Finite Group Databasebeta

← C2×#23758015 #34114256

Subgroups — C2×#23758015 #34114256

Order 20096.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#23758015C2116yesmaximal, direct product factor
2C2×#19097760C212yesmaximal, complemented
3C2×#19097759C212yesmaximal, complemented
4C2×#19097757C214yesmaximal, complemented
5C2×#19097741C214yesmaximal, complemented
6C2×#19097739C212yesmaximal, complemented
7C2×#124×C157C211yesmaximal, characteristic, complemented
8C24×D157C2212yescomplemented
9C22×Dic314C2212yescomplemented
10C22×D628C2212yes—
11C22×D628C2212yes—
12C22×C22⋊D157C2214yes—
13#19097760C22116yescomplemented
14#19097759C22116yescomplemented
15#19097757C22132yescomplemented
16#19097741C22132yescomplemented
17#19097739C22116yescomplemented
18C2×C22⋊Dic157C2212yescomplemented
19C2×#14578931C2214yescomplemented
20C2×#14578931C2214yescomplemented
21C2×#14578931C2214yescomplemented
22C2×C4×Dic157C2212yescomplemented
23C22×Q8×C157C2211yescharacteristic, complemented
24C22×D4×C157C2211yescharacteristic
25#124×C157C2218yescomplemented
26C2×C22⋊C4×C157C2214yescomplemented
27C2×C42×C157C2211yescharacteristic, complemented
28C23×D157—48no—
29C23×D157—44no—
30C23×D157C2314yes—
31C23×D157C2318yescomplemented
32C2×Dic314—44no—
33C2×Dic314C2318yescomplemented
34C2×D628—44no—
35C2×D628—44no—
36C2×D628C2318yes—
37C2×D628C2318yes—
38C2×C22⋊D157—48no—
39C2×C22⋊D157C23116yes—
40C22×Dic157C2314yes—
41C22⋊Dic157C2318yescomplemented
42#14578931C23116yescomplemented
43#14578931C23116yescomplemented
44#14578931C23116yescomplemented
45C4×Dic157C2318yescomplemented
46C24×C157C2312yes—
47C2×Q8×C157—42no—
48C2×Q8×C157C2314yescomplemented
49C2×D4×C157—42no—
50C2×D4×C157C2314yes—
51C22×C4×C157C2312yes—
52C22×C4×C157C2311yescharacteristic
53C22×C4×C157C2312yes—
54C22⋊C4×C157C23116yescomplemented
55C42×C157C2314yescomplemented
56C22×D157—28no—
57C22×D157—28no—
58C22×D157—28no—
59C22×D157—416no—
60C22×D157—48no—
61C22×D157—432no—
62C22×D157C2418yes—
63Dic314—48no—
64D628—48no—
65D628—48no—
66C22⋊D157—416no—
67C2×Dic157—28no—
68C2×Dic157C2418yes—
69C23×C157—28no—
70C23×C157—24no—
71C23×C157C2414yes—
72C23×C157C2411yescharacteristic
73Q8×C157—44no—
74D4×C157—44no—
75C2×C4×C157—24no—
76C2×C4×C157—24no—
77C2×C4×C157—22no—
78C2×C4×C157C2414yes—
79C2×C4×C157C2414yes—
80C2×C4×C157C2412yes—
81C22×C157—24no—
82C22×C157—28no—
83C22×C157—48no—
84C22×C157H(5,𝔽2)12yes—
85C22×C157H(5,𝔽2)14yes—
86C22×C157C2511yescharacteristic
87C2×D157—28no—
88C2×D157—28no—
89C2×D157—28no—
90C2×D157—432no—
91C2×D157—416no—
92C4×C157—22no—
93C4×C157—24no—
94C4×C157—24no—
95Dic157—28no—
96C2×C157—44no—
97C2×C157H(3,𝔽4)14yescomplemented
98C2×C157C2×H(5,𝔽2)11yescharacteristic
99C2×C157C2×H(5,𝔽2)12yes—
100D157—48no—
101C157C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
102C2×H(3,𝔽4)—1571nomaximal, Sylow
103H(3,𝔽4)—15716no—
104C2×#124—1572no—
105C2×#124—1574no—
106C2×#124—1572no—
107C2×#124D15711yescharacteristic, complemented
108C2×#120—1572no—
109C2×#120—1574no—
110C25—1572no—
111C22×Q8—1572no—
112C22×Q8C2×D15711yescharacteristic, complemented
113C22×D4—1574no—
114C22×D4—1572no—
115C22×D4—1572no—
116C22×D4C2×D15711yescharacteristic
117#124—15716no—
118#124—15732no—
119#124—15716no—
120#124C2×D15718yescomplemented
121#120—15716no—
122#120—15732no—
123C2×C22⋊C4—1572no—
124C2×C22⋊C4—1574no—
125C2×C22⋊C4—1574no—
126C2×C22⋊C4—1574no—
127C2×C22⋊C4C2×D15714yescomplemented
128C2×C42—1572no—
129C2×C42C2×D15711yescharacteristic, complemented
130C24—6284no—
131C24—6288no—
132C24—1574no—
133C24—1578no—
134C24C22×D15712yes—
135C2×Q8—42no—
136C2×Q8—6284no—
137C2×Q8—1578no—
138C2×Q8C22×D15714yescomplemented
139C2×D4—42no—
140C2×D4—6284no—
141C2×D4—6284no—
142C2×D4—6288no—
143C2×D4—15716no—
144C2×D4—1578no—
145C2×D4—1578no—
146C2×D4C22×D15714yes—
147C22×C4—1574no—
148C22×C4C22×D15712yes—
149C22×C4C22×D15711yescharacteristic
150C22×C4C22×D15712yes—
151C22⋊C4—1578no—
152C22⋊C4—15716no—
153C22⋊C4—15716no—
154C22⋊C4—15716no—
155C22⋊C4C22×D157116yescomplemented
156C42—1578no—
157C42C22×D15714yescomplemented
158C23—24no—
159C23—28no—
160C23—3148no—
161C23—3148no—
162C23—3148no—
163C23—6288no—
164C23—62816no—
165C23—62832no—
166C23—1578no—
167C23C23×D15714yes—
168C23C23×D15711yescharacteristic
169Q8—44no—
170Q8—6288no—
171D4—44no—
172D4—6288no—
173D4—6288no—
174D4—62816no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—3148no—
179C2×C4—1578no—
180C2×C4C23×D15714yes—
181C2×C4C23×D15712yes—
182C2×C4C23×D15714yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—3148no—
187C22—3148no—
188C22—3148no—
189C22—62832no—
190C22—62816no—
191C22#1909781612yes—
192C22#1909781714yes—
193C22C24×D15711yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—3148no—
198C2—44no—
199C2—6288no—
200C2#2375801514yesdirect product factor
201C2C2×#1909781611yescharacteristic
202C2C2×#1909781712yes—
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×#124×C157C2
socleC23×C157C24
hypercenterC2×#124D157
hypocenterC157C2×H(3,𝔽4)
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #23758015C2direct (N × H)
2. C2×#19097760C2semidirect
3. C2×#19097759C2semidirect
4. C2×#19097757C2semidirect
5. C2×#19097741C2semidirect
6. C2×#19097739C2semidirect
7. C2×#124×C157C2semidirect
8. C24×D157C22semidirect
9. C22×Dic314C22semidirect
13. #19097760C22semidirect
14. #19097759C22semidirect
15. #19097757C22semidirect
16. #19097741C22semidirect
17. #19097739C22semidirect
18. C2×C22⋊Dic157C22semidirect
19. C2×#14578931C22semidirect
20. C2×#14578931C22semidirect
21. C2×#14578931C22semidirect
22. C2×C4×Dic157C22semidirect
23. C22×Q8×C157C22semidirect
25. #124×C157C22semidirect
26. C2×C22⋊C4×C157C22semidirect
27. C2×C42×C157C22semidirect
31. C23×D157C23semidirect
33. C2×Dic314C23semidirect
41. C22⋊Dic157C23semidirect
42. #14578931C23semidirect
43. #14578931C23semidirect
44. #14578931C23semidirect
45. C4×Dic157C23semidirect
48. C2×Q8×C157C23semidirect
54. C22⋊C4×C157C23semidirect
55. C42×C157C23semidirect
97. C2×C157H(3,𝔽4)semidirect
101. C157C2×H(3,𝔽4)semidirect
107. C2×#124D157semidirect
112. C22×Q8C2×D157semidirect
120. #124C2×D157semidirect
127. C2×C22⋊C4C2×D157semidirect
129. C2×C42C2×D157semidirect
138. C2×Q8C22×D157semidirect
155. C22⋊C4C22×D157semidirect
157. C42C22×D157semidirect
200. C2#23758015direct (N × H)

Maximal subgroups

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

subgroupindex
#237580152
C2×#190977602
C2×#190977592
C2×#190977572
C2×#190977412
C2×#190977392
C2×#124×C1572
C2×H(3,𝔽4)157