Finite Group Databasebeta

← C2×#126×D157 #34113390

Subgroups — C2×#126×D157 #34113390

Order 20096.

1320 conjugacy classes of subgroups; only the 517 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 517 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×#19097766C213yesmaximal, complemented
2C2×#19097765C211yesmaximal, characteristic, complemented
3C2×#19097764C213yesmaximal, complemented
4#126×D157C2116yesmaximal, direct product factor
5C2×C4⋊C4×D157C213yesmaximal, complemented
6C2×C22⋊C4×D157C213yesmaximal
7C2×C42×D157C211yesmaximal, characteristic, complemented
8C2×#126×C157C211yesmaximal, characteristic, complemented
9C24×D157C2211yescharacteristic
10C22×C4×D157C2213yes—
11C22×C4×D157C2213yes—
12#19097766C22124yescomplemented
13#19097765C2218yescomplemented
14#19097764C22124yescomplemented
15C2×C4⋊Dic157C2213yescomplemented
16C2×#14578933C2213yescomplemented
17C2×#14578933C2213yescomplemented
18C4⋊C4×D157C22124yescomplemented
19C2×C22⋊Dic157C2213yes—
20C2×#14578931C2213yes—
21C2×#14578931C2213yes—
22C22⋊C4×D157C22124yes—
23C2×C4×Dic157C2213yescomplemented
24C42×D157C2218yescomplemented
25#126×C157C2218yescomplemented
26C2×C4⋊C4×C157C2213yescomplemented
27C2×C22⋊C4×C157C2213yes—
28C2×C42×C157C2211yescharacteristic, complemented
29C23×D157—212no—
30C23×D157C2314yes—
31C23×D157C2311yescharacteristic
32C23×D157C2311yescharacteristic
33C22×Dic157C2313yes—
34C22×Dic157C2313yes—
35C2×C4×D157—212no—
36C2×C4×D157—212no—
37C2×C4×D157C23112yes—
38C2×C4×D157C23112yes—
39C4⋊Dic157C23112yescomplemented
40#14578933C23112yescomplemented
41#14578933C23112yescomplemented
42C22⋊Dic157C23112yes—
43#14578931C23112yes—
44#14578931C23112yes—
45C4×Dic157C23112yescomplemented
46C24×C157C2311yescharacteristic
47C22×C4×C157C2313yes—
48C22×C4×C157C2313yes—
49C4⋊C4×C157C23112yescomplemented
50C22⋊C4×C157C23112yes—
51C42×C157C2314yescomplemented
52C22×D157—26no—
53C22×D157—224no—
54C22×D157—416no—
55C22×D157#41112yes—
56C22×D157C2412yes—
57C22×D157C2412yes—
58C2×Dic157—26no—
59C2×Dic157—26no—
60C2×Dic157C2416yes—
61C2×Dic157C2416yes—
62C4×D157—224no—
63C4×D157—224no—
64C23×C157—26no—
65C23×C157C2412yes—
66C23×C157C2411yescharacteristic
67C2×C4×C157—26no—
68C2×C4×C157—26no—
69C2×C4×C157C2416yes—
70C2×C4×C157C2416yes—
71C22×C157—26no—
72C22×C157—44no—
73C22×C157C2×#4116yes—
74C22×C157C2511yescharacteristic
75C2×D157—26no—
76C2×D157—44no—
77C2×D157—416no—
78C2×D157#126116yesdirect product factor
79C2×D157C2×#41112yes—
80C4×C157—26no—
81C4×C157—26no—
82Dic157—26no—
83Dic157—26no—
84C2×C157—42no—
85C2×C157C2×#12614yescomplemented
86C2×C157C22×#4113yes—
87D157—42no—
88D157C2×#12618yesdirect product factor
89C157C22×#12611yescharacteristic, complemented, Sylow
90C22×#126—1571nomaximal, Sylow
91C2×#126—15716no—
92C2×#126—1573no—
93C2×#126—1573no—
94C2×#126—1571no—
95C2×#126D15711yescharacteristic, direct product factor
96C22×C4⋊C4—1573no—
97C22×C22⋊C4—1573no—
98C22×C42—1571no—
99C25—1571no—
100C23×C4—1573no—
101C23×C4—1573no—
102#126—15724no—
103#126—15724no—
104#126—1578no—
105#126C2×D15718yesdirect product factor
106C2×C4⋊C4—1573no—
107C2×C4⋊C4—1573no—
108C2×C4⋊C4—1573no—
109C2×C4⋊C4—15724no—
110C2×C4⋊C4C2×D15713yescomplemented
111C2×C22⋊C4—1573no—
112C2×C22⋊C4—1573no—
113C2×C22⋊C4—1573no—
114C2×C22⋊C4—15724no—
115C2×C22⋊C4C2×D15713yes—
116C2×C42—1573no—
117C2×C42—1578no—
118C2×C42C2×D15711yescharacteristic, complemented
119C24—31412no—
120C24—1574no—
121C24—1571no—
122C24—1571no—
123C24C22×D15711yescharacteristic
124C22×C4—31412no—
125C22×C4—31412no—
126C22×C4—1573no—
127C22×C4—1573no—
128C22×C4—15712no—
129C22×C4—15712no—
130C22×C4C22×D15713yes—
131C22×C4C22×D15713yes—
132C4⋊C4—15712no—
133C4⋊C4—15712no—
134C4⋊C4—15712no—
135C4⋊C4C22×D157112yescomplemented
136C22⋊C4—15712no—
137C22⋊C4—15712no—
138C22⋊C4—15712no—
139C22⋊C4C22×D157112yes—
140C42—15712no—
141C42C22×D15714yescomplemented
142C23—26no—
143C23—31424no—
144C23—3146no—
145C23—62816no—
146C23—1572no—
147C23—15712no—
148C23—1572no—
149C23C23×D15712yes—
150C23C23×D15711yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—31424no—
154C2×C4—31424no—
155C2×C4—3146no—
156C2×C4—3146no—
157C2×C4—1576no—
158C2×C4—1576no—
159C2×C4C23×D15716yes—
160C2×C4C23×D15716yes—
161C22—26no—
162C22—44no—
163C22—3146no—
164C22—62816no—
165C22—6284no—
166C22—15712no—
167C22—15716no—
168C22#41×D15716yes—
169C22C24×D15711yescharacteristic
170C4—26no—
171C4—26no—
172C4—3146no—
173C4—3146no—
174C2—42no—
175C2—6282no—
176C2—1578no—
177C2#126×D15714yesdirect product factor
178C2C2×#41×D15713yes—
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×#126×C157C2
socleC23×C157C24
prime radicalC24×D157C22
hypercenterC2×#126D157
hypocenterC157C22×#126
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#19097766C2semidirect
2. C2×#19097765C2semidirect
3. C2×#19097764C2semidirect
4. #126×D157C2direct (N × H)
5. C2×C4⋊C4×D157C2semidirect
7. C2×C42×D157C2semidirect
8. C2×#126×C157C2semidirect
12. #19097766C22semidirect
13. #19097765C22semidirect
14. #19097764C22semidirect
15. C2×C4⋊Dic157C22semidirect
16. C2×#14578933C22semidirect
17. C2×#14578933C22semidirect
18. C4⋊C4×D157C22semidirect
23. C2×C4×Dic157C22semidirect
24. C42×D157C22semidirect
25. #126×C157C22semidirect
26. C2×C4⋊C4×C157C22semidirect
28. C2×C42×C157C22semidirect
39. C4⋊Dic157C23semidirect
40. #14578933C23semidirect
41. #14578933C23semidirect
45. C4×Dic157C23semidirect
49. C4⋊C4×C157C23semidirect
51. C42×C157C23semidirect
78. C2×D157#126direct (N × H)
85. C2×C157C2×#126semidirect
88. D157C2×#126direct (N × H)
89. C157C22×#126semidirect
95. C2×#126D157direct (N × H)
105. #126C2×D157direct (N × H)
110. C2×C4⋊C4C2×D157semidirect
118. C2×C42C2×D157semidirect
135. C4⋊C4C22×D157semidirect
141. C42C22×D157semidirect
177. C2#126×D157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190977662
C2×#190977652
C2×#190977642
#126×D1572
C2×C4⋊C4×D1572
C2×C22⋊C4×D1572
C2×C42×D1572
C2×#126×C1572
C22×#126157