Finite Group Databasebeta

← #20299336

Subgroups — #20299336

Order 7232.

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#16813618C211yesmaximal, characteristic, complemented
2#16813617C211yesmaximal, characteristic, complemented
3#16813615C211yesmaximal, characteristic, complemented
4#16813607C211yesmaximal, characteristic, complemented
5#16813606C211yesmaximal, characteristic, complemented
6#16813605C211yesmaximal, characteristic, complemented
7#16813596C211yesmaximal, characteristic, complemented
8#16813595C211yesmaximal, characteristic, complemented
9#16813594C211yesmaximal, characteristic, complemented
10#16813594C211yesmaximal, characteristic, complemented
11#16813593C211yesmaximal, characteristic, complemented
12#16813592C211yesmaximal, characteristic, complemented
13#16813590C211yesmaximal, characteristic, complemented
14#16813590C211yesmaximal, characteristic, complemented
15#121×C113C211yesmaximal, characteristic, complemented
16C23×D113C2211yescharacteristic
17C2×D452C2211yescharacteristic
18C2×D452C2211yescharacteristic
19C2×C22⋊D113C2211yescharacteristic
20C2×C22⋊D113C2211yescharacteristic
21C2×C22⋊D113C2211yescharacteristic
22C2×C22⋊D113C2211yescharacteristic
23C2×C22⋊D113C2211yescharacteristic
24C2×C22⋊D113C2211yescharacteristic, complemented
25C2×C22⋊D113C2211yescharacteristic, complemented
26C22×Dic113C2211yescharacteristic, complemented
27C2×C4×D113C2211yescharacteristic, complemented
28C4⋊Dic113C2211yescharacteristic, complemented
29C4⋊Dic113C2211yescharacteristic, complemented
30#14268626C2211yescharacteristic, complemented
31#14268626C2211yescharacteristic, complemented
32#14268626C2211yescharacteristic, complemented
33C22⋊Dic113C2211yescharacteristic
34C22⋊Dic113C2211yescharacteristic
35C22⋊Dic113C2211yescharacteristic, complemented
36#14268624C2211yescharacteristic
37#14268624C2211yescharacteristic
38#14268624C2211yescharacteristic
39#14268624C2211yescharacteristic
40#14268624C2211yescharacteristic
41#14268624C2211yescharacteristic
42#14268624C2211yescharacteristic, complemented
43C4×Dic113C2211yescharacteristic, complemented
44C2×D4×C113C2211yescharacteristic
45C2×D4×C113C2211yescharacteristic
46C2×D4×C113C2211yescharacteristic, complemented
47C22×C4×C113C2211yescharacteristic, complemented
48C4⋊C4×C113C2211yescharacteristic, complemented
49C22⋊C4×C113C2211yescharacteristic
50C22⋊C4×C113C2211yescharacteristic, complemented
51C23×C113C2311yescharacteristic
52C23×C113C2311yescharacteristic
53C23×C113C2311yescharacteristic
54C22×D113—41no—
55C22×D113—41no—
56C22×D113—41no—
57C22×D113C2311yescharacteristic
58C22×D113C2311yescharacteristic
59C22×D113C2311yescharacteristic
60D4×C113—41no—
61D4×C113—41no—
62D4×C113—41no—
63C2×C4×C113—41no—
64C2×C4×C113C2311yescharacteristic
65C2×C4×C113C2311yescharacteristic
66C2×C4×C113C2311yescharacteristic
67C2×C4×C113C2311yescharacteristic
68C22⋊D113—41no—
69C22⋊D113—41no—
70C22⋊D113—41no—
71C22⋊D113—41no—
72C22⋊D113—41no—
73C22⋊D113—41no—
74C22⋊D113—41no—
75C2×Dic113—41no—
76C2×Dic113C2311yescharacteristic
77C2×Dic113C2311yescharacteristic
78C2×Dic113C2311yescharacteristic
79C2×Dic113C2311yescharacteristic
80C2×Dic113C2311yescharacteristic
81D452—41no—
82D452—41no—
83C4×D113—41no—
84C22×C113—21no—
85C22×C113—21no—
86C22×C113—21no—
87C22×C113—21no—
88C22×C113—21no—
89C22×C113—21no—
90C22×C113—21no—
91C22×C113—21no—
92C22×C113—21no—
93C22×C113C2411yescharacteristic
94C2×D113—21no—
95C2×D113—21no—
96C2×D113—21no—
97C2×D113—21no—
98C2×D113—21no—
99C2×D113—21no—
100C2×D113—21no—
101C2×D113—21no—
102C2×D113—21no—
103C2×D113—44no—
104C4×C113—21no—
105C4×C113—21no—
106C4×C113—21no—
107C4×C113—21no—
108Dic113—21no—
109Dic113—21no—
110Dic113—21no—
111Dic113—21no—
112Dic113—21no—
113C2×C113—41no—
114C2×C113—41no—
115C2×C113—41no—
116C2×C113H(5,𝔽2)11yescharacteristic
117C2×C113H(5,𝔽2)11yescharacteristic
118C2×C113H(5,𝔽2)11yescharacteristic
119D113—41no—
120D113—41no—
121D113—41no—
122C113#56011yescharacteristic, complemented, Sylow
123#560—1131nomaximal, Sylow
124#127—1131no—
125#126—1131no—
126#126—1131no—
127#123—1131no—
128#123—1131no—
129#123—1131no—
130#121—1131no—
131#121—1131no—
132#121—1131no—
133#121—1131no—
134#121—1131no—
135#121D11311yescharacteristic, complemented
136#120—1131no—
137#120—1131no—
138#120—1131no—
139C24—1131no—
140C2×D4—1131no—
141C2×D4—1131no—
142C2×D4—1131no—
143C2×D4—1131no—
144C2×D4—1131no—
145C2×D4—1131no—
146C2×D4—1131no—
147C2×D4—1131no—
148C2×D4—1131no—
149C2×D4C2×D11311yescharacteristic
150C2×D4C2×D11311yescharacteristic
151C2×D4C2×D11311yescharacteristic, complemented
152C22×C4—1131no—
153C22×C4—1131no—
154C22×C4C2×D11311yescharacteristic, complemented
155C4⋊C4—1131no—
156C4⋊C4—1131no—
157C4⋊C4—1131no—
158C4⋊C4—1131no—
159C4⋊C4—1131no—
160C4⋊C4C2×D11311yescharacteristic, complemented
161C22⋊C4—1131no—
162C22⋊C4—1131no—
163C22⋊C4—1131no—
164C22⋊C4—1131no—
165C22⋊C4—1131no—
166C22⋊C4—1131no—
167C22⋊C4—1131no—
168C22⋊C4—1131no—
169C22⋊C4—1131no—
170C22⋊C4—1131no—
171C22⋊C4C2×D11311yescharacteristic
172C22⋊C4C2×D11311yescharacteristic, complemented
173C42—1131no—
174C23—1131no—
175C23—1131no—
176C23—1131no—
177C23—4521no—
178C23—4521no—
179C23—4521no—
180C23C22×D11311yescharacteristic
181C23C22×D11311yescharacteristic
182C23C22×D11311yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—4521no—
187D4—4521no—
188D4—4521no—
189D4—4521no—
190D4—4521no—
191D4—4521no—
192D4—4521no—
193D4—4521no—
194D4—4521no—
195C2×C4—41no—
196C2×C4—1131no—
197C2×C4—1131no—
198C2×C4—1131no—
199C2×C4—1131no—
200C2×C4—1131no—
201C2×C4—4521no—
202C2×C4—4521no—
203C2×C4C22×D11311yescharacteristic
204C2×C4C22×D11311yescharacteristic
205C2×C4C22×D11311yescharacteristic
206C2×C4C22×D11311yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—4524no—
217C22—2261no—
218C22—2261no—
219C22—2261no—
220C22—2261no—
221C22—2261no—
222C22—2261no—
223C22—2261no—
224C22—2261no—
225C22—2261no—
226C22C23×D11311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—2261no—
232C4—2261no—
233C4—2261no—
234C4—2261no—
235C4—2261no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—4521no—
240C2—4521no—
241C2—4521no—
242C2#1681366711yescharacteristic
243C2#1681366711yescharacteristic
244C2#1681366811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
Fitting#121×C113C2
hypercenter#121D113
hypocenterC113#560
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #16813618C2semidirect
2. #16813617C2semidirect
3. #16813615C2semidirect
4. #16813607C2semidirect
5. #16813606C2semidirect
6. #16813605C2semidirect
7. #16813596C2semidirect
8. #16813595C2semidirect
9. #16813594C2semidirect
10. #16813594C2semidirect
11. #16813593C2semidirect
12. #16813592C2semidirect
13. #16813590C2semidirect
14. #16813590C2semidirect
15. #121×C113C2semidirect
24. C2×C22⋊D113C22semidirect
25. C2×C22⋊D113C22semidirect
26. C22×Dic113C22semidirect
27. C2×C4×D113C22semidirect
28. C4⋊Dic113C22semidirect
29. C4⋊Dic113C22semidirect
30. #14268626C22semidirect
31. #14268626C22semidirect
32. #14268626C22semidirect
35. C22⋊Dic113C22semidirect
42. #14268624C22semidirect
43. C4×Dic113C22semidirect
46. C2×D4×C113C22semidirect
47. C22×C4×C113C22semidirect
48. C4⋊C4×C113C22semidirect
50. C22⋊C4×C113C22semidirect
122. C113#560semidirect
135. #121D113semidirect
151. C2×D4C2×D113semidirect
154. C22×C4C2×D113semidirect
160. C4⋊C4C2×D113semidirect
172. C22⋊C4C2×D113semidirect