Finite Group Databasebeta

← #20293259

Subgroups — #20293259

Order 6848.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×C22⋊D107C212yesmaximal, complemented
2C2×D4×D107C212yesmaximal, complemented
3#16812105C211yesmaximal, characteristic, complemented
4#16812082C212yesmaximal, complemented
5#16812081C212yesmaximal, complemented
6#16812079C212yesmaximal, complemented
7#16812077C211yesmaximal, characteristic, complemented
8(C22⋊D107)⋊C4C212yesmaximal, complemented
9#120×C107C211yesmaximal, characteristic, complemented
10C23×D107C2211yescharacteristic, complemented
11C23×D107C2212yescomplemented
12C2×D428C2212yescomplemented
13C2×D428C2211yescharacteristic, complemented
14C2×C22⋊D107—28no—
15C2×C22⋊D107C2212yes—
16C2×C22⋊D107C2212yescomplemented
17C2×C22⋊D107C2212yescomplemented
18C2×C22⋊D107C2212yescomplemented
19C2×C22⋊D107C2212yescomplemented
20D4×D107—28no—
21C22×Dic107C2212yescomplemented
22C2×C4×D107C2212yescomplemented
23Dic107⋊C4C2212yescomplemented
24C22⋊Dic107C2211yescharacteristic, complemented
25(C2×D107)⋊C4C2212yescomplemented
26(C2×D107)⋊C4C2212yescomplemented
27C4×Dic107C2211yescharacteristic, complemented
28C24×C107C2211yescharacteristic, complemented
29C2×D4×C107C2212yescomplemented
30C2×D4×C107C2211yescharacteristic, complemented
31C22⋊C4×C107C2212yescomplemented
32C22⋊C4×C107C2211yescharacteristic, complemented
33C23×C107—24no—
34C23×C107—41no—
35C23×C107C2311yescharacteristic
36C23×C107C2311yescharacteristic
37C23×C107C2312yes—
38C22×D107—24no—
39C22×D107—24no—
40C22×D107—24no—
41C22×D107—41no—
42C22×D107—42no—
43C22×D107C2312yes—
44C22×D107C2312yes—
45D4×C107—24no—
46D4×C107—41no—
47C2×C4×C107C2311yescharacteristic
48C2×C4×C107C2312yes—
49C22⋊D107—24no—
50C22⋊D107—24no—
51C22⋊D107—24no—
52C22⋊D107—24no—
53C22⋊D107D418yescomplemented
54C2×Dic107—24no—
55C2×Dic107C2312yes—
56C2×Dic107C2312yes—
57D428—24no—
58D428—41no—
59C4×D107—24no—
60C22×C107—22no—
61C22×C107—22no—
62C22×C107—22no—
63C22×C107—22no—
64C22×C107—21no—
65C22×C107—21no—
66C22×C107—44no—
67C22×C107C2×D414yescomplemented
68C22×C107C2411yescharacteristic
69C2×D107—22no—
70C2×D107—22no—
71C2×D107—22no—
72C2×D107—22no—
73C2×D107—22no—
74C2×D107—44no—
75C2×D107—48no—
76C2×D107C2×D414yescomplemented
77C4×C107—22no—
78C4×C107—21no—
79Dic107—22no—
80Dic107C2×D414yescomplemented
81C2×C107—24no—
82C2×C107—41no—
83C2×C107—41no—
84C2×C107C22×D412yes—
85C2×C107H(5,𝔽2)11yescharacteristic
86D107—24no—
87D107—42no—
88C107D4211yescharacteristic, complemented, Sylow
89D42—1071nomaximal, Sylow
90C22×D4—1072no—
91C22×D4—1072no—
92#127—1071no—
93#121—1072no—
94#121—1072no—
95#120—1071no—
96#120—1072no—
97#120D10711yescharacteristic, complemented
98C4×D4—1072no—
99C24—1071no—
100C24—1072no—
101C24C2×D10711yescharacteristic, complemented
102C2×D4—1072no—
103C2×D4—1072no—
104C2×D4—1072no—
105C2×D4—1072no—
106C2×D4—1072no—
107C2×D4—1071no—
108C2×D4—1072no—
109C2×D4—2148no—
110C2×D4—2148no—
111C2×D4C2×D10712yescomplemented
112C2×D4C2×D10711yescharacteristic, complemented
113C22×C4—1072no—
114C22×C4—1072no—
115C4⋊C4—1072no—
116C22⋊C4—1071no—
117C22⋊C4—1072no—
118C22⋊C4—1072no—
119C22⋊C4C2×D10712yescomplemented
120C22⋊C4C2×D10711yescharacteristic, complemented
121C42—1071no—
122C23—24no—
123C23—41no—
124C23—1072no—
125C23—1072no—
126C23—4282no—
127C23—4281no—
128C23—2144no—
129C23—2144no—
130C23—2144no—
131C23C22×D10711yescharacteristic
132C23C22×D10711yescharacteristic
133C23C22×D10712yes—
134D4—24no—
135D4—41no—
136D4—1078no—
137D4—4281no—
138D4—2144no—
139D4—2144no—
140D4—2144no—
141D4—2144no—
142D4—2144no—
143C2×C4—1072no—
144C2×C4—1072no—
145C2×C4—2144no—
146C2×C4—2144no—
147C2×C4C22×D10711yescharacteristic
148C2×C4C22×D10712yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—1074no—
157C22—4288no—
158C22—4284no—
159C22—2142no—
160C22—2142no—
161C22—2142no—
162C22—2142no—
163C22—2142no—
164C22D4×D10714yescomplemented
165C22C23×D10711yescharacteristic
166C4—22no—
167C4—21no—
168C4—1074no—
169C4—2142no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—4282no—
174C2—2144no—
175C2C2×D4×D10712yes—
176C2#1681215411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#120×C107C2
hypercenter#120D107
hypocenterC107D42
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D107C2semidirect
2. C2×D4×D107C2semidirect
3. #16812105C2semidirect
4. #16812082C2semidirect
5. #16812081C2semidirect
6. #16812079C2semidirect
7. #16812077C2semidirect
8. (C22⋊D107)⋊C4C2semidirect
9. #120×C107C2semidirect
10. C23×D107C22semidirect
11. C23×D107C22semidirect
12. C2×D428C22semidirect
13. C2×D428C22semidirect
16. C2×C22⋊D107C22semidirect
17. C2×C22⋊D107C22semidirect
18. C2×C22⋊D107C22semidirect
19. C2×C22⋊D107C22semidirect
21. C22×Dic107C22semidirect
22. C2×C4×D107C22semidirect
23. Dic107⋊C4C22semidirect
24. C22⋊Dic107C22semidirect
25. (C2×D107)⋊C4C22semidirect
26. (C2×D107)⋊C4C22semidirect
27. C4×Dic107C22semidirect
28. C24×C107C22semidirect
29. C2×D4×C107C22semidirect
30. C2×D4×C107C22semidirect
31. C22⋊C4×C107C22semidirect
32. C22⋊C4×C107C22semidirect
53. C22⋊D107D4semidirect
67. C22×C107C2×D4semidirect
76. C2×D107C2×D4semidirect
80. Dic107C2×D4semidirect
88. C107D42semidirect
97. #120D107semidirect
101. C24C2×D107semidirect
111. C2×D4C2×D107semidirect
112. C2×D4C2×D107semidirect
119. C22⋊C4C2×D107semidirect
120. C22⋊C4C2×D107semidirect
164. C22D4×D107semidirect