Finite Group Databasebeta

← C2×#20295767 #28377150

Subgroups — C2×#20295767 #28377150

Order 13952.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#20295767C2116yesmaximal, direct product factor
2C2×#16812696C211yesmaximal, characteristic, complemented
3C2×#16812693C211yesmaximal, characteristic, complemented
4C2×#16812684C213yesmaximal, complemented
5C2×#16812674C213yesmaximal, complemented
6C2×#16812672C213yesmaximal, complemented
7C2×#16812669C213yesmaximal, complemented
8C2×#126×C109C211yesmaximal, characteristic, complemented
9C24×D109C2211yescharacteristic
10C22×D436C2213yes—
11C22×D436C2213yes—
12C22×D436C2213yescomplemented
13C22×C22⋊D109C2213yes—
14C22×C4×D109C2213yescomplemented
15#16812696C2218yescomplemented
16#16812693C2218yescomplemented
17#16812684C22124yescomplemented
18#16812674C22124yescomplemented
19#16812672C22124yescomplemented
20#16812669C22124yescomplemented
21C2×#13182008C2213yescomplemented
22C2×#13182006C2213yes—
23C2×#13182006C2213yes—
24C2×#13182006C2213yescomplemented
25#126×C109C2218yescomplemented
26C2×C4⋊C4×C109C2213yescomplemented
27C2×C22⋊C4×C109C2213yes—
28C2×C42×C109C2211yescharacteristic, complemented
29C23×D109—46no—
30C23×D109C2311yescharacteristic
31C23×D109C2311yescharacteristic
32C23×D109C2313yes—
33C23×D109C2314yes—
34C2×D436—46no—
35C2×D436—46no—
36C2×D436—46no—
37C2×D436C23112yes—
38C2×D436C23112yes—
39C2×D436C23112yescomplemented
40C2×C22⋊D109—46no—
41C2×C22⋊D109C23112yes—
42C22×Dic109C2313yes—
43C2×C4×D109—46no—
44C2×C4×D109C23112yescomplemented
45#13182008C23112yescomplemented
46#13182006C23112yes—
47#13182006C23112yes—
48#13182006C23112yescomplemented
49C24×C109C2311yescharacteristic
50C22×C4×C109C2313yes—
51C22×C4×C109C2313yes—
52C4⋊C4×C109C23112yescomplemented
53C22⋊C4×C109C23112yes—
54C42×C109C2314yescomplemented
55C23×C109—26no—
56C23×C109C2412yes—
57C23×C109C2411yescharacteristic
58C22×D109—26no—
59C22×D109—26no—
60C22×D109—26no—
61C22×D109—26no—
62C22×D109—26no—
63C22×D109—412no—
64C22×D109—416no—
65C22×D109C2412yes—
66C22×D109C2412yes—
67C22×D109C2416yes—
68C2×C4×C109—26no—
69C2×C4×C109—26no—
70C2×C4×C109C2416yes—
71C2×C4×C109C2416yes—
72C22⋊D109—412no—
73C2×Dic109—26no—
74C2×Dic109C2416yes—
75D436—412no—
76D436—412no—
77D436—412no—
78C4×D109—412no—
79C22×C109—26no—
80C22×C109—44no—
81C22×C109H(5,𝔽2)16yes—
82C22×C109C2511yescharacteristic
83C2×D109—26no—
84C2×D109—26no—
85C2×D109—26no—
86C2×D109—26no—
87C2×D109—26no—
88C2×D109—416no—
89C2×D109—412no—
90C2×D109—44no—
91C2×D109—44no—
92C4×C109—26no—
93C4×C109—26no—
94Dic109—26no—
95C2×C109—42no—
96C2×C109#56014yescomplemented
97C2×C109C2×H(5,𝔽2)13yes—
98D109—46no—
99D109—42no—
100D109—42no—
101C2×#560—1091nomaximal, Sylow
102C109C2×#56011yescharacteristic, complemented, Sylow
103#560—10916no—
104C2×#127—1091no—
105C2×#126—1091no—
106C2×#126D10911yescharacteristic, complemented
107C2×#123—1093no—
108C2×#121—1093no—
109C2×#121—1093no—
110C2×#120—1093no—
111C25—1091no—
112C22×D4—1093no—
113C22×D4—1093no—
114C22×D4—1093no—
115C22×D4—1093no—
116C23×C4—1093no—
117#127—1098no—
118#126—1098no—
119#126C2×D10918yescomplemented
120#123—10924no—
121#121—10924no—
122#121—10924no—
123#120—10924no—
124C2×C4⋊C4—1093no—
125C2×C4⋊C4C2×D10913yescomplemented
126C2×C22⋊C4—1093no—
127C2×C22⋊C4—1093no—
128C2×C22⋊C4—1093no—
129C2×C22⋊C4C2×D10913yes—
130C2×C42C2×D10911yescharacteristic, complemented
131C24—1094no—
132C24—1091no—
133C24—1091no—
134C24—1093no—
135C24—4366no—
136C24C22×D10911yescharacteristic
137C2×D4—10912no—
138C2×D4—10912no—
139C2×D4—10912no—
140C2×D4—10912no—
141C2×D4—4366no—
142C2×D4—4366no—
143C2×D4—4366no—
144C2×D4—4366no—
145C22×C4—1093no—
146C22×C4—10912no—
147C22×C4—4366no—
148C22×C4C22×D10913yes—
149C22×C4C22×D10913yes—
150C4⋊C4—10912no—
151C4⋊C4C22×D109112yescomplemented
152C22⋊C4—10912no—
153C22⋊C4—10912no—
154C22⋊C4—10912no—
155C22⋊C4C22×D109112yes—
156C42C22×D10914yescomplemented
157C23—26no—
158C23—1092no—
159C23—1092no—
160C23—1096no—
161C23—43612no—
162C23—43616no—
163C23—2186no—
164C23—2186no—
165C23—2186no—
166C23—2186no—
167C23—2186no—
168C23C23×D10912yes—
169C23C23×D10911yescharacteristic
170D4—43612no—
171D4—43612no—
172D4—43612no—
173D4—43612no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—1096no—
177C2×C4—43612no—
178C2×C4—2186no—
179C2×C4C23×D10916yes—
180C2×C4C23×D10916yes—
181C22—26no—
182C22—44no—
183C22—43616no—
184C22—4364no—
185C22—4364no—
186C22—43612no—
187C22—2186no—
188C22—2186no—
189C22—2186no—
190C22—2186no—
191C22—2186no—
192C22#1681274516yes—
193C22C24×D10911yescharacteristic
194C4—26no—
195C4—26no—
196C4—2186no—
197C2—42no—
198C2—4362no—
199C2—4362no—
200C2—4366no—
201C2#2029576714yesdirect product factor
202C2C2×#1681274513yes—
subgroupG/H
centerC23C23×D109
commutator, cosocleC22×C109C25
FrattiniC22C24×D109
FittingC2×#126×C109C2
socleC23×C109C24
hypercenterC2×#126D109
hypocenterC109C2×#560
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20295767C2direct (N × H)
2. C2×#16812696C2semidirect
3. C2×#16812693C2semidirect
4. C2×#16812684C2semidirect
5. C2×#16812674C2semidirect
6. C2×#16812672C2semidirect
7. C2×#16812669C2semidirect
8. C2×#126×C109C2semidirect
12. C22×D436C22semidirect
14. C22×C4×D109C22semidirect
15. #16812696C22semidirect
16. #16812693C22semidirect
17. #16812684C22semidirect
18. #16812674C22semidirect
19. #16812672C22semidirect
20. #16812669C22semidirect
21. C2×#13182008C22semidirect
24. C2×#13182006C22semidirect
25. #126×C109C22semidirect
26. C2×C4⋊C4×C109C22semidirect
28. C2×C42×C109C22semidirect
39. C2×D436C23semidirect
44. C2×C4×D109C23semidirect
45. #13182008C23semidirect
48. #13182006C23semidirect
52. C4⋊C4×C109C23semidirect
54. C42×C109C23semidirect
96. C2×C109#560semidirect
102. C109C2×#560semidirect
106. C2×#126D109semidirect
119. #126C2×D109semidirect
125. C2×C4⋊C4C2×D109semidirect
130. C2×C42C2×D109semidirect
151. C4⋊C4C22×D109semidirect
156. C42C22×D109semidirect
201. C2#20295767direct (N × H)

Maximal subgroups

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

subgroupindex
#202957672
C2×#168126962
C2×#168126932
C2×#168126842
C2×#168126742
C2×#168126722
C2×#168126692
C2×#126×C1092
C2×#560109