Finite Group Databasebeta

← #31771237

Subgroups — #31771237

Order 17216.

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⋊D269C212yesmaximal, complemented
2C2×D4×D269C212yesmaximal, complemented
3#22575449C211yesmaximal, characteristic, complemented
4#22575426C212yesmaximal, complemented
5#22575425C212yesmaximal, complemented
6#22575423C212yesmaximal, complemented
7#22575421C211yesmaximal, characteristic, complemented
8#22575413C212yesmaximal, complemented
9#120×C269C211yesmaximal, characteristic, complemented
10C23×D269C2211yescharacteristic, complemented
11C23×D269C2212yescomplemented
12C2×D1076C2212yescomplemented
13C2×D1076C2211yescharacteristic, complemented
14C2×C22⋊D269—28no—
15C2×C22⋊D269C2212yes—
16C2×C22⋊D269C2212yescomplemented
17C2×C22⋊D269C2212yescomplemented
18C2×C22⋊D269C2212yescomplemented
19C2×C22⋊D269C2212yescomplemented
20D4×D269—28no—
21C22×Dic269C2212yescomplemented
22C2×C4×D269C2212yescomplemented
23#16864381C2212yescomplemented
24C22⋊Dic269C2211yescharacteristic, complemented
25#16864379C2212yescomplemented
26#16864379C2212yescomplemented
27C4×Dic269C2211yescharacteristic, complemented
28C24×C269C2211yescharacteristic, complemented
29C2×D4×C269C2212yescomplemented
30C2×D4×C269C2211yescharacteristic, complemented
31C22⋊C4×C269C2212yescomplemented
32C22⋊C4×C269C2211yescharacteristic, complemented
33C22×D269—24no—
34C22×D269—24no—
35C22×D269—24no—
36C22×D269—41no—
37C22×D269—42no—
38C22×D269C2312yes—
39C22×D269C2312yes—
40D1076—24no—
41D1076—41no—
42C22⋊D269—24no—
43C22⋊D269—24no—
44C22⋊D269—24no—
45C22⋊D269—24no—
46C22⋊D269D418yescomplemented
47C2×Dic269—24no—
48C2×Dic269C2312yes—
49C2×Dic269C2312yes—
50C4×D269—24no—
51C23×C269—24no—
52C23×C269—41no—
53C23×C269C2311yescharacteristic
54C23×C269C2311yescharacteristic
55C23×C269C2312yes—
56D4×C269—24no—
57D4×C269—41no—
58C2×C4×C269C2311yescharacteristic
59C2×C4×C269C2312yes—
60C22×C269—22no—
61C22×C269—22no—
62C22×C269—22no—
63C22×C269—22no—
64C22×C269—21no—
65C22×C269—21no—
66C22×C269—44no—
67C22×C269C2×D414yescomplemented
68C22×C269C2411yescharacteristic
69C2×D269—22no—
70C2×D269—22no—
71C2×D269—22no—
72C2×D269—22no—
73C2×D269—22no—
74C2×D269—44no—
75C2×D269—48no—
76C2×D269C2×D414yescomplemented
77C4×C269—22no—
78C4×C269—21no—
79Dic269—22no—
80Dic269C2×D414yescomplemented
81C2×C269—24no—
82C2×C269—41no—
83C2×C269—41no—
84C2×C269C22×D412yes—
85C2×C269H(5,𝔽2)11yescharacteristic
86D269—24no—
87D269—42no—
88C269D4211yescharacteristic, complemented, Sylow
89D42—2691nomaximal, Sylow
90C22×D4—2692no—
91C22×D4—2692no—
92#127—2691no—
93#121—2692no—
94#121—2692no—
95#120—2691no—
96#120—2692no—
97#120D26911yescharacteristic, complemented
98C4×D4—2692no—
99C24—2691no—
100C24—2692no—
101C24C2×D26911yescharacteristic, complemented
102C2×D4—2692no—
103C2×D4—2692no—
104C2×D4—2692no—
105C2×D4—2692no—
106C2×D4—2692no—
107C2×D4—2692no—
108C2×D4—2691no—
109C2×D4—5388no—
110C2×D4—5388no—
111C2×D4C2×D26912yescomplemented
112C2×D4C2×D26911yescharacteristic, complemented
113C22×C4—2692no—
114C22×C4—2692no—
115C4⋊C4—2692no—
116C22⋊C4—2691no—
117C22⋊C4—2692no—
118C22⋊C4—2692no—
119C22⋊C4C2×D26912yescomplemented
120C22⋊C4C2×D26911yescharacteristic, complemented
121C42—2691no—
122C23—24no—
123C23—41no—
124C23—2692no—
125C23—2692no—
126C23—5384no—
127C23—5384no—
128C23—5384no—
129C23—10762no—
130C23—10761no—
131C23C22×D26911yescharacteristic
132C23C22×D26911yescharacteristic
133C23C22×D26912yes—
134D4—24no—
135D4—41no—
136D4—2698no—
137D4—5384no—
138D4—5384no—
139D4—5384no—
140D4—5384no—
141D4—5384no—
142D4—10761no—
143C2×C4—2692no—
144C2×C4—2692no—
145C2×C4—5384no—
146C2×C4—5384no—
147C2×C4C22×D26911yescharacteristic
148C2×C4C22×D26912yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—2694no—
157C22—5382no—
158C22—5382no—
159C22—5382no—
160C22—5382no—
161C22—5382no—
162C22—10768no—
163C22—10764no—
164C22D4×D26914yescomplemented
165C22C23×D26911yescharacteristic
166C4—22no—
167C4—21no—
168C4—2694no—
169C4—5382no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—5384no—
174C2—10762no—
175C2C2×D4×D26912yes—
176C2#2257549811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D269
commutator, socle, cosocleC22×C269C24
Fitting#120×C269C2
hypercenter#120D269
hypocenterC269D42
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D269C2semidirect
2. C2×D4×D269C2semidirect
3. #22575449C2semidirect
4. #22575426C2semidirect
5. #22575425C2semidirect
6. #22575423C2semidirect
7. #22575421C2semidirect
8. #22575413C2semidirect
9. #120×C269C2semidirect
10. C23×D269C22semidirect
11. C23×D269C22semidirect
12. C2×D1076C22semidirect
13. C2×D1076C22semidirect
16. C2×C22⋊D269C22semidirect
17. C2×C22⋊D269C22semidirect
18. C2×C22⋊D269C22semidirect
19. C2×C22⋊D269C22semidirect
21. C22×Dic269C22semidirect
22. C2×C4×D269C22semidirect
23. #16864381C22semidirect
24. C22⋊Dic269C22semidirect
25. #16864379C22semidirect
26. #16864379C22semidirect
27. C4×Dic269C22semidirect
28. C24×C269C22semidirect
29. C2×D4×C269C22semidirect
30. C2×D4×C269C22semidirect
31. C22⋊C4×C269C22semidirect
32. C22⋊C4×C269C22semidirect
46. C22⋊D269D4semidirect
67. C22×C269C2×D4semidirect
76. C2×D269C2×D4semidirect
80. Dic269C2×D4semidirect
88. C269D42semidirect
97. #120D269semidirect
101. C24C2×D269semidirect
111. C2×D4C2×D269semidirect
112. C2×D4C2×D269semidirect
119. C22⋊C4C2×D269semidirect
120. C22⋊C4C2×D269semidirect
164. C22D4×D269semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D2692
C2×D4×D2692
#225754492
#225754262
#225754252
#225754232
#225754212
#225754132
#120×C2692
D42269