Finite Group Databasebeta

← C2×#20293370 #28350582

Subgroups — C2×#20293370 #28350582

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#20293370C2116yesmaximal, direct product factor
2C2×#16812096C213yesmaximal, complemented
3C2×#16812095C216yesmaximal, complemented
4C2×#16812078C212yesmaximal, complemented
5C2×#16812077C213yesmaximal, complemented
6C2×#120×C107C211yesmaximal, characteristic, complemented
7C24×D107C2211yescharacteristic, complemented
8C22×Dic214C2213yescomplemented
9C22×C22⋊D107C2216yes—
10#16812096C22124yescomplemented
11#16812095C22148yescomplemented
12#16812078C22116yescomplemented
13#16812077C22124yescomplemented
14C2×C22⋊Dic107C2216yescomplemented
15C2×C22⋊Dic107C2213yescomplemented
16C2×(C2×D107)⋊C4C2216yescomplemented
17C2×C4×Dic107C2213yescomplemented
18C25×C107C2211yescharacteristic, complemented
19C22×D4×C107C2213yes—
20#120×C107C2218yescomplemented
21C2×C22⋊C4×C107C2213yescomplemented
22C23×D107—46no—
23C23×D107C2312yes—
24C23×D107C2314yescomplemented
25C2×Dic214—46no—
26C2×Dic214C23112yescomplemented
27C2×C22⋊D107—412no—
28C2×C22⋊D107C23124yes—
29C22×Dic107C2316yes—
30C22⋊Dic107C23112yescomplemented
31C22⋊Dic107C23124yescomplemented
32(C2×D107)⋊C4C23124yescomplemented
33C4×Dic107C23112yescomplemented
34C24×C107—46no—
35C24×C107C2311yescharacteristic
36C24×C107C2313yes—
37C24×C107C2314yescomplemented
38C2×D4×C107—46no—
39C2×D4×C107C23112yes—
40C22×C4×C107C2313yes—
41C22⋊C4×C107C23112yescomplemented
42C23×C107—26no—
43C23×C107—26no—
44C23×C107—212no—
45C23×C107—412no—
46C23×C107—416no—
47C23×C107C2416yes—
48C23×C107C2412yes—
49C23×C107C2411yescharacteristic
50C22×D107—212no—
51C22×D107—412no—
52C22×D107—416no—
53C22×D107C2414yes—
54D4×C107—412no—
55C2×C4×C107—26no—
56C2×C4×C107C2416yes—
57C22⋊D107—424no—
58C2×Dic107—212no—
59C2×Dic107C24112yes—
60Dic214—412no—
61C22×C107—212no—
62C22×C107—26no—
63C22×C107—26no—
64C22×C107—416no—
65C22×C107—44no—
66C22×C107—412no—
67C22×C107H(5,𝔽2)16yes—
68C22×C107C2511yescharacteristic
69C2×D107—212no—
70C2×D107—416no—
71C2×D107—48no—
72C4×C107—26no—
73Dic107—212no—
74C2×C107—42no—
75C2×C107—46no—
76C2×C107H(3,𝔽4)14yescomplemented
77C2×C107C2×H(5,𝔽2)13yes—
78D107—44no—
79C2×H(3,𝔽4)—1071nomaximal, Sylow
80C107C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
81H(3,𝔽4)—10716no—
82C2×#124—1073no—
83C2×#124—1076no—
84C2×#120—1072no—
85C2×#120—1073no—
86C2×#120D10711yescharacteristic, complemented
87C25—1071no—
88C25C2×D10711yescharacteristic, complemented
89C22×Q8—1073no—
90C22×D4—1076no—
91C22×D4C2×D10713yes—
92#124—10724no—
93#124—10748no—
94#120—10716no—
95#120—10724no—
96#120C2×D10718yescomplemented
97C2×C22⋊C4—1073no—
98C2×C22⋊C4—1076no—
99C2×C22⋊C4—1076no—
100C2×C22⋊C4C2×D10713yescomplemented
101C2×C42—1073no—
102C24—46no—
103C24—1074no—
104C24—1072no—
105C24—4286no—
106C24C22×D10711yescharacteristic
107C24C22×D10713yes—
108C24C22×D10714yescomplemented
109C2×Q8—10712no—
110C2×Q8—4286no—
111C2×D4—46no—
112C2×D4—10724no—
113C2×D4—42812no—
114C2×D4C22×D107112yes—
115C22×C4—1076no—
116C22×C4C22×D10713yes—
117C22⋊C4—10712no—
118C22⋊C4—10724no—
119C22⋊C4—10724no—
120C22⋊C4C22×D107112yescomplemented
121C42—10712no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—1074no—
128C23—42812no—
129C23—42816no—
130C23—21412no—
131C23C23×D10712yes—
132C23C23×D10716yes—
133C23C23×D10711yescharacteristic
134Q8—42812no—
135D4—412no—
136D4—42824no—
137C2×C4—26no—
138C2×C4—10712no—
139C2×C4—21412no—
140C2×C4C23×D10716yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—42816no—
148C22—4288no—
149C22—21412no—
150C22#1681215416yes—
151C22C24×D10711yescharacteristic
152C4—26no—
153C4—21412no—
154C2—46no—
155C2—42no—
156C2—4284no—
157C2#2029337014yesdirect product factor
158C2C2×#1681215413yes—
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×#120×C107C2
socleC23×C107C24
hypercenterC2×#120D107
hypocenterC107C2×H(3,𝔽4)
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #20293370C2direct (N × H)
2. C2×#16812096C2semidirect
3. C2×#16812095C2semidirect
4. C2×#16812078C2semidirect
5. C2×#16812077C2semidirect
6. C2×#120×C107C2semidirect
7. C24×D107C22semidirect
8. C22×Dic214C22semidirect
10. #16812096C22semidirect
11. #16812095C22semidirect
12. #16812078C22semidirect
13. #16812077C22semidirect
14. C2×C22⋊Dic107C22semidirect
15. C2×C22⋊Dic107C22semidirect
16. C2×(C2×D107)⋊C4C22semidirect
17. C2×C4×Dic107C22semidirect
18. C25×C107C22semidirect
20. #120×C107C22semidirect
21. C2×C22⋊C4×C107C22semidirect
24. C23×D107C23semidirect
26. C2×Dic214C23semidirect
30. C22⋊Dic107C23semidirect
31. C22⋊Dic107C23semidirect
32. (C2×D107)⋊C4C23semidirect
33. C4×Dic107C23semidirect
37. C24×C107C23semidirect
41. C22⋊C4×C107C23semidirect
76. C2×C107H(3,𝔽4)semidirect
80. C107C2×H(3,𝔽4)semidirect
86. C2×#120D107semidirect
88. C25C2×D107semidirect
96. #120C2×D107semidirect
100. C2×C22⋊C4C2×D107semidirect
108. C24C22×D107semidirect
120. C22⋊C4C22×D107semidirect
157. C2#20293370direct (N × H)

Maximal subgroups

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

subgroupindex
#202933702
C2×#168120962
C2×#168120952
C2×#168120782
C2×#168120772
C2×#120×C1072
C2×H(3,𝔽4)107