Finite Group Databasebeta

← C2×#126×D97 #27154990

Subgroups — C2×#126×D97 #27154990

Order 12416.

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×#15690805C213yesmaximal, complemented
2C2×#15690804C211yesmaximal, characteristic, complemented
3C2×#15690803C213yesmaximal, complemented
4#126×D97C2116yesmaximal, direct product factor
5C2×C4⋊C4×D97C213yesmaximal, complemented
6C2×C22⋊C4×D97C213yesmaximal
7C2×C42×D97C211yesmaximal, characteristic, complemented
8C2×#126×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C22×C4×D97C2213yes—
11C22×C4×D97C2213yes—
12#15690805C22124yescomplemented
13#15690804C2218yescomplemented
14#15690803C22124yescomplemented
15C2×C4⋊Dic97C2213yescomplemented
16C2×#13095490C2213yescomplemented
17C2×#13095490C2213yescomplemented
18C4⋊C4×D97C22124yescomplemented
19C2×C22⋊Dic97C2213yes—
20C2×#13095488C2213yes—
21C2×#13095488C2213yes—
22C22⋊C4×D97C22124yes—
23C2×C4×Dic97C2213yescomplemented
24C42×D97C2218yescomplemented
25#126×C97C2218yescomplemented
26C2×C4⋊C4×C97C2213yescomplemented
27C2×C22⋊C4×C97C2213yes—
28C2×C42×C97C2211yescharacteristic, complemented
29C23×D97—212no—
30C23×D97C2314yes—
31C23×D97C2311yescharacteristic
32C23×D97C2311yescharacteristic
33C22×Dic97C2313yes—
34C22×Dic97C2313yes—
35C2×C4×D97—212no—
36C2×C4×D97—212no—
37C2×C4×D97C23112yes—
38C2×C4×D97C23112yes—
39C4⋊Dic97C23112yescomplemented
40#13095490C23112yescomplemented
41#13095490C23112yescomplemented
42C22⋊Dic97C23112yes—
43#13095488C23112yes—
44#13095488C23112yes—
45C4×Dic97C23112yescomplemented
46C24×C97C2311yescharacteristic
47C22×C4×C97C2313yes—
48C22×C4×C97C2313yes—
49C4⋊C4×C97C23112yescomplemented
50C22⋊C4×C97C23112yes—
51C42×C97C2314yescomplemented
52C23×C97—26no—
53C23×C97C2412yes—
54C23×C97C2411yescharacteristic
55C22×D97—26no—
56C22×D97—224no—
57C22×D97—416no—
58C22×D97#41112yes—
59C22×D97C2412yes—
60C22×D97C2412yes—
61C2×C4×C97—26no—
62C2×C4×C97—26no—
63C2×C4×C97C2416yes—
64C2×C4×C97C2416yes—
65C2×Dic97—26no—
66C2×Dic97—26no—
67C2×Dic97C2416yes—
68C2×Dic97C2416yes—
69C4×D97—224no—
70C4×D97—224no—
71C22×C97—26no—
72C22×C97—44no—
73C22×C97C2×#4116yes—
74C22×C97C2511yescharacteristic
75C2×D97—26no—
76C2×D97—44no—
77C2×D97—416no—
78C2×D97#126116yesdirect product factor
79C2×D97C2×#41112yes—
80C4×C97—26no—
81C4×C97—26no—
82Dic97—26no—
83Dic97—26no—
84C2×C97—42no—
85C2×C97C2×#12614yescomplemented
86C2×C97C22×#4113yes—
87D97—42no—
88D97C2×#12618yesdirect product factor
89C22×#126—971nomaximal, Sylow
90C97C22×#12611yescharacteristic, complemented, Sylow
91C2×#126—9716no—
92C2×#126—973no—
93C2×#126—973no—
94C2×#126—971no—
95C2×#126D9711yescharacteristic, direct product factor
96C22×C4⋊C4—973no—
97C22×C22⋊C4—973no—
98C22×C42—971no—
99C25—971no—
100C23×C4—973no—
101C23×C4—973no—
102#126—9724no—
103#126—9724no—
104#126—978no—
105#126C2×D9718yesdirect product factor
106C2×C4⋊C4—973no—
107C2×C4⋊C4—973no—
108C2×C4⋊C4—973no—
109C2×C4⋊C4—9724no—
110C2×C4⋊C4C2×D9713yescomplemented
111C2×C22⋊C4—973no—
112C2×C22⋊C4—973no—
113C2×C22⋊C4—973no—
114C2×C22⋊C4—9724no—
115C2×C22⋊C4C2×D9713yes—
116C2×C42—973no—
117C2×C42—978no—
118C2×C42C2×D9711yescharacteristic, complemented
119C24—971no—
120C24—974no—
121C24—971no—
122C24—19412no—
123C24C22×D9711yescharacteristic
124C22×C4—973no—
125C22×C4—973no—
126C22×C4—9712no—
127C22×C4—9712no—
128C22×C4—19412no—
129C22×C4—19412no—
130C22×C4C22×D9713yes—
131C22×C4C22×D9713yes—
132C4⋊C4—9712no—
133C4⋊C4—9712no—
134C4⋊C4—9712no—
135C4⋊C4C22×D97112yescomplemented
136C22⋊C4—9712no—
137C22⋊C4—9712no—
138C22⋊C4—9712no—
139C22⋊C4C22×D97112yes—
140C42—9712no—
141C42C22×D9714yescomplemented
142C23—26no—
143C23—9712no—
144C23—972no—
145C23—972no—
146C23—38816no—
147C23—19424no—
148C23—1946no—
149C23C23×D9712yes—
150C23C23×D9711yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—976no—
154C2×C4—976no—
155C2×C4—19424no—
156C2×C4—19424no—
157C2×C4—1946no—
158C2×C4—1946no—
159C2×C4C23×D9716yes—
160C2×C4C23×D9716yes—
161C22—26no—
162C22—44no—
163C22—9716no—
164C22—9712no—
165C22—38816no—
166C22—3884no—
167C22—1946no—
168C22#41×D9716yes—
169C22C24×D9711yescharacteristic
170C4—26no—
171C4—26no—
172C4—1946no—
173C4—1946no—
174C2—42no—
175C2—978no—
176C2—3882no—
177C2#126×D9714yesdirect product factor
178C2C2×#41×D9713yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#126×C97C2
socleC23×C97C24
prime radicalC24×D97C22
hypercenterC2×#126D97
hypocenterC97C22×#126
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#15690805C2semidirect
2. C2×#15690804C2semidirect
3. C2×#15690803C2semidirect
4. #126×D97C2direct (N × H)
5. C2×C4⋊C4×D97C2semidirect
7. C2×C42×D97C2semidirect
8. C2×#126×C97C2semidirect
12. #15690805C22semidirect
13. #15690804C22semidirect
14. #15690803C22semidirect
15. C2×C4⋊Dic97C22semidirect
16. C2×#13095490C22semidirect
17. C2×#13095490C22semidirect
18. C4⋊C4×D97C22semidirect
23. C2×C4×Dic97C22semidirect
24. C42×D97C22semidirect
25. #126×C97C22semidirect
26. C2×C4⋊C4×C97C22semidirect
28. C2×C42×C97C22semidirect
39. C4⋊Dic97C23semidirect
40. #13095490C23semidirect
41. #13095490C23semidirect
45. C4×Dic97C23semidirect
49. C4⋊C4×C97C23semidirect
51. C42×C97C23semidirect
78. C2×D97#126direct (N × H)
85. C2×C97C2×#126semidirect
88. D97C2×#126direct (N × H)
90. C97C22×#126semidirect
95. C2×#126D97direct (N × H)
105. #126C2×D97direct (N × H)
110. C2×C4⋊C4C2×D97semidirect
118. C2×C42C2×D97semidirect
135. C4⋊C4C22×D97semidirect
141. C42C22×D97semidirect
177. C2#126×D97direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908052
C2×#156908042
C2×#156908032
#126×D972
C2×C4⋊C4×D972
C2×C22⋊C4×D972
C2×C42×D972
C2×#126×C972
C22×#12697