Finite Group Databasebeta

← C2×#126×D89 #26033506

Subgroups — C2×#126×D89 #26033506

Order 11392.

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×#15668007C213yesmaximal, complemented
2C2×#15668006C211yesmaximal, characteristic, complemented
3C2×#15668005C213yesmaximal, complemented
4#126×D89C2116yesmaximal, direct product factor
5C2×C4⋊C4×D89C213yesmaximal, complemented
6C2×C22⋊C4×D89C213yesmaximal
7C2×C42×D89C211yesmaximal, characteristic, complemented
8C2×#126×C89C211yesmaximal, characteristic, complemented
9C24×D89C2211yescharacteristic
10C22×C4×D89C2213yes—
11C22×C4×D89C2213yes—
12#15668007C22124yescomplemented
13#15668006C2218yescomplemented
14#15668005C22124yescomplemented
15C2×C4⋊Dic89C2213yescomplemented
16C2×#13083553C2213yescomplemented
17C2×#13083553C2213yescomplemented
18C4⋊C4×D89C22124yescomplemented
19C2×C22⋊Dic89C2213yes—
20C2×#13083551C2213yes—
21C2×#13083551C2213yes—
22C22⋊C4×D89C22124yes—
23C2×C4×Dic89C2213yescomplemented
24C42×D89C2218yescomplemented
25#126×C89C2218yescomplemented
26C2×C4⋊C4×C89C2213yescomplemented
27C2×C22⋊C4×C89C2213yes—
28C2×C42×C89C2211yescharacteristic, complemented
29C23×D89—212no—
30C23×D89C2314yes—
31C23×D89C2311yescharacteristic
32C23×D89C2311yescharacteristic
33C22×Dic89C2313yes—
34C22×Dic89C2313yes—
35C2×C4×D89—212no—
36C2×C4×D89—212no—
37C2×C4×D89C23112yes—
38C2×C4×D89C23112yes—
39C4⋊Dic89C23112yescomplemented
40#13083553C23112yescomplemented
41#13083553C23112yescomplemented
42C22⋊Dic89C23112yes—
43#13083551C23112yes—
44#13083551C23112yes—
45C4×Dic89C23112yescomplemented
46C24×C89C2311yescharacteristic
47C22×C4×C89C2313yes—
48C22×C4×C89C2313yes—
49C4⋊C4×C89C23112yescomplemented
50C22⋊C4×C89C23112yes—
51C42×C89C2314yescomplemented
52C23×C89—26no—
53C23×C89C2412yes—
54C23×C89C2411yescharacteristic
55C22×D89—26no—
56C22×D89—224no—
57C22×D89—416no—
58C22×D89#41112yes—
59C22×D89C2412yes—
60C22×D89C2412yes—
61C2×C4×C89—26no—
62C2×C4×C89—26no—
63C2×C4×C89C2416yes—
64C2×C4×C89C2416yes—
65C2×Dic89—26no—
66C2×Dic89—26no—
67C2×Dic89C2416yes—
68C2×Dic89C2416yes—
69C4×D89—224no—
70C4×D89—224no—
71C22×C89—26no—
72C22×C89—44no—
73C22×C89C2×#4116yes—
74C22×C89C2511yescharacteristic
75C2×D89—26no—
76C2×D89—44no—
77C2×D89—416no—
78C2×D89#126116yesdirect product factor
79C2×D89C2×#41112yes—
80C4×C89—26no—
81C4×C89—26no—
82Dic89—26no—
83Dic89—26no—
84C2×C89—42no—
85C2×C89C2×#12614yescomplemented
86C2×C89C22×#4113yes—
87D89—42no—
88D89C2×#12618yesdirect product factor
89C22×#126—891nomaximal, Sylow
90C89C22×#12611yescharacteristic, complemented, Sylow
91C2×#126—8916no—
92C2×#126—893no—
93C2×#126—893no—
94C2×#126—891no—
95C2×#126D8911yescharacteristic, direct product factor
96C22×C4⋊C4—893no—
97C22×C22⋊C4—893no—
98C22×C42—891no—
99C25—891no—
100C23×C4—893no—
101C23×C4—893no—
102#126—8924no—
103#126—8924no—
104#126—898no—
105#126C2×D8918yesdirect product factor
106C2×C4⋊C4—893no—
107C2×C4⋊C4—893no—
108C2×C4⋊C4—893no—
109C2×C4⋊C4—8924no—
110C2×C4⋊C4C2×D8913yescomplemented
111C2×C22⋊C4—893no—
112C2×C22⋊C4—893no—
113C2×C22⋊C4—893no—
114C2×C22⋊C4—8924no—
115C2×C22⋊C4C2×D8913yes—
116C2×C42—893no—
117C2×C42—898no—
118C2×C42C2×D8911yescharacteristic, complemented
119C24—891no—
120C24—894no—
121C24—891no—
122C24—17812no—
123C24C22×D8911yescharacteristic
124C22×C4—893no—
125C22×C4—893no—
126C22×C4—8912no—
127C22×C4—8912no—
128C22×C4—17812no—
129C22×C4—17812no—
130C22×C4C22×D8913yes—
131C22×C4C22×D8913yes—
132C4⋊C4—8912no—
133C4⋊C4—8912no—
134C4⋊C4—8912no—
135C4⋊C4C22×D89112yescomplemented
136C22⋊C4—8912no—
137C22⋊C4—8912no—
138C22⋊C4—8912no—
139C22⋊C4C22×D89112yes—
140C42—8912no—
141C42C22×D8914yescomplemented
142C23—26no—
143C23—8912no—
144C23—892no—
145C23—892no—
146C23—35616no—
147C23—17824no—
148C23—1786no—
149C23C23×D8912yes—
150C23C23×D8911yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—896no—
154C2×C4—896no—
155C2×C4—17824no—
156C2×C4—17824no—
157C2×C4—1786no—
158C2×C4—1786no—
159C2×C4C23×D8916yes—
160C2×C4C23×D8916yes—
161C22—26no—
162C22—44no—
163C22—8912no—
164C22—8916no—
165C22—35616no—
166C22—3564no—
167C22—1786no—
168C22#41×D8916yes—
169C22C24×D8911yescharacteristic
170C4—26no—
171C4—26no—
172C4—1786no—
173C4—1786no—
174C2—42no—
175C2—898no—
176C2—3562no—
177C2#126×D8914yesdirect product factor
178C2C2×#41×D8913yes—
subgroupG/H
centerC23C23×D89
commutator, cosocleC22×C89C25
FrattiniC22C24×D89
FittingC2×#126×C89C2
socleC23×C89C24
prime radicalC24×D89C22
hypercenterC2×#126D89
hypocenterC89C22×#126
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#15668007C2semidirect
2. C2×#15668006C2semidirect
3. C2×#15668005C2semidirect
4. #126×D89C2direct (N × H)
5. C2×C4⋊C4×D89C2semidirect
7. C2×C42×D89C2semidirect
8. C2×#126×C89C2semidirect
12. #15668007C22semidirect
13. #15668006C22semidirect
14. #15668005C22semidirect
15. C2×C4⋊Dic89C22semidirect
16. C2×#13083553C22semidirect
17. C2×#13083553C22semidirect
18. C4⋊C4×D89C22semidirect
23. C2×C4×Dic89C22semidirect
24. C42×D89C22semidirect
25. #126×C89C22semidirect
26. C2×C4⋊C4×C89C22semidirect
28. C2×C42×C89C22semidirect
39. C4⋊Dic89C23semidirect
40. #13083553C23semidirect
41. #13083553C23semidirect
45. C4×Dic89C23semidirect
49. C4⋊C4×C89C23semidirect
51. C42×C89C23semidirect
78. C2×D89#126direct (N × H)
85. C2×C89C2×#126semidirect
88. D89C2×#126direct (N × H)
90. C89C22×#126semidirect
95. C2×#126D89direct (N × H)
105. #126C2×D89direct (N × H)
110. C2×C4⋊C4C2×D89semidirect
118. C2×C42C2×D89semidirect
135. C4⋊C4C22×D89semidirect
141. C42C22×D89semidirect
177. C2#126×D89direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156680072
C2×#156680062
C2×#156680052
#126×D892
C2×C4⋊C4×D892
C2×C22⋊C4×D892
C2×C42×D892
C2×#126×C892
C22×#12689