Finite Group Databasebeta

← #20295658

Subgroups — #20295658

Order 6976.

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⋊D109C212yesmaximal, complemented
2C2×D4×D109C212yesmaximal, complemented
3#16812695C211yesmaximal, characteristic, complemented
4#16812672C212yesmaximal, complemented
5#16812671C212yesmaximal, complemented
6#16812669C212yesmaximal, complemented
7#16812667C211yesmaximal, characteristic, complemented
8#16812659C212yesmaximal, complemented
9#120×C109C211yesmaximal, characteristic, complemented
10C23×D109C2211yescharacteristic, complemented
11C23×D109C2212yescomplemented
12C2×D436C2212yescomplemented
13C2×D436C2211yescharacteristic, complemented
14C2×C22⋊D109—28no—
15C2×C22⋊D109C2212yes—
16C2×C22⋊D109C2212yescomplemented
17C2×C22⋊D109C2212yescomplemented
18C2×C22⋊D109C2212yescomplemented
19C2×C22⋊D109C2212yescomplemented
20D4×D109—28no—
21C22×Dic109C2212yescomplemented
22C2×C4×D109C2212yescomplemented
23#13182008C2212yescomplemented
24C22⋊Dic109C2211yescharacteristic, complemented
25#13182006C2212yescomplemented
26#13182006C2212yescomplemented
27C4×Dic109C2211yescharacteristic, complemented
28C24×C109C2211yescharacteristic, complemented
29C2×D4×C109C2212yescomplemented
30C2×D4×C109C2211yescharacteristic, complemented
31C22⋊C4×C109C2212yescomplemented
32C22⋊C4×C109C2211yescharacteristic, complemented
33C23×C109—24no—
34C23×C109—41no—
35C23×C109C2311yescharacteristic
36C23×C109C2311yescharacteristic
37C23×C109C2312yes—
38C22×D109—24no—
39C22×D109—24no—
40C22×D109—24no—
41C22×D109—41no—
42C22×D109—42no—
43C22×D109C2312yes—
44C22×D109C2312yes—
45D4×C109—24no—
46D4×C109—41no—
47C2×C4×C109C2311yescharacteristic
48C2×C4×C109C2312yes—
49C22⋊D109—24no—
50C22⋊D109—24no—
51C22⋊D109—24no—
52C22⋊D109—24no—
53C22⋊D109D418yescomplemented
54C2×Dic109—24no—
55C2×Dic109C2312yes—
56C2×Dic109C2312yes—
57D436—24no—
58D436—41no—
59C4×D109—24no—
60C22×C109—22no—
61C22×C109—22no—
62C22×C109—22no—
63C22×C109—22no—
64C22×C109—21no—
65C22×C109—21no—
66C22×C109—44no—
67C22×C109C2×D414yescomplemented
68C22×C109C2411yescharacteristic
69C2×D109—22no—
70C2×D109—22no—
71C2×D109—22no—
72C2×D109—22no—
73C2×D109—22no—
74C2×D109—44no—
75C2×D109—48no—
76C2×D109C2×D414yescomplemented
77C4×C109—22no—
78C4×C109—21no—
79Dic109—22no—
80Dic109C2×D414yescomplemented
81C2×C109—24no—
82C2×C109—41no—
83C2×C109—41no—
84C2×C109C22×D412yes—
85C2×C109H(5,𝔽2)11yescharacteristic
86D109—24no—
87D109—42no—
88C109D4211yescharacteristic, complemented, Sylow
89D42—1091nomaximal, Sylow
90C22×D4—1092no—
91C22×D4—1092no—
92#127—1091no—
93#121—1092no—
94#121—1092no—
95#120—1091no—
96#120—1092no—
97#120D10911yescharacteristic, complemented
98C4×D4—1092no—
99C24—1091no—
100C24—1092no—
101C24C2×D10911yescharacteristic, complemented
102C2×D4—1092no—
103C2×D4—1092no—
104C2×D4—1092no—
105C2×D4—1092no—
106C2×D4—1092no—
107C2×D4—1091no—
108C2×D4—1092no—
109C2×D4—2188no—
110C2×D4—2188no—
111C2×D4C2×D10912yescomplemented
112C2×D4C2×D10911yescharacteristic, complemented
113C22×C4—1092no—
114C22×C4—1092no—
115C4⋊C4—1092no—
116C22⋊C4—1091no—
117C22⋊C4—1092no—
118C22⋊C4—1092no—
119C22⋊C4C2×D10912yescomplemented
120C22⋊C4C2×D10911yescharacteristic, complemented
121C42—1091no—
122C23—24no—
123C23—41no—
124C23—1092no—
125C23—1092no—
126C23—4362no—
127C23—4361no—
128C23—2184no—
129C23—2184no—
130C23—2184no—
131C23C22×D10911yescharacteristic
132C23C22×D10911yescharacteristic
133C23C22×D10912yes—
134D4—24no—
135D4—41no—
136D4—1098no—
137D4—4361no—
138D4—2184no—
139D4—2184no—
140D4—2184no—
141D4—2184no—
142D4—2184no—
143C2×C4—1092no—
144C2×C4—1092no—
145C2×C4—2184no—
146C2×C4—2184no—
147C2×C4C22×D10911yescharacteristic
148C2×C4C22×D10912yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—1094no—
157C22—4368no—
158C22—4364no—
159C22—2182no—
160C22—2182no—
161C22—2182no—
162C22—2182no—
163C22—2182no—
164C22D4×D10914yescomplemented
165C22C23×D10911yescharacteristic
166C4—22no—
167C4—21no—
168C4—1094no—
169C4—2182no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—4362no—
174C2—2184no—
175C2C2×D4×D10912yes—
176C2#1681274411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D109
commutator, socleC22×C109C24
Fitting#120×C109C2
hypercenter#120D109
hypocenterC109D42
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D109C2semidirect
2. C2×D4×D109C2semidirect
3. #16812695C2semidirect
4. #16812672C2semidirect
5. #16812671C2semidirect
6. #16812669C2semidirect
7. #16812667C2semidirect
8. #16812659C2semidirect
9. #120×C109C2semidirect
10. C23×D109C22semidirect
11. C23×D109C22semidirect
12. C2×D436C22semidirect
13. C2×D436C22semidirect
16. C2×C22⋊D109C22semidirect
17. C2×C22⋊D109C22semidirect
18. C2×C22⋊D109C22semidirect
19. C2×C22⋊D109C22semidirect
21. C22×Dic109C22semidirect
22. C2×C4×D109C22semidirect
23. #13182008C22semidirect
24. C22⋊Dic109C22semidirect
25. #13182006C22semidirect
26. #13182006C22semidirect
27. C4×Dic109C22semidirect
28. C24×C109C22semidirect
29. C2×D4×C109C22semidirect
30. C2×D4×C109C22semidirect
31. C22⋊C4×C109C22semidirect
32. C22⋊C4×C109C22semidirect
53. C22⋊D109D4semidirect
67. C22×C109C2×D4semidirect
76. C2×D109C2×D4semidirect
80. Dic109C2×D4semidirect
88. C109D42semidirect
97. #120D109semidirect
101. C24C2×D109semidirect
111. C2×D4C2×D109semidirect
112. C2×D4C2×D109semidirect
119. C22⋊C4C2×D109semidirect
120. C22⋊C4C2×D109semidirect
164. C22D4×D109semidirect