Finite Group Databasebeta

← #31771171

Subgroups — #31771171

Order 17216.

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

normal subgroups only

This group has 334 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
1C2×#16864402C211yesmaximal, characteristic, complemented
2C2×D4×D269C211yesmaximal, characteristic, complemented
3#22575449C212yesmaximal, complemented
4#22575439C212yesmaximal, complemented
5#22575428C211yesmaximal, characteristic, complemented
6#22575427C211yesmaximal, characteristic, complemented
7#22575424C211yesmaximal, characteristic, complemented
8#22575423C212yesmaximal, complemented
9#22575421C212yesmaximal, complemented
10#22575411C211yesmaximal, characteristic, complemented
11#121×C269C211yesmaximal, characteristic, complemented
12C23×D269C2212yes—
13#16864402—24no—
14C2×Dic538C2211yescharacteristic, complemented
15C2×D1076C2211yescharacteristic
16C2×D1076C2211yescharacteristic
17C2×D1076C2212yescomplemented
18C2×C22⋊D269C2211yescharacteristic
19C2×C22⋊D269C2212yes—
20C2×C22⋊D269C2211yescharacteristic, complemented
21C2×C22⋊D269C2212yescomplemented
22C2×C22⋊D269C2212yescomplemented
23D4×D269—24no—
24C2×C4×D269C2211yescharacteristic
25C2×C4×D269C2211yescharacteristic, complemented
26C4⋊Dic269C2211yescharacteristic, complemented
27C22⋊Dic269C2212yescomplemented
28#16864379C2212yes—
29#16864379C2212yescomplemented
30#16864379C2212yescomplemented
31C4×Dic269C2212yescomplemented
32C2×D4×C269C2211yescharacteristic
33C2×D4×C269C2212yescomplemented
34C22×C4×C269C2211yescharacteristic, complemented
35C4⋊C4×C269C2211yescharacteristic, complemented
36C22⋊C4×C269C2212yescomplemented
37C22×D269—24no—
38C22×D269—42no—
39C22×D269—42no—
40C22×D269C2312yes—
41C22×D269C2311yescharacteristic
42C22×D269C2311yescharacteristic
43Dic538—22no—
44D1076—22no—
45D1076—22no—
46D1076—42no—
47C22⋊D269—22no—
48C22⋊D269—24no—
49C22⋊D269—22no—
50C22⋊D269—42no—
51C22⋊D269—42no—
52C2×Dic269C2312yes—
53C2×Dic269C2311yescharacteristic
54C2×Dic269C2311yescharacteristic
55C4×D269—22no—
56C4×D269D414yescomplemented
57C23×C269C2312yes—
58C23×C269C2311yescharacteristic
59D4×C269—22no—
60D4×C269—42no—
61C2×C4×C269—22no—
62C2×C4×C269C2311yescharacteristic
63C2×C4×C269C2312yes—
64C2×C4×C269C2311yescharacteristic
65C22×C269—21no—
66C22×C269—22no—
67C22×C269—21no—
68C22×C269—22no—
69C22×C269—21no—
70C22×C269—22no—
71C22×C269C2411yescharacteristic
72C2×D269—21no—
73C2×D269—21no—
74C2×D269—21no—
75C2×D269—22no—
76C2×D269—21no—
77C2×D269—22no—
78C2×D269—21no—
79C2×D269—22no—
80C2×D269—48no—
81C2×D269C2×D412yes—
82C4×C269—21no—
83C4×C269—22no—
84C4×C269C2×D412yes—
85Dic269—21no—
86Dic269—22no—
87Dic269C2×D412yes—
88C2×C269—42no—
89C2×C269—41no—
90C2×C269C22×D411yescharacteristic
91C2×C269H(5,𝔽2)11yescharacteristic
92C2×C269H(5,𝔽2)11yescharacteristic
93D269—22no—
94D269—41no—
95D269—42no—
96C269#53511yescharacteristic, complemented, Sylow
97#535—2691nomaximal, Sylow
98C2×#41—2691no—
99C22×D4—2691no—
100#127—2692no—
101#124—2692no—
102#121—2691no—
103#121—2691no—
104#121—2691no—
105#121D26911yescharacteristic, complemented
106#120—2692no—
107#120—2692no—
108#117—2691no—
109C24—2692no—
110#41—5384no—
111C2×Q8—2691no—
112C2×D4—2691no—
113C2×D4—2692no—
114C2×D4—2691no—
115C2×D4—2692no—
116C2×D4—2692no—
117C2×D4—2691no—
118C2×D4—2691no—
119C2×D4—2692no—
120C2×D4—5384no—
121C2×D4C2×D26911yescharacteristic
122C2×D4C2×D26912yescomplemented
123C22×C4—2691no—
124C22×C4—2691no—
125C22×C4C2×D26911yescharacteristic, complemented
126C4⋊C4—2691no—
127C4⋊C4C2×D26911yescharacteristic, complemented
128C22⋊C4—2692no—
129C22⋊C4—2692no—
130C22⋊C4—2692no—
131C22⋊C4—2692no—
132C22⋊C4C2×D26912yescomplemented
133C42—2692no—
134C23—2691no—
135C23—2692no—
136C23—2691no—
137C23—5384no—
138C23—10762no—
139C23—10762no—
140C23C22×D26912yes—
141C23C22×D26911yescharacteristic
142Q8—5382no—
143D4—22no—
144D4—42no—
145D4—5382no—
146D4—5384no—
147D4—5382no—
148D4—5382no—
149D4—5382no—
150D4—10762no—
151D4—10762no—
152D4—10762no—
153C2×C4—22no—
154C2×C4—2691no—
155C2×C4—2694no—
156C2×C4—2692no—
157C2×C4—2691no—
158C2×C4—5382no—
159C2×C4C22×D26912yes—
160C2×C4C22×D26911yescharacteristic
161C2×C4C22×D26911yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—2692no—
169C22—5381no—
170C22—5381no—
171C22—5381no—
172C22—5382no—
173C22—5381no—
174C22—5382no—
175C22—5381no—
176C22—5382no—
177C22—10768no—
178C22C23×D26911yescharacteristic
179C4—21no—
180C4—22no—
181C4—2692no—
182C4—5381no—
183C4—5382no—
184C4D4×D26912yes—
185C2—42no—
186C2—41no—
187C2—5382no—
188C2—10762no—
189C2—10761no—
190C2C2×D4×D26911yescharacteristic
191C2#2257549811yescharacteristic
192C2#2257549911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D269
commutator, socle, cosocleC22×C269C24
Fitting#121×C269C2
hypercenter#121D269
hypocenterC269#535
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16864402C2semidirect
2. C2×D4×D269C2semidirect
3. #22575449C2semidirect
4. #22575439C2semidirect
5. #22575428C2semidirect
6. #22575427C2semidirect
7. #22575424C2semidirect
8. #22575423C2semidirect
9. #22575421C2semidirect
10. #22575411C2semidirect
11. #121×C269C2semidirect
14. C2×Dic538C22semidirect
17. C2×D1076C22semidirect
20. C2×C22⋊D269C22semidirect
21. C2×C22⋊D269C22semidirect
22. C2×C22⋊D269C22semidirect
25. C2×C4×D269C22semidirect
26. C4⋊Dic269C22semidirect
27. C22⋊Dic269C22semidirect
29. #16864379C22semidirect
30. #16864379C22semidirect
31. C4×Dic269C22semidirect
33. C2×D4×C269C22semidirect
34. C22×C4×C269C22semidirect
35. C4⋊C4×C269C22semidirect
36. C22⋊C4×C269C22semidirect
56. C4×D269D4semidirect
96. C269#535semidirect
105. #121D269semidirect
122. C2×D4C2×D269semidirect
125. C22×C4C2×D269semidirect
127. C4⋊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
C2×#168644022
C2×D4×D2692
#225754492
#225754392
#225754282
#225754272
#225754242
#225754232
#225754212
#225754112
#121×C2692
#535269