Finite Group Databasebeta

← #31771187

Subgroups — #31771187

Order 17216.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#22575448C211yesmaximal, characteristic, complemented
2#22575446C211yesmaximal, characteristic, complemented
3#22575440C211yesmaximal, characteristic, complemented
4#22575439C211yesmaximal, characteristic, complemented
5#22575436C211yesmaximal, characteristic, complemented
6#22575434C211yesmaximal, characteristic, complemented
7#22575431C211yesmaximal, characteristic, complemented
8#22575425C211yesmaximal, characteristic, complemented
9#22575424C211yesmaximal, characteristic, complemented
10#22575421C211yesmaximal, characteristic, complemented
11#22575421C211yesmaximal, characteristic, complemented
12C4×C22⋊D269C211yesmaximal, characteristic, complemented
13#22575413C211yesmaximal, characteristic, complemented
14C22⋊C4×D269C211yesmaximal, characteristic, complemented
15#121×C269C211yesmaximal, characteristic, complemented
16C23×D269C2211yescharacteristic
17C2×Dic538C2211yescharacteristic, complemented
18C2×C22⋊D269C2211yescharacteristic
19C2×C22⋊D269C2211yescharacteristic
20C2×C22⋊D269C2211yescharacteristic
21C2×C22⋊D269C2211yescharacteristic
22C22×Dic269C2211yescharacteristic, complemented
23C2×C4×D269C2211yescharacteristic
24C2×C4×D269C2211yescharacteristic
25C4⋊Dic269C2211yescharacteristic, complemented
26#16864381C2211yescharacteristic, complemented
27#16864381C2211yescharacteristic, complemented
28#16864381C2211yescharacteristic, complemented
29#16864381C2211yescharacteristic, complemented
30C22⋊Dic269C2211yescharacteristic
31C22⋊Dic269C2211yescharacteristic
32C22⋊Dic269C2211yescharacteristic
33C22⋊Dic269C2211yescharacteristic, complemented
34C22⋊Dic269C2211yescharacteristic, complemented
35C22⋊Dic269C2211yescharacteristic, complemented
36#16864379C2211yescharacteristic
37#16864379C2211yescharacteristic
38#16864379C2211yescharacteristic
39#16864379C2211yescharacteristic
40#16864379C2211yescharacteristic
41#16864379C2211yescharacteristic
42C4×Dic269C2211yescharacteristic, complemented
43C4×Dic269C2211yescharacteristic, complemented
44C2×D4×C269C2211yescharacteristic
45C2×D4×C269C2211yescharacteristic
46C2×D4×C269C2211yescharacteristic, complemented
47C22×C4×C269C2211yescharacteristic, complemented
48C4⋊C4×C269C2211yescharacteristic, complemented
49C22⋊C4×C269C2211yescharacteristic
50C22⋊C4×C269C2211yescharacteristic, complemented
51C22×D269—22no—
52C22×D269—41no—
53C22×D269—41no—
54C22×D269C2311yescharacteristic
55C22×D269C2311yescharacteristic
56Dic538—41no—
57C22⋊D269—22no—
58C22⋊D269—22no—
59C22⋊D269—41no—
60C22⋊D269—41no—
61C2×Dic269—41no—
62C2×Dic269C2311yescharacteristic
63C2×Dic269C2311yescharacteristic
64C2×Dic269C2311yescharacteristic
65C2×Dic269C2311yescharacteristic
66C2×Dic269C2311yescharacteristic
67C2×Dic269C2311yescharacteristic
68C4×D269—41no—
69C4×D269—41no—
70C23×C269C2311yescharacteristic
71C23×C269C2311yescharacteristic
72C23×C269C2311yescharacteristic
73D4×C269—41no—
74D4×C269—41no—
75D4×C269—41no—
76C2×C4×C269—41no—
77C2×C4×C269C2311yescharacteristic
78C2×C4×C269C2311yescharacteristic
79C2×C4×C269C2311yescharacteristic
80C2×C4×C269C2311yescharacteristic
81C22×C269—21no—
82C22×C269—21no—
83C22×C269—21no—
84C22×C269—21no—
85C22×C269—21no—
86C22×C269—21no—
87C22×C269—21no—
88C22×C269—21no—
89C22×C269—21no—
90C22×C269C2411yescharacteristic
91C2×D269—21no—
92C2×D269—21no—
93C2×D269—21no—
94C2×D269—21no—
95C2×D269—21no—
96C2×D269—44no—
97C2×D269#4112yes—
98C4×C269—21no—
99C4×C269—21no—
100C4×C269—21no—
101C4×C269—21no—
102Dic269—21no—
103Dic269—21no—
104Dic269—21no—
105Dic269—21no—
106Dic269—21no—
107Dic269—21no—
108C2×C269—41no—
109C2×C269—41no—
110C2×C269—41no—
111C2×C269C2×#4111yescharacteristic
112C2×C269H(5,𝔽2)11yescharacteristic
113C2×C269H(5,𝔽2)11yescharacteristic
114D269—22no—
115D269—41no—
116C269#53811yescharacteristic, complemented, Sylow
117#538—2691nomaximal, Sylow
118#126—2691no—
119#126—2691no—
120#124—2691no—
121#124—2691no—
122#123—2691no—
123#123—2691no—
124C22⋊Q8—2691no—
125#121—2691no—
126#121—2691no—
127#121D26911yescharacteristic, complemented
128#120—2691no—
129#120—2691no—
130C4×D4—2691no—
131C4×D4—2691no—
132C2×C22⋊C4—2691no—
133C24—2691no—
134C2×Q8—2691no—
135C2×D4—2691no—
136C2×D4—2691no—
137C2×D4—2691no—
138C2×D4—2691no—
139C2×D4C2×D26911yescharacteristic
140C2×D4C2×D26911yescharacteristic
141C2×D4C2×D26911yescharacteristic, complemented
142C22×C4—2691no—
143C22×C4—2691no—
144C22×C4—2691no—
145C22×C4C2×D26911yescharacteristic, complemented
146C4⋊C4—2691no—
147C4⋊C4—2691no—
148C4⋊C4—2691no—
149C4⋊C4—2691no—
150C4⋊C4—2691no—
151C4⋊C4C2×D26911yescharacteristic, complemented
152C22⋊C4—2691no—
153C22⋊C4—2691no—
154C22⋊C4—2691no—
155C22⋊C4—2691no—
156C22⋊C4—2691no—
157C22⋊C4—2691no—
158C22⋊C4—2691no—
159C22⋊C4—2691no—
160C22⋊C4—2691no—
161C22⋊C4—2691no—
162C22⋊C4—2691no—
163C22⋊C4—2691no—
164C22⋊C4C2×D26911yescharacteristic
165C22⋊C4C2×D26911yescharacteristic, complemented
166C42—2691no—
167C42—2691no—
168C23—2691no—
169C23—2691no—
170C23—5382no—
171C23—10761no—
172C23—10761no—
173C23C22×D26911yescharacteristic
174C23C22×D26911yescharacteristic
175C23C22×D26911yescharacteristic
176Q8—10761no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—5382no—
181D4—5382no—
182D4—10761no—
183D4—10761no—
184C2×C4—41no—
185C2×C4—2691no—
186C2×C4—2691no—
187C2×C4—2691no—
188C2×C4—2691no—
189C2×C4—2691no—
190C2×C4—2691no—
191C2×C4—10761no—
192C2×C4—10761no—
193C2×C4—10761no—
194C2×C4C22×D26911yescharacteristic
195C2×C4C22×D26911yescharacteristic
196C2×C4C22×D26911yescharacteristic
197C2×C4C22×D26911yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—2692no—
208C22—5381no—
209C22—5381no—
210C22—5381no—
211C22—5381no—
212C22—5381no—
213C22—10764no—
214C22C23×D26911yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—5381no—
220C4—5381no—
221C4—5381no—
222C4—5381no—
223C4—5381no—
224C4—5381no—
225C2—41no—
226C2—41no—
227C2—41no—
228C2—5382no—
229C2—10761no—
230C2#41×D26911yescharacteristic
231C2#2257549811yescharacteristic
232C2#2257549811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D269
commutator, socle, cosocleC22×C269C24
Fitting#121×C269C2
hypercenter#121D269
hypocenterC269#538
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22575448C2semidirect
2. #22575446C2semidirect
3. #22575440C2semidirect
4. #22575439C2semidirect
5. #22575436C2semidirect
6. #22575434C2semidirect
7. #22575431C2semidirect
8. #22575425C2semidirect
9. #22575424C2semidirect
10. #22575421C2semidirect
11. #22575421C2semidirect
12. C4×C22⋊D269C2semidirect
13. #22575413C2semidirect
14. C22⋊C4×D269C2semidirect
15. #121×C269C2semidirect
17. C2×Dic538C22semidirect
22. C22×Dic269C22semidirect
25. C4⋊Dic269C22semidirect
26. #16864381C22semidirect
27. #16864381C22semidirect
28. #16864381C22semidirect
29. #16864381C22semidirect
33. C22⋊Dic269C22semidirect
34. C22⋊Dic269C22semidirect
35. C22⋊Dic269C22semidirect
42. C4×Dic269C22semidirect
43. C4×Dic269C22semidirect
46. C2×D4×C269C22semidirect
47. C22×C4×C269C22semidirect
48. C4⋊C4×C269C22semidirect
50. C22⋊C4×C269C22semidirect
116. C269#538semidirect
127. #121D269semidirect
141. C2×D4C2×D269semidirect
145. C22×C4C2×D269semidirect
151. C4⋊C4C2×D269semidirect
165. C22⋊C4C2×D269semidirect

Maximal subgroups

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

subgroupindex
#225754482
#225754462
#225754402
#225754392
#225754362
#225754342
#225754312
#225754252
#225754242
#225754212
C4×C22⋊D2692
#225754132
C22⋊C4×D2692
#121×C2692
#538269