Finite Group Databasebeta

← C2×#24923706 #36337262

Subgroups — C2×#24923706 #36337262

Order 21376.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19128855C211yesmaximal, characteristic, complemented
2C2×D8×D167C211yesmaximal, characteristic, complemented
3#24923706C218yesmaximal, direct product factor
4C2×#19128775C211yesmaximal, characteristic, complemented
5C2×D2672C211yesmaximal, characteristic, complemented
6C2×#19128773C211yesmaximal, characteristic, complemented
7C2×#19128771C211yesmaximal, characteristic, complemented
8C2×SD32×C167C211yesmaximal, characteristic, complemented
9C2×#14581951—21no—
10C2×D4×D167—21no—
11#19128855—22no—
12#19128855C2214yescomplemented
13C2×#14581937—21no—
14C2×D1336C2211yescharacteristic
15C2×D1336C2211yescharacteristic, complemented
16C2×#14581935—21no—
17D8×D167—22no—
18D8×D167C2214yescomplemented
19C2×C8×D167C2211yescharacteristic
20#19128775C2214yescomplemented
21D2672C2214yescomplemented
22#19128773C2214yescomplemented
23#19128771C2214yescomplemented
24C2×C167⋊C16C2211yescharacteristic, complemented
25C2×Q16×C167C2211yescharacteristic, complemented
26C2×D8×C167C2211yescharacteristic
27SD32×C167C2214yescomplemented
28C2×C16×C167C2211yescharacteristic, complemented
29C23×D167—41no—
30#14581951—24no—
31#14581951—42no—
32C2×D668—21no—
33C2×D668—21no—
34C2×D668—41no—
35C2×C22⋊D167—41no—
36D4×D167—24no—
37D4×D167—42no—
38C2×C4×D167—41no—
39C2×C4×D167D411yescharacteristic
40#14581937—24no—
41D1336—21no—
42D1336—21no—
43D1336C2312yes—
44D1336C2312yescomplemented
45#14581935—24no—
46C2×C167⋊C8D411yescharacteristic
47C8×D167D412yes—
48C8×D167C2312yes—
49C167⋊C16C2312yescomplemented
50C2×Q8×C167—21no—
51C2×D4×C167—21no—
52Q16×C167—21no—
53Q16×C167C2312yescomplemented
54D8×C167—21no—
55D8×C167C2312yes—
56C2×C8×C167C2311yescharacteristic
57C16×C167C2312yescomplemented
58C22×D167—41no—
59C22×D167—41no—
60C22×D167—44no—
61C22×D167—84no—
62C22×D167D811yescharacteristic
63D668—22no—
64D668—22no—
65D668—44no—
66D668—41no—
67D668—41no—
68C22⋊D167—44no—
69C2×Dic167D811yescharacteristic
70C4×D167—44no—
71C4×D167D812yes—
72C4×D167C2×D412yes—
73C167⋊C8C2×D412yes—
74C23×C167—41no—
75Q8×C167—22no—
76Q8×C167—41no—
77D4×C167—22no—
78D4×C167—41no—
79C2×C4×C167—41no—
80C2×C4×C167C2×D411yescharacteristic
81C8×C167C2×D411yescharacteristic
82C8×C167C2411yescharacteristic
83C22×C167—42no—
84C22×C167—82no—
85C22×C167C2×D811yescharacteristic
86C2×D167—22no—
87C2×D167—42no—
88C2×D167—42no—
89C2×D167—88no—
90C2×D167—82no—
91C2×D167—82no—
92C2×D167C2×D812yes—
93C4×C167—42no—
94C4×C167C2×D811yescharacteristic
95C4×C167C22×D411yescharacteristic
96Dic167C2×D812yes—
97C2×C167—82no—
98C2×C167#50912yescomplemented
99C2×C167C22×D811yescharacteristic
100D167—22no—
101D167—82no—
102D167—82no—
103C167C2×#50911yescharacteristic, complemented, Sylow
104C2×#509—1671nomaximal, Sylow
105C2×#135—1671no—
106C22×D8—1671no—
107#509—1678no—
108C2×SD32—1671no—
109C2×SD32D16711yescharacteristic, complemented
110C2×D16—1671no—
111C2×D16—1671no—
112C2×M32—1671no—
113C2×#41—3341no—
114C22×D4—3341no—
115#135—3342no—
116#135—1674no—
117C2×Q16C2×D16711yescharacteristic, complemented
118C2×SD16—3341no—
119C2×D8—3342no—
120C2×D8—3341no—
121C2×D8—1674no—
122C2×D8—1671no—
123C2×D8—1671no—
124C2×D8C2×D16711yescharacteristic
125C22×C8—1671no—
126SD32—1674no—
127SD32C2×D16714yescomplemented
128D16—1674no—
129D16—1674no—
130M32—1674no—
131C2×C16—1671no—
132C2×C16C2×D16711yescharacteristic, complemented
133C24—6681no—
134#41—3344no—
135#41—6682no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—3344no—
139C2×D4—3341no—
140C2×D4—3341no—
141C2×D4—6681no—
142C2×D4—6682no—
143C2×D4—6681no—
144C22×C4—6681no—
145C22×C4—1671no—
146Q16—21no—
147Q16C22×D16712yescomplemented
148SD16—3344no—
149D8—21no—
150D8—3344no—
151D8—3341no—
152D8—3341no—
153D8—1672no—
154D8—1672no—
155D8C22×D16712yes—
156C2×C8—1671no—
157C2×C8—1672no—
158C2×C8—1672no—
159C2×C8C22×D16711yescharacteristic
160C16—1672no—
161C16C22×D16712yescomplemented
162C23—41no—
163C23—13364no—
164C23—6684no—
165C23—6681no—
166C23—6681no—
167C23—1671no—
168Q8—22no—
169Q8—41no—
170D4—22no—
171D4—41no—
172D4—3342no—
173D4—3342no—
174D4—6684no—
175D4—6684no—
176D4—6681no—
177D4—6681no—
178C2×C4—41no—
179C2×C4—6684no—
180C2×C4—1671no—
181C2×C4—1672no—
182C2×C4—1672no—
183C2×C4D4×D16711yescharacteristic
184C8—1672no—
185C8D4×D16711yescharacteristic
186C8C23×D16711yescharacteristic
187C22—42no—
188C22—82no—
189C22—13368no—
190C22—13362no—
191C22—13362no—
192C22—3342no—
193C22—6682no—
194C22—6682no—
195C22—1672no—
196C22D8×D16711yescharacteristic
197C4—42no—
198C4—1672no—
199C4D8×D16711yescharacteristic
200C4C2×D4×D16711yescharacteristic
201C2—82no—
202C2—13362no—
203C2—13362no—
204C2—3342no—
205C2#2492370612yesdirect product factor
206C2C2×D8×D16711yescharacteristic
subgroupG/H
centerC22D8×D167
commutator, cosocleC8×C167C24
FrattiniC8C23×D167
FittingC2×SD32×C167C2
socleC22×C167C2×D8
hypercenterC2×SD32D167
hypocenterC167C2×#509
LCSG ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. C2×#19128855C2semidirect
2. C2×D8×D167C2semidirect
3. #24923706C2direct (N × H)
4. C2×#19128775C2semidirect
5. C2×D2672C2semidirect
6. C2×#19128773C2semidirect
7. C2×#19128771C2semidirect
8. C2×SD32×C167C2semidirect
12. #19128855C22semidirect
15. C2×D1336C22semidirect
18. D8×D167C22semidirect
20. #19128775C22semidirect
21. D2672C22semidirect
22. #19128773C22semidirect
23. #19128771C22semidirect
24. C2×C167⋊C16C22semidirect
25. C2×Q16×C167C22semidirect
27. SD32×C167C22semidirect
28. C2×C16×C167C22semidirect
44. D1336C23semidirect
49. C167⋊C16C23semidirect
53. Q16×C167C23semidirect
57. C16×C167C23semidirect
98. C2×C167#509semidirect
103. C167C2×#509semidirect
109. C2×SD32D167semidirect
117. C2×Q16C2×D167semidirect
127. SD32C2×D167semidirect
132. C2×C16C2×D167semidirect
147. Q16C22×D167semidirect
161. C16C22×D167semidirect
205. C2#24923706direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191288552
C2×D8×D1672
#249237062
C2×#191287752
C2×D26722
C2×#191287732
C2×#191287712
C2×SD32×C1672
C2×#509167