Finite Group Databasebeta

← C2×#24923831 #36337441

Subgroups — C2×#24923831 #36337441

Order 21376.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C23×C22⋊D167C212yesmaximal, complemented
2#24923831C2116yesmaximal, direct product factor
3C2×#19128816C214yesmaximal, complemented
4C2×#19128806C214yesmaximal, complemented
5C2×#19128802C214yesmaximal, complemented
6C22×C22⋊C4×C167C211yesmaximal, characteristic, complemented
7C24×D167C2212yescomplemented
8C22×D668C2214yescomplemented
9C22×C22⋊D167—216no—
10C22×C22⋊D167C2214yes—
11C22×C22⋊D167C2214yescomplemented
12C22×C22⋊D167C22116yescomplemented
13C23×Dic167C2212yescomplemented
14#19128816C22132yescomplemented
15#19128806C22132yescomplemented
16#19128802C22132yescomplemented
17C2×C4⋊Dic167C2214yescomplemented
18C2×(C2×D167)⋊C4C2218yescomplemented
19C25×C167C2211yescharacteristic
20C23×C4×C167C2212yescomplemented
21C2×C22⋊C4×C167C2214yescomplemented
22C2×C22⋊C4×C167C2218yescomplemented
23C23×D167—28no—
24C23×D167—48no—
25C23×D167C2314yes—
26C23×D167C2318yescomplemented
27C2×D668—48no—
28C2×D668C23116yescomplemented
29C2×C22⋊D167—216no—
30C2×C22⋊D167—216no—
31C2×C22⋊D167—264no—
32C2×C22⋊D167C23116yes—
33C2×C22⋊D167C23116yescomplemented
34C22×Dic167—28no—
35C22×Dic167C2314yes—
36C22×Dic167C2318yescomplemented
37C4⋊Dic167C23116yescomplemented
38(C2×D167)⋊C4C23132yescomplemented
39C24×C167—44no—
40C24×C167D418yescomplemented
41C24×C167C2312yes—
42C24×C167C2311yescharacteristic
43C24×C167C2314yes—
44C22×C4×C167—44no—
45C22×C4×C167C2314yes—
46C22×C4×C167C2318yescomplemented
47C22⋊C4×C167C23116yescomplemented
48C22×D167—28no—
49C22×D167—216no—
50C22×D167—216no—
51C22×D167—416no—
52C22×D167—432no—
53C22×D167C2418yes—
54D668—416no—
55C22⋊D167—232no—
56C22⋊D167—232no—
57C2×Dic167—28no—
58C2×Dic167—216no—
59C2×Dic167C2418yes—
60C23×C167—24no—
61C23×C167—28no—
62C23×C167—416no—
63C23×C167—48no—
64C23×C167C2×D418yes—
65C23×C167C2×D414yes—
66C23×C167C2×D4116yescomplemented
67C23×C167C2414yes—
68C23×C167C2412yes—
69C23×C167C2411yescharacteristic
70C2×C4×C167—28no—
71C2×C4×C167—48no—
72C2×C4×C167C2418yes—
73C22×C167—28no—
74C22×C167—216no—
75C22×C167—24no—
76C22×C167—28no—
77C22×C167—416no—
78C22×C167C22×D418yes—
79C22×C167C22×D414yes—
80C22×C167C22×D412yes—
81C22×C167H(5,𝔽2)14yes—
82C22×C167C2511yescharacteristic
83C2×D167—28no—
84C2×D167—216no—
85C2×D167—432no—
86C2×D167—416no—
87C4×C167—28no—
88Dic167—28no—
89C2×C167—28no—
90C2×C167—24no—
91C2×C167#53414yescomplemented
92C2×C167C23×D411yescharacteristic
93C2×C167C2×H(5,𝔽2)12yes—
94D167—48no—
95C167C2×#53411yescharacteristic, complemented, Sylow
96C2×#534—1671nomaximal, Sylow
97C23×D4—1672no—
98#534—16716no—
99C2×#123—1674no—
100C2×#121—1674no—
101C2×#120—1674no—
102C22×C22⋊C4D16711yescharacteristic, complemented
103C25—1672no—
104C25C2×D16711yescharacteristic
105C22×D4—33416no—
106C22×D4—1674no—
107C22×D4—1674no—
108C22×D4—16716no—
109C22×D4—1674no—
110C23×C4—1672no—
111C23×C4C2×D16712yescomplemented
112#123—16732no—
113#121—16732no—
114#120—16732no—
115C2×C4⋊C4—1674no—
116C2×C22⋊C4—1678no—
117C2×C22⋊C4C2×D16714yescomplemented
118C2×C22⋊C4C2×D16718yescomplemented
119C24—44no—
120C24—3348no—
121C24—6688no—
122C24—1674no—
123C24—1678no—
124C24D66818yescomplemented
125C24C22×D16712yes—
126C24C22×D16711yescharacteristic
127C24C22×D16714yes—
128C2×D4—33416no—
129C2×D4—33416no—
130C2×D4—33464no—
131C2×D4—6688no—
132C2×D4—16716no—
133C2×D4—16716no—
134C2×D4—16716no—
135C22×C4—44no—
136C22×C4—3348no—
137C22×C4—1678no—
138C22×C4—1674no—
139C22×C4C22×D16714yes—
140C22×C4C22×D16718yescomplemented
141C4⋊C4—16716no—
142C22⋊C4—16732no—
143C22⋊C4C22×D167116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—33416no—
149C23—3348no—
150C23—33416no—
151C23—66816no—
152C23—66832no—
153C23—1678no—
154C23C2×D66818yes—
155C23C2×D66814yes—
156C23C2×D668116yescomplemented
157C23C23×D16714yes—
158C23C23×D16712yes—
159C23C23×D16711yescharacteristic
160D4—33432no—
161D4—33432no—
162D4—66816no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—33416no—
166C2×C4—3348no—
167C2×C4—1678no—
168C2×C4C23×D16718yes—
169C22—216no—
170C22—28no—
171C22—24no—
172C22—28no—
173C22—416no—
174C22—3348no—
175C22—33416no—
176C22—66832no—
177C22—66816no—
178C22C22×D66818yes—
179C22C22×D66814yes—
180C22C22×D66812yes—
181C22#1912887714yes—
182C22C24×D16711yescharacteristic
183C4—28no—
184C4—3348no—
185C2—28no—
186C2—24no—
187C2—6688no—
188C2#2492383114yesdirect product factor
189C2C23×D66811yescharacteristic
190C2C2×#1912887712yes—
subgroupG/H
centerC23C23×D167
commutator, cosocleC22×C167C25
FrattiniC22C24×D167
FittingC22×C22⋊C4×C167C2
socleC23×C167C24
hypercenterC22×C22⋊C4D167
hypocenterC167C2×#534
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D167C2semidirect
2. #24923831C2direct (N × H)
3. C2×#19128816C2semidirect
4. C2×#19128806C2semidirect
5. C2×#19128802C2semidirect
6. C22×C22⋊C4×C167C2semidirect
7. C24×D167C22semidirect
8. C22×D668C22semidirect
11. C22×C22⋊D167C22semidirect
12. C22×C22⋊D167C22semidirect
13. C23×Dic167C22semidirect
14. #19128816C22semidirect
15. #19128806C22semidirect
16. #19128802C22semidirect
17. C2×C4⋊Dic167C22semidirect
18. C2×(C2×D167)⋊C4C22semidirect
20. C23×C4×C167C22semidirect
21. C2×C22⋊C4×C167C22semidirect
22. C2×C22⋊C4×C167C22semidirect
26. C23×D167C23semidirect
28. C2×D668C23semidirect
33. C2×C22⋊D167C23semidirect
36. C22×Dic167C23semidirect
37. C4⋊Dic167C23semidirect
38. (C2×D167)⋊C4C23semidirect
40. C24×C167D4semidirect
46. C22×C4×C167C23semidirect
47. C22⋊C4×C167C23semidirect
66. C23×C167C2×D4semidirect
91. C2×C167#534semidirect
95. C167C2×#534semidirect
102. C22×C22⋊C4D167semidirect
111. C23×C4C2×D167semidirect
117. C2×C22⋊C4C2×D167semidirect
118. C2×C22⋊C4C2×D167semidirect
124. C24D668semidirect
140. C22×C4C22×D167semidirect
143. C22⋊C4C22×D167semidirect
156. C23C2×D668semidirect
188. C2#24923831direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1672
#249238312
C2×#191288162
C2×#191288062
C2×#191288022
C22×C22⋊C4×C1672
C2×#534167