C22×D167C2×D416yes—
61C22×D167C2×D416yescomplemented
62C22×D167#4112yes—
63D668—224no—
64C22⋊D167—248no—
65C2×Dic167—26no—
66C2×Dic167—23no—
67C2×Dic167C2×D416yescomplemented
68C4×D167—224no—
69C4×D167—44no—
70C23×C167—23no—
71C23×C167—26no—
72C23×C167—26no—
73C23×C167—23no—
74C23×C167C2×D416yescomplemented
75C23×C167C2411yescharacteristic
76D4×C167—224no—
77C2×C4×C167—26no—
78C2×C4×C167—23no—
79C2×C4×C167C2×D416yescomplemented
80C22×C167—212no—
81C22×C167—26no—
82C22×C167—43no—
83C22×C167—46no—
84C22×C167—43no—
85C22×C167C22×D413yes—
86C22×C167C22×D413yes—
87C22×C167C2×#4111yescharacteristic
88C2×D167—212no—
89C2×D167—26no—
90C2×D167—43no—
91C2×D167—46no—
92C2×D167—43no—
93C2×D167—448no—
94C2×D167#120112yes—
95C2×D167#121112yes—
96C2×D167#12714yes—
97C4×C167—26no—
98C4×C167—41no—
99Dic167—26no—
100Dic167—41no—
101C2×C167—46no—
102C2×C167C2×#12013yes—
103C2×C167C2×#12113yes—
104C2×C167C2×#12711yescharacteristic
105D167—46no—
106D167#39218yesdirect product factor
107C167C2×#39211yescharacteristic, complemented, Sylow
108C2×#392—1671nomaximal, Sylow
109C23×D4—1673no—
110C22×C22⋊C4—1673no—
111#392—1673no—
112#392—1673no—
113#392—1671no—
114#392D16711yescharacteristic, direct product factor
115C2×#95—1671no—
116C25—1673no—
117C22×D4—33424no—
118C22×D4—33412no—
119C22×D4—1676no—
120C22×D4—1673no—
121C22×D4C2×D16713yescomplemented
122C23×C4—1671no—
123C23×C4—1673no—
124C2×C22⋊C4—33412no—
125C2×C22⋊C4—1673no—
126C2×C22⋊C4—1673no—
127C2×C22⋊C4—1673no—
128C2×C22⋊C4C2×D16713yescomplemented
129#95—1673no—
130#95C2×D16711yescharacteristic, complemented
131C24—3346no—
132C24—33412no—
133C24—33412no—
134C24—3346no—
135C24—16712no—
136C24—1673no—
137C24—1671no—
138C24C22×D16713yes—
139C2×D4—212no—
140C2×D4—26no—
141C2×D4—33412no—
142C2×D4—33496no—
143C2×D4—33412no—
144C2×D4—33412no—
145C2×D4—33412no—
146C2×D4—3346no—
147C22×C4—33412no—
148C22×C4—3346no—
149C22×C4—1671no—
150C22×C4—1673no—
151C22×C4—16712no—
152C22×C4C22×D16711yescharacteristic
153C22×C4C22×D16713yes—
154C22⋊C4—26no—
155C22⋊C4—3346no—
156C22⋊C4—3346no—
157C22⋊C4—3346no—
158C23—23no—
159C23—26no—
160C23—26no—
161C23—23no—
162C23—33448no—
163C23—33424no—
164C23—3343no—
165C23—3346no—
166C23—3346no—
167C23—3343no—
168C23—66812no—
169C23—66824no—
170C23—66812no—
171C23—1676no—
172C23—1676no—
173C23—1676no—
174C23—1672no—
175C23D4×D16716yescomplemented
176C23C23×D16711yescharacteristic
177D4—224no—
178D4—33424no—
179D4—33448no—
180C2×C4—26no—
181C2×C4—23no—
182C2×C4—33424no—
183C2×C4—3346no—
184C2×C4—3343no—
185C2×C4—6684no—
186C2×C4—1676no—
187C2×C4D4×D16716yescomplemented
188C22—212no—
189C22—26no—
190C22—43no—
191C22—46no—
192C22—43no—
193C22—33412no—
194C22—3346no—
195C22—66848no—
196C22—6683no—
197C22—6686no—
198C22—6683no—
199C22—16712no—
200C22—16712no—
201C22—1674no—
202C22C2×D4×D16713yes—
203C22C2×D4×D16713yes—
204C22#41×D16711yescharacteristic
205C4—26no—
206C4—41no—
207C4—3346no—
208C4—6681no—
209C2—46no—
210C2—6686no—
211C2—1678no—
212C2#120×D16713yes—
213C2#121×D16713yes—
214C2#127×D16711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D167
commutator, socle, cosocleC23×C167C24
Fitting#392×C167C2
hypercenter#392D167
hypocenterC167C2×#392
LCSG ▷ C23×C167 ▷ C167
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D167C2semidirect
2. C2×C22⋊C4×D167C2semidirect
3. #24923179C2semidirect
4. #24923178C2semidirect
5. #24923177C2semidirect
6. #95×D167C2semidirect
7. #392×C167C2semidirect
9. C22×D668C22semidirect
10. C22×C22⋊D167C22semidirect
15. C2×C22⋊Dic167C22semidirect
16. C2×(C2×D167)⋊C4C22semidirect
17. C2×(C2×D167)⋊C4C22semidirect
19. C167⋊#95C22semidirect
20. C22×D4×C167C22semidirect
21. C2×C22⋊C4×C167C22semidirect
22. #95×C167C22semidirect
27. C23×D167D4semidirect
40. C2×C4×D167D4semidirect
61. C22×D167C2×D4semidirect
67. C2×Dic167C2×D4semidirect
74. C23×C167C2×D4semidirect
79. C2×C4×C167C2×D4semidirect
106. D167#392direct (N × H)
107. C167C2×#392semidirect
114. #392D167direct (N × H)
121. C22×D4C2×D167semidirect
128. C2×C22⋊C4C2×D167semidirect
130. #95C2×D167semidirect
175. C23D4×D167semidirect
187. C2×C4D4×D167semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1672
C2×C22⋊C4×D1672
#249231792
#249231782
#249231772
#95×D1672
#392×C1672
C2×#392167