Finite Group Databasebeta

← C2×#126×D101 #27184386

Subgroups — C2×#126×D101 #27184386

Order 12928.

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×#15692250C213yesmaximal, complemented
2C2×#15692249C211yesmaximal, characteristic, complemented
3C2×#15692248C213yesmaximal, complemented
4#126×D101C2116yesmaximal, direct product factor
5C2×C4⋊C4×D101C213yesmaximal, complemented
6C2×C22⋊C4×D101C213yesmaximal
7C2×C42×D101C211yesmaximal, characteristic, complemented
8C2×#126×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic
10C22×C4×D101C2213yes—
11C22×C4×D101C2213yes—
12#15692250C22124yescomplemented
13#15692249C2218yescomplemented
14#15692248C22124yescomplemented
15C2×C4⋊Dic101C2213yescomplemented
16C2×#13108053C2213yescomplemented
17C2×#13108053C2213yescomplemented
18C4⋊C4×D101C22124yescomplemented
19C2×C22⋊Dic101C2213yes—
20C2×#13108051C2213yes—
21C2×#13108051C2213yes—
22C22⋊C4×D101C22124yes—
23C2×C4×Dic101C2213yescomplemented
24C42×D101C2218yescomplemented
25#126×C101C2218yescomplemented
26C2×C4⋊C4×C101C2213yescomplemented
27C2×C22⋊C4×C101C2213yes—
28C2×C42×C101C2211yescharacteristic, complemented
29C23×D101—212no—
30C23×D101C2314yes—
31C23×D101C2311yescharacteristic
32C23×D101C2311yescharacteristic
33C22×Dic101C2313yes—
34C22×Dic101C2313yes—
35C2×C4×D101—212no—
36C2×C4×D101—212no—
37C2×C4×D101C23112yes—
38C2×C4×D101C23112yes—
39C4⋊Dic101C23112yescomplemented
40#13108053C23112yescomplemented
41#13108053C23112yescomplemented
42C22⋊Dic101C23112yes—
43#13108051C23112yes—
44#13108051C23112yes—
45C4×Dic101C23112yescomplemented
46C24×C101C2311yescharacteristic
47C22×C4×C101C2313yes—
48C22×C4×C101C2313yes—
49C4⋊C4×C101C23112yescomplemented
50C22⋊C4×C101C23112yes—
51C42×C101C2314yescomplemented
52C23×C101—26no—
53C23×C101C2412yes—
54C23×C101C2411yescharacteristic
55C22×D101—26no—
56C22×D101—224no—
57C22×D101—416no—
58C22×D101#41112yes—
59C22×D101C2412yes—
60C22×D101C2412yes—
61C2×C4×C101—26no—
62C2×C4×C101—26no—
63C2×C4×C101C2416yes—
64C2×C4×C101C2416yes—
65C2×Dic101—26no—
66C2×Dic101—26no—
67C2×Dic101C2416yes—
68C2×Dic101C2416yes—
69C4×D101—224no—
70C4×D101—224no—
71C22×C101—26no—
72C22×C101—44no—
73C22×C101C2×#4116yes—
74C22×C101C2511yescharacteristic
75C2×D101—26no—
76C2×D101—44no—
77C2×D101—416no—
78C2×D101#126116yesdirect product factor
79C2×D101C2×#41112yes—
80C4×C101—26no—
81C4×C101—26no—
82Dic101—26no—
83Dic101—26no—
84C2×C101—42no—
85C2×C101C2×#12614yescomplemented
86C2×C101C22×#4113yes—
87D101—42no—
88D101C2×#12618yesdirect product factor
89C22×#126—1011nomaximal, Sylow
90C101C22×#12611yescharacteristic, complemented, Sylow
91C2×#126—10116no—
92C2×#126—1013no—
93C2×#126—1013no—
94C2×#126—1011no—
95C2×#126D10111yescharacteristic, direct product factor
96C22×C4⋊C4—1013no—
97C22×C22⋊C4—1013no—
98C22×C42—1011no—
99C25—1011no—
100C23×C4—1013no—
101C23×C4—1013no—
102#126—10124no—
103#126—10124no—
104#126—1018no—
105#126C2×D10118yesdirect product factor
106C2×C4⋊C4—1013no—
107C2×C4⋊C4—1013no—
108C2×C4⋊C4—1013no—
109C2×C4⋊C4—10124no—
110C2×C4⋊C4C2×D10113yescomplemented
111C2×C22⋊C4—1013no—
112C2×C22⋊C4—1013no—
113C2×C22⋊C4—1013no—
114C2×C22⋊C4—10124no—
115C2×C22⋊C4C2×D10113yes—
116C2×C42—1013no—
117C2×C42—1018no—
118C2×C42C2×D10111yescharacteristic, complemented
119C24—1011no—
120C24—1014no—
121C24—1011no—
122C24—20212no—
123C24C22×D10111yescharacteristic
124C22×C4—1013no—
125C22×C4—1013no—
126C22×C4—10112no—
127C22×C4—10112no—
128C22×C4—20212no—
129C22×C4—20212no—
130C22×C4C22×D10113yes—
131C22×C4C22×D10113yes—
132C4⋊C4—10112no—
133C4⋊C4—10112no—
134C4⋊C4—10112no—
135C4⋊C4C22×D101112yescomplemented
136C22⋊C4—10112no—
137C22⋊C4—10112no—
138C22⋊C4—10112no—
139C22⋊C4C22×D101112yes—
140C42—10112no—
141C42C22×D10114yescomplemented
142C23—26no—
143C23—10112no—
144C23—1012no—
145C23—1012no—
146C23—40416no—
147C23—20224no—
148C23—2026no—
149C23C23×D10112yes—
150C23C23×D10111yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—1016no—
154C2×C4—1016no—
155C2×C4—20224no—
156C2×C4—20224no—
157C2×C4—2026no—
158C2×C4—2026no—
159C2×C4C23×D10116yes—
160C2×C4C23×D10116yes—
161C22—26no—
162C22—44no—
163C22—10112no—
164C22—10116no—
165C22—40416no—
166C22—4044no—
167C22—2026no—
168C22#41×D10116yes—
169C22C24×D10111yescharacteristic
170C4—26no—
171C4—26no—
172C4—2026no—
173C4—2026no—
174C2—42no—
175C2—1018no—
176C2—4042no—
177C2#126×D10114yesdirect product factor
178C2C2×#41×D10113yes—
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#126×C101C2
socleC23×C101C24
prime radicalC24×D101C22
hypercenterC2×#126D101
hypocenterC101C22×#126
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#15692250C2semidirect
2. C2×#15692249C2semidirect
3. C2×#15692248C2semidirect
4. #126×D101C2direct (N × H)
5. C2×C4⋊C4×D101C2semidirect
7. C2×C42×D101C2semidirect
8. C2×#126×C101C2semidirect
12. #15692250C22semidirect
13. #15692249C22semidirect
14. #15692248C22semidirect
15. C2×C4⋊Dic101C22semidirect
16. C2×#13108053C22semidirect
17. C2×#13108053C22semidirect
18. C4⋊C4×D101C22semidirect
23. C2×C4×Dic101C22semidirect
24. C42×D101C22semidirect
25. #126×C101C22semidirect
26. C2×C4⋊C4×C101C22semidirect
28. C2×C42×C101C22semidirect
39. C4⋊Dic101C23semidirect
40. #13108053C23semidirect
41. #13108053C23semidirect
45. C4×Dic101C23semidirect
49. C4⋊C4×C101C23semidirect
51. C42×C101C23semidirect
78. C2×D101#126direct (N × H)
85. C2×C101C2×#126semidirect
88. D101C2×#126direct (N × H)
90. C101C22×#126semidirect
95. C2×#126D101direct (N × H)
105. #126C2×D101direct (N × H)
110. C2×C4⋊C4C2×D101semidirect
118. C2×C42C2×D101semidirect
135. C4⋊C4C22×D101semidirect
141. C42C22×D101semidirect
177. C2#126×D101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156922502
C2×#156922492
C2×#156922482
#126×D1012
C2×C4⋊C4×D1012
C2×C22⋊C4×D1012
C2×C42×D1012
C2×#126×C1012
C22×#126101