Finite Group Databasebeta

← C2×#126×D23 #15686651

Subgroups — C2×#126×D23 #15686651

Order 2944.

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×#10625069C213yesmaximal, complemented
2C2×#10625047C211yesmaximal, characteristic, complemented
3C2×#10625050C213yesmaximal, complemented
4#126×D23C2116yesmaximal, direct product factor
5C2×C4⋊C4×D23C213yesmaximal, complemented
6C2×C22⋊C4×D23C213yesmaximal
7C2×C42×D23C211yesmaximal, characteristic, complemented
8C2×C23×#126C211yesmaximal, characteristic, complemented
9C24×D23C2211yescharacteristic
10C22×C4×D23C2213yes—
11C22×C4×D23C2213yes—
12C23×#126C2218yescomplemented
13C2×C4⋊C4×C23C2213yescomplemented
14C2×C22⋊C4×C23C2213yes—
15C2×C42×C23C2211yescharacteristic, complemented
16C2×C22⋊Dic23C2213yes—
17C2×(C2×D23)⋊C4C2213yes—
18C2×(C2×D23)⋊C4C2213yes—
19C2×C4⋊Dic23C2213yescomplemented
20C2×Dic23⋊C4C2213yescomplemented
21C2×Dic23⋊C4C2213yescomplemented
22C2×C4×Dic23C2213yescomplemented
23#10625069C22124yescomplemented
24C4⋊C4×D23C22124yescomplemented
25C22⋊C4×D23C22124yes—
26#10625050C22124yescomplemented
27#10625047C2218yescomplemented
28C42×D23C2218yescomplemented
29C24×C23C2311yescharacteristic
30C23×D23—212no—
31C23×D23C2314yes—
32C23×D23C2311yescharacteristic
33C23×D23C2311yescharacteristic
34C22×C4×C23C2313yes—
35C22×C4×C23C2313yes—
36C22×Dic23C2313yes—
37C22×Dic23C2313yes—
38C2×C4×D23—212no—
39C2×C4×D23—212no—
40C2×C4×D23C23112yes—
41C2×C4×D23C23112yes—
42C4⋊C4×C23C23112yescomplemented
43C22⋊C4×C23C23112yes—
44C42×C23C2314yescomplemented
45C22⋊Dic23C23112yes—
46(C2×D23)⋊C4C23112yes—
47(C2×D23)⋊C4C23112yes—
48C4⋊Dic23C23112yescomplemented
49Dic23⋊C4C23112yescomplemented
50Dic23⋊C4C23112yescomplemented
51C4×Dic23C23112yescomplemented
52C23×C23—26no—
53C23×C23C2412yes—
54C23×C23C2411yescharacteristic
55C22×D23—26no—
56C22×D23—224no—
57C22×D23—416no—
58C22×D23#41112yes—
59C22×D23C2412yes—
60C22×D23C2412yes—
61C2×C4×C23—26no—
62C2×C4×C23—26no—
63C2×C4×C23C2416yes—
64C2×C4×C23C2416yes—
65C2×Dic23—26no—
66C2×Dic23—26no—
67C2×Dic23C2416yes—
68C2×Dic23C2416yes—
69C4×D23—224no—
70C4×D23—224no—
71C22×#126—231nomaximal, Sylow
72C22×C23—26no—
73C22×C23—44no—
74C22×C23C2×#4116yes—
75C22×C23C2511yescharacteristic
76C2×D23—26no—
77C2×D23—44no—
78C2×D23—416no—
79C2×D23#126116yesdirect product factor
80C2×D23C2×#41112yes—
81C4×C23—26no—
82C4×C23—26no—
83Dic23—26no—
84Dic23—26no—
85C2×#126—2316no—
86C2×#126—233no—
87C2×#126—233no—
88C2×#126—231no—
89C2×#126D2311yescharacteristic, direct product factor
90C22×C4⋊C4—233no—
91C22×C22⋊C4—233no—
92C22×C42—231no—
93C2×C23—42no—
94C2×C23C2×#12614yescomplemented
95C2×C23C22×#4113yes—
96D23—42no—
97D23C2×#12618yesdirect product factor
98C25—231no—
99C23×C4—233no—
100C23×C4—233no—
101#126—2324no—
102#126—2324no—
103#126—238no—
104#126C2×D2318yesdirect product factor
105C2×C4⋊C4—233no—
106C2×C4⋊C4—233no—
107C2×C4⋊C4—233no—
108C2×C4⋊C4—2324no—
109C2×C4⋊C4C2×D2313yescomplemented
110C2×C22⋊C4—233no—
111C2×C22⋊C4—233no—
112C2×C22⋊C4—233no—
113C2×C22⋊C4—2324no—
114C2×C22⋊C4C2×D2313yes—
115C2×C42—233no—
116C2×C42—238no—
117C2×C42C2×D2311yescharacteristic, complemented
118C23C22×#12611yescharacteristic, complemented, Sylow
119C24—231no—
120C24—234no—
121C24—231no—
122C24—4612no—
123C24C22×D2311yescharacteristic
124C22×C4—233no—
125C22×C4—233no—
126C22×C4—2312no—
127C22×C4—2312no—
128C22×C4—4612no—
129C22×C4—4612no—
130C22×C4C22×D2313yes—
131C22×C4C22×D2313yes—
132C4⋊C4—2312no—
133C4⋊C4—2312no—
134C4⋊C4—2312no—
135C4⋊C4C22×D23112yescomplemented
136C22⋊C4—2312no—
137C22⋊C4—2312no—
138C22⋊C4—2312no—
139C22⋊C4C22×D23112yes—
140C42—2312no—
141C42C22×D2314yescomplemented
142C23—26no—
143C23—2312no—
144C23—232no—
145C23—232no—
146C23—4624no—
147C23—466no—
148C23—9216no—
149C23C23×D2312yes—
150C23C23×D2311yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—236no—
154C2×C4—236no—
155C2×C4—4624no—
156C2×C4—4624no—
157C2×C4—466no—
158C2×C4—466no—
159C2×C4C23×D2316yes—
160C2×C4C23×D2316yes—
161C22—26no—
162C22—44no—
163C22—2316no—
164C22—2312no—
165C22—466no—
166C22—9216no—
167C22—924no—
168C22#41×D2316yes—
169C22C24×D2311yescharacteristic
170C4—26no—
171C4—26no—
172C4—466no—
173C4—466no—
174C2—42no—
175C2—238no—
176C2—922no—
177C2#126×D2314yesdirect product factor
178C2C2×#41×D2313yes—
subgroupG/H
centerC23C23×D23
commutator, cosocleC22×C23C25
FrattiniC22C24×D23
FittingC2×C23×#126C2
socleC23×C23C24
prime radicalC24×D23C22
hypercenterC2×#126D23
hypocenterC23C22×#126
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#10625069C2semidirect
2. C2×#10625047C2semidirect
3. C2×#10625050C2semidirect
4. #126×D23C2direct (N × H)
5. C2×C4⋊C4×D23C2semidirect
7. C2×C42×D23C2semidirect
8. C2×C23×#126C2semidirect
12. C23×#126C22semidirect
13. C2×C4⋊C4×C23C22semidirect
15. C2×C42×C23C22semidirect
19. C2×C4⋊Dic23C22semidirect
20. C2×Dic23⋊C4C22semidirect
21. C2×Dic23⋊C4C22semidirect
22. C2×C4×Dic23C22semidirect
23. #10625069C22semidirect
24. C4⋊C4×D23C22semidirect
26. #10625050C22semidirect
27. #10625047C22semidirect
28. C42×D23C22semidirect
42. C4⋊C4×C23C23semidirect
44. C42×C23C23semidirect
48. C4⋊Dic23C23semidirect
49. Dic23⋊C4C23semidirect
50. Dic23⋊C4C23semidirect
51. C4×Dic23C23semidirect
79. C2×D23#126direct (N × H)
89. C2×#126D23direct (N × H)
94. C2×C23C2×#126semidirect
97. D23C2×#126direct (N × H)
104. #126C2×D23direct (N × H)
109. C2×C4⋊C4C2×D23semidirect
117. C2×C42C2×D23semidirect
118. C23C22×#126semidirect
135. C4⋊C4C22×D23semidirect
141. C42C22×D23semidirect
177. C2#126×D23direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106250692
C2×#106250472
C2×#106250502
#126×D232
C2×C4⋊C4×D232
C2×C22⋊C4×D232
C2×C42×D232
C2×C23×#1262
C22×#12623