Finite Group Databasebeta

← C2×#20293369 #28350581

Subgroups — C2×#20293369 #28350581

Order 13696.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 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#20293369C2116yesmaximal, direct product factor
2C2×#16812098C212yesmaximal, complemented
3C2×#16812097C212yesmaximal, complemented
4C2×#16812095C214yesmaximal, complemented
5C2×#16812079C214yesmaximal, complemented
6C2×#16812077C212yesmaximal, complemented
7C2×#124×C107C211yesmaximal, characteristic, complemented
8C24×D107C2212yescomplemented
9C22×Dic214C2212yescomplemented
10C22×D428C2212yes—
11C22×D428C2212yes—
12C22×C22⋊D107C2214yes—
13#16812098C22116yescomplemented
14#16812097C22116yescomplemented
15#16812095C22132yescomplemented
16#16812079C22132yescomplemented
17#16812077C22116yescomplemented
18C2×C22⋊Dic107C2212yescomplemented
19C2×(C2×D107)⋊C4C2214yescomplemented
20C2×(C2×D107)⋊C4C2214yescomplemented
21C2×(C2×D107)⋊C4C2214yescomplemented
22C2×C4×Dic107C2212yescomplemented
23C22×Q8×C107C2211yescharacteristic, complemented
24C22×D4×C107C2211yescharacteristic
25#124×C107C2218yescomplemented
26C2×C22⋊C4×C107C2214yescomplemented
27C2×C42×C107C2211yescharacteristic, complemented
28C23×D107—48no—
29C23×D107—44no—
30C23×D107C2314yes—
31C23×D107C2318yescomplemented
32C2×Dic214—44no—
33C2×Dic214C2318yescomplemented
34C2×D428—44no—
35C2×D428—44no—
36C2×D428C2318yes—
37C2×D428C2318yes—
38C2×C22⋊D107—48no—
39C2×C22⋊D107C23116yes—
40C22×Dic107C2314yes—
41C22⋊Dic107C2318yescomplemented
42(C2×D107)⋊C4C23116yescomplemented
43(C2×D107)⋊C4C23116yescomplemented
44(C2×D107)⋊C4C23116yescomplemented
45C4×Dic107C2318yescomplemented
46C24×C107C2312yes—
47C2×Q8×C107—42no—
48C2×Q8×C107C2314yescomplemented
49C2×D4×C107—42no—
50C2×D4×C107C2314yes—
51C22×C4×C107C2312yes—
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2312yes—
54C22⋊C4×C107C23116yescomplemented
55C42×C107C2314yescomplemented
56C23×C107—24no—
57C23×C107—28no—
58C23×C107C2414yes—
59C23×C107C2411yescharacteristic
60C22×D107—28no—
61C22×D107—28no—
62C22×D107—28no—
63C22×D107—48no—
64C22×D107—416no—
65C22×D107—432no—
66C22×D107C2418yes—
67Q8×C107—44no—
68D4×C107—44no—
69C2×C4×C107—22no—
70C2×C4×C107—24no—
71C2×C4×C107—24no—
72C2×C4×C107C2414yes—
73C2×C4×C107C2412yes—
74C2×C4×C107C2414yes—
75C22⋊D107—416no—
76C2×Dic107—28no—
77C2×Dic107C2418yes—
78D428—48no—
79D428—48no—
80Dic214—48no—
81C22×C107—28no—
82C22×C107—24no—
83C22×C107—48no—
84C22×C107H(5,𝔽2)12yes—
85C22×C107H(5,𝔽2)14yes—
86C22×C107C2511yescharacteristic
87C2×D107—28no—
88C2×D107—28no—
89C2×D107—28no—
90C2×D107—432no—
91C2×D107—416no—
92C4×C107—24no—
93C4×C107—24no—
94C4×C107—22no—
95Dic107—28no—
96C2×C107—44no—
97C2×C107H(3,𝔽4)14yescomplemented
98C2×C107C2×H(5,𝔽2)11yescharacteristic
99C2×C107C2×H(5,𝔽2)12yes—
100D107—48no—
101C2×H(3,𝔽4)—1071nomaximal, Sylow
102C107C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
103H(3,𝔽4)—10716no—
104C2×#124—1072no—
105C2×#124—1074no—
106C2×#124—1072no—
107C2×#124D10711yescharacteristic, complemented
108C2×#120—1072no—
109C2×#120—1074no—
110C25—1072no—
111C22×Q8—1072no—
112C22×Q8C2×D10711yescharacteristic, complemented
113C22×D4—1074no—
114C22×D4—1072no—
115C22×D4—1072no—
116C22×D4C2×D10711yescharacteristic
117#124—10716no—
118#124—10732no—
119#124—10716no—
120#124C2×D10718yescomplemented
121#120—10716no—
122#120—10732no—
123C2×C22⋊C4—1072no—
124C2×C22⋊C4—1074no—
125C2×C22⋊C4—1074no—
126C2×C22⋊C4—1074no—
127C2×C22⋊C4C2×D10714yescomplemented
128C2×C42—1072no—
129C2×C42C2×D10711yescharacteristic, complemented
130C24—1078no—
131C24—1074no—
132C24—4284no—
133C24—4288no—
134C24C22×D10712yes—
135C2×Q8—42no—
136C2×Q8—1078no—
137C2×Q8—4284no—
138C2×Q8C22×D10714yescomplemented
139C2×D4—42no—
140C2×D4—10716no—
141C2×D4—1078no—
142C2×D4—1078no—
143C2×D4—4284no—
144C2×D4—4284no—
145C2×D4—4288no—
146C2×D4C22×D10714yes—
147C22×C4—1074no—
148C22×C4C22×D10712yes—
149C22×C4C22×D10711yescharacteristic
150C22×C4C22×D10712yes—
151C22⋊C4—1078no—
152C22⋊C4—10716no—
153C22⋊C4—10716no—
154C22⋊C4—10716no—
155C22⋊C4C22×D107116yescomplemented
156C42—1078no—
157C42C22×D10714yescomplemented
158C23—24no—
159C23—28no—
160C23—1078no—
161C23—4288no—
162C23—42816no—
163C23—42832no—
164C23—2148no—
165C23—2148no—
166C23—2148no—
167C23C23×D10714yes—
168C23C23×D10711yescharacteristic
169Q8—44no—
170Q8—4288no—
171D4—44no—
172D4—4288no—
173D4—4288no—
174D4—42816no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—1078no—
179C2×C4—2148no—
180C2×C4C23×D10714yes—
181C2×C4C23×D10712yes—
182C2×C4C23×D10714yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—42832no—
187C22—42816no—
188C22—2148no—
189C22—2148no—
190C22—2148no—
191C22#1681215412yes—
192C22#1681215514yes—
193C22C24×D10711yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—2148no—
198C2—44no—
199C2—4288no—
200C2#2029336914yesdirect product factor
201C2C2×#1681215411yescharacteristic
202C2C2×#1681215512yes—
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×#124×C107C2
socleC23×C107C24
hypercenterC2×#124D107
hypocenterC107C2×H(3,𝔽4)
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20293369C2direct (N × H)
2. C2×#16812098C2semidirect
3. C2×#16812097C2semidirect
4. C2×#16812095C2semidirect
5. C2×#16812079C2semidirect
6. C2×#16812077C2semidirect
7. C2×#124×C107C2semidirect
8. C24×D107C22semidirect
9. C22×Dic214C22semidirect
13. #16812098C22semidirect
14. #16812097C22semidirect
15. #16812095C22semidirect
16. #16812079C22semidirect
17. #16812077C22semidirect
18. C2×C22⋊Dic107C22semidirect
19. C2×(C2×D107)⋊C4C22semidirect
20. C2×(C2×D107)⋊C4C22semidirect
21. C2×(C2×D107)⋊C4C22semidirect
22. C2×C4×Dic107C22semidirect
23. C22×Q8×C107C22semidirect
25. #124×C107C22semidirect
26. C2×C22⋊C4×C107C22semidirect
27. C2×C42×C107C22semidirect
31. C23×D107C23semidirect
33. C2×Dic214C23semidirect
41. C22⋊Dic107C23semidirect
42. (C2×D107)⋊C4C23semidirect
43. (C2×D107)⋊C4C23semidirect
44. (C2×D107)⋊C4C23semidirect
45. C4×Dic107C23semidirect
48. C2×Q8×C107C23semidirect
54. C22⋊C4×C107C23semidirect
55. C42×C107C23semidirect
97. C2×C107H(3,𝔽4)semidirect
102. C107C2×H(3,𝔽4)semidirect
107. C2×#124D107semidirect
112. C22×Q8C2×D107semidirect
120. #124C2×D107semidirect
127. C2×C22⋊C4C2×D107semidirect
129. C2×C42C2×D107semidirect
138. C2×Q8C22×D107semidirect
155. C22⋊C4C22×D107semidirect
157. C42C22×D107semidirect
200. C2#20293369direct (N × H)

Maximal subgroups

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

subgroupindex
#202933692
C2×#168120982
C2×#168120972
C2×#168120952
C2×#168120792
C2×#168120772
C2×#124×C1072
C2×H(3,𝔽4)107