Finite Group Databasebeta

← C2×#126×D109 #28376286

Subgroups — C2×#126×D109 #28376286

Order 13952.

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×#16812694C213yesmaximal, complemented
2C2×#16812693C211yesmaximal, characteristic, complemented
3C2×#16812692C213yesmaximal, complemented
4#126×D109C2116yesmaximal, direct product factor
5C2×C4⋊C4×D109C213yesmaximal, complemented
6C2×C22⋊C4×D109C213yesmaximal
7C2×C42×D109C211yesmaximal, characteristic, complemented
8C2×#126×C109C211yesmaximal, characteristic, complemented
9C24×D109C2211yescharacteristic
10C22×C4×D109C2213yes—
11C22×C4×D109C2213yes—
12#16812694C22124yescomplemented
13#16812693C2218yescomplemented
14#16812692C22124yescomplemented
15C2×C4⋊Dic109C2213yescomplemented
16C2×#13182008C2213yescomplemented
17C2×#13182008C2213yescomplemented
18C4⋊C4×D109C22124yescomplemented
19C2×C22⋊Dic109C2213yes—
20C2×#13182006C2213yes—
21C2×#13182006C2213yes—
22C22⋊C4×D109C22124yes—
23C2×C4×Dic109C2213yescomplemented
24C42×D109C2218yescomplemented
25#126×C109C2218yescomplemented
26C2×C4⋊C4×C109C2213yescomplemented
27C2×C22⋊C4×C109C2213yes—
28C2×C42×C109C2211yescharacteristic, complemented
29C23×D109—212no—
30C23×D109C2314yes—
31C23×D109C2311yescharacteristic
32C23×D109C2311yescharacteristic
33C22×Dic109C2313yes—
34C22×Dic109C2313yes—
35C2×C4×D109—212no—
36C2×C4×D109—212no—
37C2×C4×D109C23112yes—
38C2×C4×D109C23112yes—
39C4⋊Dic109C23112yescomplemented
40#13182008C23112yescomplemented
41#13182008C23112yescomplemented
42C22⋊Dic109C23112yes—
43#13182006C23112yes—
44#13182006C23112yes—
45C4×Dic109C23112yescomplemented
46C24×C109C2311yescharacteristic
47C22×C4×C109C2313yes—
48C22×C4×C109C2313yes—
49C4⋊C4×C109C23112yescomplemented
50C22⋊C4×C109C23112yes—
51C42×C109C2314yescomplemented
52C23×C109—26no—
53C23×C109C2412yes—
54C23×C109C2411yescharacteristic
55C22×D109—26no—
56C22×D109—224no—
57C22×D109—416no—
58C22×D109#41112yes—
59C22×D109C2412yes—
60C22×D109C2412yes—
61C2×C4×C109—26no—
62C2×C4×C109—26no—
63C2×C4×C109C2416yes—
64C2×C4×C109C2416yes—
65C2×Dic109—26no—
66C2×Dic109—26no—
67C2×Dic109C2416yes—
68C2×Dic109C2416yes—
69C4×D109—224no—
70C4×D109—224no—
71C22×C109—26no—
72C22×C109—44no—
73C22×C109C2×#4116yes—
74C22×C109C2511yescharacteristic
75C2×D109—26no—
76C2×D109—44no—
77C2×D109—416no—
78C2×D109#126116yesdirect product factor
79C2×D109C2×#41112yes—
80C4×C109—26no—
81C4×C109—26no—
82Dic109—26no—
83Dic109—26no—
84C2×C109—42no—
85C2×C109C2×#12614yescomplemented
86C2×C109C22×#4113yes—
87D109—42no—
88D109C2×#12618yesdirect product factor
89C22×#126—1091nomaximal, Sylow
90C109C22×#12611yescharacteristic, complemented, Sylow
91C2×#126—10916no—
92C2×#126—1093no—
93C2×#126—1093no—
94C2×#126—1091no—
95C2×#126D10911yescharacteristic, direct product factor
96C22×C4⋊C4—1093no—
97C22×C22⋊C4—1093no—
98C22×C42—1091no—
99C25—1091no—
100C23×C4—1093no—
101C23×C4—1093no—
102#126—10924no—
103#126—10924no—
104#126—1098no—
105#126C2×D10918yesdirect product factor
106C2×C4⋊C4—1093no—
107C2×C4⋊C4—1093no—
108C2×C4⋊C4—1093no—
109C2×C4⋊C4—10924no—
110C2×C4⋊C4C2×D10913yescomplemented
111C2×C22⋊C4—1093no—
112C2×C22⋊C4—1093no—
113C2×C22⋊C4—1093no—
114C2×C22⋊C4—10924no—
115C2×C22⋊C4C2×D10913yes—
116C2×C42—1093no—
117C2×C42—1098no—
118C2×C42C2×D10911yescharacteristic, complemented
119C24—1091no—
120C24—1094no—
121C24—1091no—
122C24—21812no—
123C24C22×D10911yescharacteristic
124C22×C4—1093no—
125C22×C4—1093no—
126C22×C4—10912no—
127C22×C4—10912no—
128C22×C4—21812no—
129C22×C4—21812no—
130C22×C4C22×D10913yes—
131C22×C4C22×D10913yes—
132C4⋊C4—10912no—
133C4⋊C4—10912no—
134C4⋊C4—10912no—
135C4⋊C4C22×D109112yescomplemented
136C22⋊C4—10912no—
137C22⋊C4—10912no—
138C22⋊C4—10912no—
139C22⋊C4C22×D109112yes—
140C42—10912no—
141C42C22×D10914yescomplemented
142C23—26no—
143C23—10912no—
144C23—1092no—
145C23—1092no—
146C23—43616no—
147C23—21824no—
148C23—2186no—
149C23C23×D10912yes—
150C23C23×D10911yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—1096no—
154C2×C4—1096no—
155C2×C4—21824no—
156C2×C4—21824no—
157C2×C4—2186no—
158C2×C4—2186no—
159C2×C4C23×D10916yes—
160C2×C4C23×D10916yes—
161C22—26no—
162C22—44no—
163C22—10916no—
164C22—10912no—
165C22—43616no—
166C22—4364no—
167C22—2186no—
168C22#41×D10916yes—
169C22C24×D10911yescharacteristic
170C4—26no—
171C4—26no—
172C4—2186no—
173C4—2186no—
174C2—42no—
175C2—1098no—
176C2—4362no—
177C2#126×D10914yesdirect product factor
178C2C2×#41×D10913yes—
subgroupG/H
centerC23C23×D109
commutator, cosocleC22×C109C25
FrattiniC22C24×D109
FittingC2×#126×C109C2
socleC23×C109C24
prime radicalC24×D109C22
hypercenterC2×#126D109
hypocenterC109C22×#126
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#16812694C2semidirect
2. C2×#16812693C2semidirect
3. C2×#16812692C2semidirect
4. #126×D109C2direct (N × H)
5. C2×C4⋊C4×D109C2semidirect
7. C2×C42×D109C2semidirect
8. C2×#126×C109C2semidirect
12. #16812694C22semidirect
13. #16812693C22semidirect
14. #16812692C22semidirect
15. C2×C4⋊Dic109C22semidirect
16. C2×#13182008C22semidirect
17. C2×#13182008C22semidirect
18. C4⋊C4×D109C22semidirect
23. C2×C4×Dic109C22semidirect
24. C42×D109C22semidirect
25. #126×C109C22semidirect
26. C2×C4⋊C4×C109C22semidirect
28. C2×C42×C109C22semidirect
39. C4⋊Dic109C23semidirect
40. #13182008C23semidirect
41. #13182008C23semidirect
45. C4×Dic109C23semidirect
49. C4⋊C4×C109C23semidirect
51. C42×C109C23semidirect
78. C2×D109#126direct (N × H)
85. C2×C109C2×#126semidirect
88. D109C2×#126direct (N × H)
90. C109C22×#126semidirect
95. C2×#126D109direct (N × H)
105. #126C2×D109direct (N × H)
110. C2×C4⋊C4C2×D109semidirect
118. C2×C42C2×D109semidirect
135. C4⋊C4C22×D109semidirect
141. C42C22×D109semidirect
177. C2#126×D109direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168126942
C2×#168126932
C2×#168126922
#126×D1092
C2×C4⋊C4×D1092
C2×C22⋊C4×D1092
C2×C42×D1092
C2×#126×C1092
C22×#126109