Finite Group Databasebeta

← #31771343

Subgroups — #31771343

Order 17216.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#22575449C211yesmaximal, characteristic, complemented
2#22575448C211yesmaximal, characteristic, complemented
3#22575446C211yesmaximal, characteristic, complemented
4#22575438C211yesmaximal, characteristic, complemented
5#22575437C211yesmaximal, characteristic, complemented
6#22575436C211yesmaximal, characteristic, complemented
7#22575427C211yesmaximal, characteristic, complemented
8#22575426C211yesmaximal, characteristic, complemented
9#22575425C211yesmaximal, characteristic, complemented
10#22575425C211yesmaximal, characteristic, complemented
11#22575424C211yesmaximal, characteristic, complemented
12#22575423C211yesmaximal, characteristic, complemented
13#22575421C211yesmaximal, characteristic, complemented
14#22575421C211yesmaximal, characteristic, complemented
15#121×C269C211yesmaximal, characteristic, complemented
16C23×D269C2211yescharacteristic
17C2×D1076C2211yescharacteristic
18C2×D1076C2211yescharacteristic
19C2×C22⋊D269C2211yescharacteristic
20C2×C22⋊D269C2211yescharacteristic
21C2×C22⋊D269C2211yescharacteristic
22C2×C22⋊D269C2211yescharacteristic
23C2×C22⋊D269C2211yescharacteristic
24C2×C22⋊D269C2211yescharacteristic, complemented
25C2×C22⋊D269C2211yescharacteristic, complemented
26C22×Dic269C2211yescharacteristic, complemented
27C2×C4×D269C2211yescharacteristic, complemented
28C4⋊Dic269C2211yescharacteristic, complemented
29C4⋊Dic269C2211yescharacteristic, complemented
30#16864381C2211yescharacteristic, complemented
31#16864381C2211yescharacteristic, complemented
32#16864381C2211yescharacteristic, complemented
33C22⋊Dic269C2211yescharacteristic
34C22⋊Dic269C2211yescharacteristic
35C22⋊Dic269C2211yescharacteristic, complemented
36#16864379C2211yescharacteristic
37#16864379C2211yescharacteristic
38#16864379C2211yescharacteristic
39#16864379C2211yescharacteristic
40#16864379C2211yescharacteristic
41#16864379C2211yescharacteristic
42#16864379C2211yescharacteristic, 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—41no—
52C22×D269—41no—
53C22×D269—41no—
54C22×D269C2311yescharacteristic
55C22×D269C2311yescharacteristic
56C22×D269C2311yescharacteristic
57D1076—41no—
58D1076—41no—
59C22⋊D269—41no—
60C22⋊D269—41no—
61C22⋊D269—41no—
62C22⋊D269—41no—
63C22⋊D269—41no—
64C22⋊D269—41no—
65C22⋊D269—41no—
66C2×Dic269—41no—
67C2×Dic269C2311yescharacteristic
68C2×Dic269C2311yescharacteristic
69C2×Dic269C2311yescharacteristic
70C2×Dic269C2311yescharacteristic
71C2×Dic269C2311yescharacteristic
72C4×D269—41no—
73C23×C269C2311yescharacteristic
74C23×C269C2311yescharacteristic
75C23×C269C2311yescharacteristic
76D4×C269—41no—
77D4×C269—41no—
78D4×C269—41no—
79C2×C4×C269—41no—
80C2×C4×C269C2311yescharacteristic
81C2×C4×C269C2311yescharacteristic
82C2×C4×C269C2311yescharacteristic
83C2×C4×C269C2311yescharacteristic
84C22×C269—21no—
85C22×C269—21no—
86C22×C269—21no—
87C22×C269—21no—
88C22×C269—21no—
89C22×C269—21no—
90C22×C269—21no—
91C22×C269—21no—
92C22×C269—21no—
93C22×C269C2411yescharacteristic
94C2×D269—21no—
95C2×D269—21no—
96C2×D269—21no—
97C2×D269—21no—
98C2×D269—21no—
99C2×D269—21no—
100C2×D269—21no—
101C2×D269—21no—
102C2×D269—21no—
103C2×D269—44no—
104C4×C269—21no—
105C4×C269—21no—
106C4×C269—21no—
107C4×C269—21no—
108Dic269—21no—
109Dic269—21no—
110Dic269—21no—
111Dic269—21no—
112Dic269—21no—
113C2×C269—41no—
114C2×C269—41no—
115C2×C269—41no—
116C2×C269H(5,𝔽2)11yescharacteristic
117C2×C269H(5,𝔽2)11yescharacteristic
118C2×C269H(5,𝔽2)11yescharacteristic
119D269—41no—
120D269—41no—
121D269—41no—
122C269#56011yescharacteristic, complemented, Sylow
123#560—2691nomaximal, Sylow
124#127—2691no—
125#126—2691no—
126#126—2691no—
127#123—2691no—
128#123—2691no—
129#123—2691no—
130#121—2691no—
131#121—2691no—
132#121—2691no—
133#121—2691no—
134#121—2691no—
135#121D26911yescharacteristic, complemented
136#120—2691no—
137#120—2691no—
138#120—2691no—
139C24—2691no—
140C2×D4—2691no—
141C2×D4—2691no—
142C2×D4—2691no—
143C2×D4—2691no—
144C2×D4—2691no—
145C2×D4—2691no—
146C2×D4—2691no—
147C2×D4—2691no—
148C2×D4—2691no—
149C2×D4C2×D26911yescharacteristic
150C2×D4C2×D26911yescharacteristic
151C2×D4C2×D26911yescharacteristic, complemented
152C22×C4—2691no—
153C22×C4—2691no—
154C22×C4C2×D26911yescharacteristic, complemented
155C4⋊C4—2691no—
156C4⋊C4—2691no—
157C4⋊C4—2691no—
158C4⋊C4—2691no—
159C4⋊C4—2691no—
160C4⋊C4C2×D26911yescharacteristic, complemented
161C22⋊C4—2691no—
162C22⋊C4—2691no—
163C22⋊C4—2691no—
164C22⋊C4—2691no—
165C22⋊C4—2691no—
166C22⋊C4—2691no—
167C22⋊C4—2691no—
168C22⋊C4—2691no—
169C22⋊C4—2691no—
170C22⋊C4—2691no—
171C22⋊C4C2×D26911yescharacteristic
172C22⋊C4C2×D26911yescharacteristic, complemented
173C42—2691no—
174C23—2691no—
175C23—2691no—
176C23—2691no—
177C23—10761no—
178C23—10761no—
179C23—10761no—
180C23C22×D26911yescharacteristic
181C23C22×D26911yescharacteristic
182C23C22×D26911yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—10761no—
187D4—10761no—
188D4—10761no—
189D4—10761no—
190D4—10761no—
191D4—10761no—
192D4—10761no—
193D4—10761no—
194D4—10761no—
195C2×C4—41no—
196C2×C4—2691no—
197C2×C4—2691no—
198C2×C4—2691no—
199C2×C4—2691no—
200C2×C4—2691no—
201C2×C4—10761no—
202C2×C4—10761no—
203C2×C4C22×D26911yescharacteristic
204C2×C4C22×D26911yescharacteristic
205C2×C4C22×D26911yescharacteristic
206C2×C4C22×D26911yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—5381no—
217C22—5381no—
218C22—5381no—
219C22—5381no—
220C22—5381no—
221C22—5381no—
222C22—5381no—
223C22—5381no—
224C22—5381no—
225C22—10764no—
226C22C23×D26911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—5381no—
232C4—5381no—
233C4—5381no—
234C4—5381no—
235C4—5381no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—10761no—
240C2—10761no—
241C2—10761no—
242C2#2257549811yescharacteristic
243C2#2257549811yescharacteristic
244C2#2257549911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D269
commutator, socle, cosocleC22×C269C24
Fitting#121×C269C2
hypercenter#121D269
hypocenterC269#560
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22575449C2semidirect
2. #22575448C2semidirect
3. #22575446C2semidirect
4. #22575438C2semidirect
5. #22575437C2semidirect
6. #22575436C2semidirect
7. #22575427C2semidirect
8. #22575426C2semidirect
9. #22575425C2semidirect
10. #22575425C2semidirect
11. #22575424C2semidirect
12. #22575423C2semidirect
13. #22575421C2semidirect
14. #22575421C2semidirect
15. #121×C269C2semidirect
24. C2×C22⋊D269C22semidirect
25. C2×C22⋊D269C22semidirect
26. C22×Dic269C22semidirect
27. C2×C4×D269C22semidirect
28. C4⋊Dic269C22semidirect
29. C4⋊Dic269C22semidirect
30. #16864381C22semidirect
31. #16864381C22semidirect
32. #16864381C22semidirect
35. C22⋊Dic269C22semidirect
42. #16864379C22semidirect
43. C4×Dic269C22semidirect
46. C2×D4×C269C22semidirect
47. C22×C4×C269C22semidirect
48. C4⋊C4×C269C22semidirect
50. C22⋊C4×C269C22semidirect
122. C269#560semidirect
135. #121D269semidirect
151. C2×D4C2×D269semidirect
154. C22×C4C2×D269semidirect
160. C4⋊C4C2×D269semidirect
172. C22⋊C4C2×D269semidirect

Maximal subgroups

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

subgroupindex
#225754492
#225754482
#225754462
#225754382
#225754372
#225754362
#225754272
#225754262
#225754252
#225754242
#225754232
#225754212
#121×C2692
#560269