Finite Group Databasebeta

← #31771165

Subgroups — #31771165

Order 17216.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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⋊D269C211yesmaximal, characteristic, complemented
2C2×D4×D269C211yesmaximal, characteristic, complemented
3#22575437C211yesmaximal, characteristic, complemented
4#22575435C212yesmaximal, complemented
5#22575434C211yesmaximal, characteristic, complemented
6#22575426C212yesmaximal, complemented
7#22575424C212yesmaximal, complemented
8#22575422C211yesmaximal, characteristic, complemented
9#22575421C212yesmaximal, complemented
10C22⋊C4×D269C211yesmaximal, characteristic, complemented
11#120×C269C211yesmaximal, characteristic, complemented
12C23×D269C2211yescharacteristic
13C23×D269C2211yescharacteristic, complemented
14C2×D1076C2211yescharacteristic, complemented
15C2×C22⋊D269—24no—
16C2×C22⋊D269C2212yes—
17C2×C22⋊D269C2211yescharacteristic
18C2×C22⋊D269C2211yescharacteristic, complemented
19C2×C22⋊D269C2212yescomplemented
20C2×C22⋊D269C2212yescomplemented
21D4×D269—24no—
22C22×Dic269C2211yescharacteristic, complemented
23C2×C4×D269C2211yescharacteristic, complemented
24C2×C4×D269C2212yescomplemented
25C4⋊Dic269C2212yescomplemented
26#16864381C2212yescomplemented
27C22⋊Dic269C2211yescharacteristic, complemented
28C22⋊Dic269C2212yescomplemented
29C22⋊Dic269C2212yescomplemented
30#16864379C2212yescomplemented
31#16864379C2212yescomplemented
32C24×C269C2211yescharacteristic, complemented
33C2×D4×C269C2211yescharacteristic
34C2×D4×C269C2212yescomplemented
35C22⋊C4×C269C2211yescharacteristic, complemented
36C22⋊C4×C269C2212yescomplemented
37C22×D269—22no—
38C22×D269—42no—
39C22×D269—42no—
40C22×D269D414yescomplemented
41C22×D269C2311yescharacteristic
42C22×D269C2311yescharacteristic
43C22×D269C2311yescharacteristic
44D1076—22no—
45C22⋊D269—22no—
46C22⋊D269—22no—
47C22⋊D269—24no—
48C22⋊D269—24no—
49C22⋊D269—42no—
50C2×Dic269—22no—
51C2×Dic269C2312yes—
52C2×Dic269C2312yes—
53C2×Dic269C2311yescharacteristic
54C4×D269—22no—
55C4×D269—42no—
56C23×C269—22no—
57C23×C269—42no—
58C23×C269C2312yes—
59C23×C269C2311yescharacteristic
60C23×C269C2311yescharacteristic
61D4×C269—22no—
62D4×C269—42no—
63C2×C4×C269C2311yescharacteristic
64C2×C4×C269C2312yes—
65C22×C269—22no—
66C22×C269—21no—
67C22×C269—22no—
68C22×C269—22no—
69C22×C269—22no—
70C22×C269—22no—
71C22×C269—44no—
72C22×C269C2×D412yes—
73C22×C269C2411yescharacteristic
74C2×D269—22no—
75C2×D269—22no—
76C2×D269—21no—
77C2×D269—22no—
78C2×D269—44no—
79C2×D269—44no—
80C2×D269C2×D412yes—
81C2×D269C2×D412yes—
82C4×C269—21no—
83C4×C269—22no—
84Dic269—21no—
85Dic269—22no—
86Dic269—22no—
87C2×C269—22no—
88C2×C269—41no—
89C2×C269—42no—
90C2×C269C22×D411yescharacteristic
91C2×C269H(5,𝔽2)12yes—
92D269—22no—
93D269—22no—
94D269—41no—
95C269#53411yescharacteristic, complemented, Sylow
96#534—2691nomaximal, Sylow
97C22×D4—2691no—
98C22×D4—2691no—
99#123—2691no—
100#123—2692no—
101#123—2691no—
102#121—2692no—
103#121—2692no—
104#120—2691no—
105#120—2692no—
106#120D26911yescharacteristic, complemented
107C2×C22⋊C4—2691no—
108C24—2691no—
109C24—2691no—
110C24C2×D26911yescharacteristic, complemented
111C2×D4—2692no—
112C2×D4—2691no—
113C2×D4—2691no—
114C2×D4—2692no—
115C2×D4—2692no—
116C2×D4—2691no—
117C2×D4—5384no—
118C2×D4—5384no—
119C2×D4C2×D26911yescharacteristic
120C2×D4C2×D26912yescomplemented
121C22×C4—2691no—
122C22×C4—2691no—
123C22×C4—2692no—
124C4⋊C4—2692no—
125C4⋊C4—2692no—
126C22⋊C4—2691no—
127C22⋊C4—2692no—
128C22⋊C4—2692no—
129C22⋊C4—2692no—
130C22⋊C4—2692no—
131C22⋊C4C2×D26911yescharacteristic, complemented
132C22⋊C4C2×D26912yescomplemented
133C23—22no—
134C23—42no—
135C23—2691no—
136C23—2691no—
137C23—2694no—
138C23—2691no—
139C23—5382no—
140C23—10762no—
141C23—10762no—
142C23C22×D26911yescharacteristic
143C23C22×D26912yes—
144C23C22×D26911yescharacteristic
145D4—22no—
146D4—42no—
147D4—5382no—
148D4—5382no—
149D4—5384no—
150D4—5384no—
151D4—5382no—
152D4—10762no—
153C2×C4—2692no—
154C2×C4—2691no—
155C2×C4—2692no—
156C2×C4—5382no—
157C2×C4—5382no—
158C2×C4—10762no—
159C2×C4C22×D26912yes—
160C2×C4C22×D26911yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—2692no—
169C22—2692no—
170C22—5382no—
171C22—5382no—
172C22—5381no—
173C22—5382no—
174C22—10764no—
175C22—10764no—
176C22D4×D26912yes—
177C22C23×D26911yescharacteristic
178C4—21no—
179C4—22no—
180C4—5381no—
181C4—5382no—
182C4—5382no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—5382no—
187C2—5382no—
188C2—10761no—
189C2C2×D4×D26911yescharacteristic
190C2#2257549812yes—
subgroupG/H
center, FrattiniC22C23×D269
commutator, socle, cosocleC22×C269C24
Fitting#120×C269C2
hypercenter#120D269
hypocenterC269#534
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. #22575437C2semidirect
4. #22575435C2semidirect
5. #22575434C2semidirect
6. #22575426C2semidirect
7. #22575424C2semidirect
8. #22575422C2semidirect
9. #22575421C2semidirect
10. C22⋊C4×D269C2semidirect
11. #120×C269C2semidirect
13. C23×D269C22semidirect
14. C2×D1076C22semidirect
18. C2×C22⋊D269C22semidirect
19. C2×C22⋊D269C22semidirect
20. C2×C22⋊D269C22semidirect
22. C22×Dic269C22semidirect
23. C2×C4×D269C22semidirect
24. C2×C4×D269C22semidirect
25. C4⋊Dic269C22semidirect
26. #16864381C22semidirect
27. C22⋊Dic269C22semidirect
28. C22⋊Dic269C22semidirect
29. C22⋊Dic269C22semidirect
30. #16864379C22semidirect
31. #16864379C22semidirect
32. C24×C269C22semidirect
34. C2×D4×C269C22semidirect
35. C22⋊C4×C269C22semidirect
36. C22⋊C4×C269C22semidirect
40. C22×D269D4semidirect
95. C269#534semidirect
106. #120D269semidirect
110. C24C2×D269semidirect
120. C2×D4C2×D269semidirect
131. C22⋊C4C2×D269semidirect
132. C22⋊C4C2×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
#225754372
#225754352
#225754342
#225754262
#225754242
#225754222
#225754212
C22⋊C4×D2692
#120×C2692
#534269