Finite Group Databasebeta

← C2×#23758014 #34114254

Subgroups — C2×#23758014 #34114254

Order 20096.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 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#23758014C2116yesmaximal, direct product factor
2C2×#19097768C211yesmaximal, characteristic, complemented
3C2×#19097765C211yesmaximal, characteristic, complemented
4C2×#19097756C213yesmaximal, complemented
5C2×#19097746C213yesmaximal, complemented
6C2×#19097744C213yesmaximal, complemented
7C2×#19097741C213yesmaximal, complemented
8C2×#126×C157C211yesmaximal, characteristic, complemented
9C24×D157C2211yescharacteristic
10C22×D628C2213yes—
11C22×D628C2213yes—
12C22×D628C2213yescomplemented
13C22×C22⋊D157C2213yes—
14C22×C4×D157C2213yescomplemented
15#19097768C2218yescomplemented
16#19097765C2218yescomplemented
17#19097756C22124yescomplemented
18#19097746C22124yescomplemented
19#19097744C22124yescomplemented
20#19097741C22124yescomplemented
21C2×#14578933C2213yescomplemented
22C2×#14578931C2213yes—
23C2×#14578931C2213yes—
24C2×#14578931C2213yescomplemented
25#126×C157C2218yescomplemented
26C2×C4⋊C4×C157C2213yescomplemented
27C2×C22⋊C4×C157C2213yes—
28C2×C42×C157C2211yescharacteristic, complemented
29C23×D157—46no—
30C23×D157C2311yescharacteristic
31C23×D157C2311yescharacteristic
32C23×D157C2313yes—
33C23×D157C2314yes—
34C2×D628—46no—
35C2×D628—46no—
36C2×D628—46no—
37C2×D628C23112yes—
38C2×D628C23112yes—
39C2×D628C23112yescomplemented
40C2×C22⋊D157—46no—
41C2×C22⋊D157C23112yes—
42C22×Dic157C2313yes—
43C2×C4×D157—46no—
44C2×C4×D157C23112yescomplemented
45#14578933C23112yescomplemented
46#14578931C23112yes—
47#14578931C23112yes—
48#14578931C23112yescomplemented
49C24×C157C2311yescharacteristic
50C22×C4×C157C2313yes—
51C22×C4×C157C2313yes—
52C4⋊C4×C157C23112yescomplemented
53C22⋊C4×C157C23112yes—
54C42×C157C2314yescomplemented
55C22×D157—26no—
56C22×D157—26no—
57C22×D157—26no—
58C22×D157—26no—
59C22×D157—26no—
60C22×D157—412no—
61C22×D157—416no—
62C22×D157C2416yes—
63C22×D157C2412yes—
64C22×D157C2412yes—
65D628—412no—
66D628—412no—
67D628—412no—
68C22⋊D157—412no—
69C2×Dic157—26no—
70C2×Dic157C2416yes—
71C4×D157—412no—
72C23×C157—26no—
73C23×C157C2412yes—
74C23×C157C2411yescharacteristic
75C2×C4×C157—26no—
76C2×C4×C157—26no—
77C2×C4×C157C2416yes—
78C2×C4×C157C2416yes—
79C22×C157—26no—
80C22×C157—44no—
81C22×C157H(5,𝔽2)16yes—
82C22×C157C2511yescharacteristic
83C2×D157—26no—
84C2×D157—26no—
85C2×D157—26no—
86C2×D157—26no—
87C2×D157—26no—
88C2×D157—416no—
89C2×D157—412no—
90C2×D157—44no—
91C2×D157—44no—
92C4×C157—26no—
93C4×C157—26no—
94Dic157—26no—
95C2×C157—42no—
96C2×C157#56014yescomplemented
97C2×C157C2×H(5,𝔽2)13yes—
98D157—46no—
99D157—42no—
100D157—42no—
101C157C2×#56011yescharacteristic, complemented, Sylow
102C2×#560—1571nomaximal, Sylow
103#560—15716no—
104C2×#127—1571no—
105C2×#126—1571no—
106C2×#126D15711yescharacteristic, complemented
107C2×#123—1573no—
108C2×#121—1573no—
109C2×#121—1573no—
110C2×#120—1573no—
111C25—1571no—
112C22×D4—1573no—
113C22×D4—1573no—
114C22×D4—1573no—
115C22×D4—1573no—
116C23×C4—1573no—
117#127—1578no—
118#126—1578no—
119#126C2×D15718yescomplemented
120#123—15724no—
121#121—15724no—
122#121—15724no—
123#120—15724no—
124C2×C4⋊C4—1573no—
125C2×C4⋊C4C2×D15713yescomplemented
126C2×C22⋊C4—1573no—
127C2×C22⋊C4—1573no—
128C2×C22⋊C4—1573no—
129C2×C22⋊C4C2×D15713yes—
130C2×C42C2×D15711yescharacteristic, complemented
131C24—6286no—
132C24—1571no—
133C24—1571no—
134C24—1573no—
135C24—1574no—
136C24C22×D15711yescharacteristic
137C2×D4—6286no—
138C2×D4—6286no—
139C2×D4—6286no—
140C2×D4—6286no—
141C2×D4—15712no—
142C2×D4—15712no—
143C2×D4—15712no—
144C2×D4—15712no—
145C22×C4—6286no—
146C22×C4—1573no—
147C22×C4—15712no—
148C22×C4C22×D15713yes—
149C22×C4C22×D15713yes—
150C4⋊C4—15712no—
151C4⋊C4C22×D157112yescomplemented
152C22⋊C4—15712no—
153C22⋊C4—15712no—
154C22⋊C4—15712no—
155C22⋊C4C22×D157112yes—
156C42C22×D15714yescomplemented
157C23—26no—
158C23—3146no—
159C23—3146no—
160C23—3146no—
161C23—3146no—
162C23—3146no—
163C23—62812no—
164C23—62816no—
165C23—1572no—
166C23—1572no—
167C23—1576no—
168C23C23×D15712yes—
169C23C23×D15711yescharacteristic
170D4—62812no—
171D4—62812no—
172D4—62812no—
173D4—62812no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—3146no—
177C2×C4—62812no—
178C2×C4—1576no—
179C2×C4C23×D15716yes—
180C2×C4C23×D15716yes—
181C22—26no—
182C22—44no—
183C22—3146no—
184C22—3146no—
185C22—3146no—
186C22—3146no—
187C22—3146no—
188C22—62816no—
189C22—6284no—
190C22—6284no—
191C22—62812no—
192C22#1909781716yes—
193C22C24×D15711yescharacteristic
194C4—26no—
195C4—26no—
196C4—3146no—
197C2—42no—
198C2—6282no—
199C2—6282no—
200C2—6286no—
201C2#2375801414yesdirect product factor
202C2C2×#1909781713yes—
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×#126×C157C2
socleC23×C157C24
hypercenterC2×#126D157
hypocenterC157C2×#560
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #23758014C2direct (N × H)
2. C2×#19097768C2semidirect
3. C2×#19097765C2semidirect
4. C2×#19097756C2semidirect
5. C2×#19097746C2semidirect
6. C2×#19097744C2semidirect
7. C2×#19097741C2semidirect
8. C2×#126×C157C2semidirect
12. C22×D628C22semidirect
14. C22×C4×D157C22semidirect
15. #19097768C22semidirect
16. #19097765C22semidirect
17. #19097756C22semidirect
18. #19097746C22semidirect
19. #19097744C22semidirect
20. #19097741C22semidirect
21. C2×#14578933C22semidirect
24. C2×#14578931C22semidirect
25. #126×C157C22semidirect
26. C2×C4⋊C4×C157C22semidirect
28. C2×C42×C157C22semidirect
39. C2×D628C23semidirect
44. C2×C4×D157C23semidirect
45. #14578933C23semidirect
48. #14578931C23semidirect
52. C4⋊C4×C157C23semidirect
54. C42×C157C23semidirect
96. C2×C157#560semidirect
101. C157C2×#560semidirect
106. C2×#126D157semidirect
119. #126C2×D157semidirect
125. C2×C4⋊C4C2×D157semidirect
130. C2×C42C2×D157semidirect
151. C4⋊C4C22×D157semidirect
156. C42C22×D157semidirect
201. C2#23758014direct (N × H)

Maximal subgroups

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

subgroupindex
#237580142
C2×#190977682
C2×#190977652
C2×#190977562
C2×#190977462
C2×#190977442
C2×#190977412
C2×#126×C1572
C2×#560157