Finite Group Databasebeta

← C2×#20293368 #28350579

Subgroups — C2×#20293368 #28350579

Order 13696.

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#20293368C2116yesmaximal, direct product factor
2C2×#16812106C211yesmaximal, characteristic, complemented
3C2×#16812103C211yesmaximal, characteristic, complemented
4C2×#16812094C213yesmaximal, complemented
5C2×#16812084C213yesmaximal, complemented
6C2×#16812082C213yesmaximal, complemented
7C2×#16812079C213yesmaximal, complemented
8C2×#126×C107C211yesmaximal, characteristic, complemented
9C24×D107C2211yescharacteristic
10C22×D428C2213yes—
11C22×D428C2213yes—
12C22×D428C2213yescomplemented
13C22×C22⋊D107C2213yes—
14C22×C4×D107C2213yescomplemented
15#16812106C2218yescomplemented
16#16812103C2218yescomplemented
17#16812094C22124yescomplemented
18#16812084C22124yescomplemented
19#16812082C22124yescomplemented
20#16812079C22124yescomplemented
21C2×Dic107⋊C4C2213yescomplemented
22C2×(C2×D107)⋊C4C2213yes—
23C2×(C2×D107)⋊C4C2213yes—
24C2×(C2×D107)⋊C4C2213yescomplemented
25#126×C107C2218yescomplemented
26C2×C4⋊C4×C107C2213yescomplemented
27C2×C22⋊C4×C107C2213yes—
28C2×C42×C107C2211yescharacteristic, complemented
29C23×D107—46no—
30C23×D107C2311yescharacteristic
31C23×D107C2311yescharacteristic
32C23×D107C2313yes—
33C23×D107C2314yes—
34C2×D428—46no—
35C2×D428—46no—
36C2×D428—46no—
37C2×D428C23112yes—
38C2×D428C23112yes—
39C2×D428C23112yescomplemented
40C2×C22⋊D107—46no—
41C2×C22⋊D107C23112yes—
42C22×Dic107C2313yes—
43C2×C4×D107—46no—
44C2×C4×D107C23112yescomplemented
45Dic107⋊C4C23112yescomplemented
46(C2×D107)⋊C4C23112yes—
47(C2×D107)⋊C4C23112yes—
48(C2×D107)⋊C4C23112yescomplemented
49C24×C107C2311yescharacteristic
50C22×C4×C107C2313yes—
51C22×C4×C107C2313yes—
52C4⋊C4×C107C23112yescomplemented
53C22⋊C4×C107C23112yes—
54C42×C107C2314yescomplemented
55C23×C107—26no—
56C23×C107C2412yes—
57C23×C107C2411yescharacteristic
58C22×D107—26no—
59C22×D107—26no—
60C22×D107—26no—
61C22×D107—26no—
62C22×D107—26no—
63C22×D107—412no—
64C22×D107—416no—
65C22×D107C2412yes—
66C22×D107C2412yes—
67C22×D107C2416yes—
68C2×C4×C107—26no—
69C2×C4×C107—26no—
70C2×C4×C107C2416yes—
71C2×C4×C107C2416yes—
72C22⋊D107—412no—
73C2×Dic107—26no—
74C2×Dic107C2416yes—
75D428—412no—
76D428—412no—
77D428—412no—
78C4×D107—412no—
79C22×C107—26no—
80C22×C107—44no—
81C22×C107H(5,𝔽2)16yes—
82C22×C107C2511yescharacteristic
83C2×D107—26no—
84C2×D107—26no—
85C2×D107—26no—
86C2×D107—26no—
87C2×D107—26no—
88C2×D107—416no—
89C2×D107—412no—
90C2×D107—44no—
91C2×D107—44no—
92C4×C107—26no—
93C4×C107—26no—
94Dic107—26no—
95C2×C107—42no—
96C2×C107#56014yescomplemented
97C2×C107C2×H(5,𝔽2)13yes—
98D107—46no—
99D107—42no—
100D107—42no—
101C2×#560—1071nomaximal, Sylow
102C107C2×#56011yescharacteristic, complemented, Sylow
103#560—10716no—
104C2×#127—1071no—
105C2×#126—1071no—
106C2×#126D10711yescharacteristic, complemented
107C2×#123—1073no—
108C2×#121—1073no—
109C2×#121—1073no—
110C2×#120—1073no—
111C25—1071no—
112C22×D4—1073no—
113C22×D4—1073no—
114C22×D4—1073no—
115C22×D4—1073no—
116C23×C4—1073no—
117#127—1078no—
118#126—1078no—
119#126C2×D10718yescomplemented
120#123—10724no—
121#121—10724no—
122#121—10724no—
123#120—10724no—
124C2×C4⋊C4—1073no—
125C2×C4⋊C4C2×D10713yescomplemented
126C2×C22⋊C4—1073no—
127C2×C22⋊C4—1073no—
128C2×C22⋊C4—1073no—
129C2×C22⋊C4C2×D10713yes—
130C2×C42C2×D10711yescharacteristic, complemented
131C24—1074no—
132C24—1071no—
133C24—1071no—
134C24—1073no—
135C24—4286no—
136C24C22×D10711yescharacteristic
137C2×D4—10712no—
138C2×D4—10712no—
139C2×D4—10712no—
140C2×D4—10712no—
141C2×D4—4286no—
142C2×D4—4286no—
143C2×D4—4286no—
144C2×D4—4286no—
145C22×C4—1073no—
146C22×C4—10712no—
147C22×C4—4286no—
148C22×C4C22×D10713yes—
149C22×C4C22×D10713yes—
150C4⋊C4—10712no—
151C4⋊C4C22×D107112yescomplemented
152C22⋊C4—10712no—
153C22⋊C4—10712no—
154C22⋊C4—10712no—
155C22⋊C4C22×D107112yes—
156C42C22×D10714yescomplemented
157C23—26no—
158C23—1072no—
159C23—1072no—
160C23—1076no—
161C23—42812no—
162C23—42816no—
163C23—2146no—
164C23—2146no—
165C23—2146no—
166C23—2146no—
167C23—2146no—
168C23C23×D10712yes—
169C23C23×D10711yescharacteristic
170D4—42812no—
171D4—42812no—
172D4—42812no—
173D4—42812no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—1076no—
177C2×C4—42812no—
178C2×C4—2146no—
179C2×C4C23×D10716yes—
180C2×C4C23×D10716yes—
181C22—26no—
182C22—44no—
183C22—42816no—
184C22—4284no—
185C22—4284no—
186C22—42812no—
187C22—2146no—
188C22—2146no—
189C22—2146no—
190C22—2146no—
191C22—2146no—
192C22#1681215516yes—
193C22C24×D10711yescharacteristic
194C4—26no—
195C4—26no—
196C4—2146no—
197C2—42no—
198C2—4282no—
199C2—4282no—
200C2—4286no—
201C2#2029336814yesdirect product factor
202C2C2×#1681215513yes—
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×#126×C107C2
socleC23×C107C24
hypercenterC2×#126D107
hypocenterC107C2×#560
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20293368C2direct (N × H)
2. C2×#16812106C2semidirect
3. C2×#16812103C2semidirect
4. C2×#16812094C2semidirect
5. C2×#16812084C2semidirect
6. C2×#16812082C2semidirect
7. C2×#16812079C2semidirect
8. C2×#126×C107C2semidirect
12. C22×D428C22semidirect
14. C22×C4×D107C22semidirect
15. #16812106C22semidirect
16. #16812103C22semidirect
17. #16812094C22semidirect
18. #16812084C22semidirect
19. #16812082C22semidirect
20. #16812079C22semidirect
21. C2×Dic107⋊C4C22semidirect
24. C2×(C2×D107)⋊C4C22semidirect
25. #126×C107C22semidirect
26. C2×C4⋊C4×C107C22semidirect
28. C2×C42×C107C22semidirect
39. C2×D428C23semidirect
44. C2×C4×D107C23semidirect
45. Dic107⋊C4C23semidirect
48. (C2×D107)⋊C4C23semidirect
52. C4⋊C4×C107C23semidirect
54. C42×C107C23semidirect
96. C2×C107#560semidirect
102. C107C2×#560semidirect
106. C2×#126D107semidirect
119. #126C2×D107semidirect
125. C2×C4⋊C4C2×D107semidirect
130. C2×C42C2×D107semidirect
151. C4⋊C4C22×D107semidirect
156. C42C22×D107semidirect
201. C2#20293368direct (N × H)

Maximal subgroups

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

subgroupindex
#202933682
C2×#168121062
C2×#168121032
C2×#168120942
C2×#168120842
C2×#168120822
C2×#168120792
C2×#126×C1072
C2×#560107