Finite Group Databasebeta

← #20299187

Subgroups — #20299187

Order 7232.

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#16813617C211yesmaximal, characteristic, complemented
2#16813610C211yesmaximal, characteristic, complemented
3#16813608C211yesmaximal, characteristic, complemented
4#16813607C211yesmaximal, characteristic, complemented
5#16813604C211yesmaximal, characteristic, complemented
6#16813600C211yesmaximal, characteristic, complemented
7#16813597C211yesmaximal, characteristic, complemented
8#16813597C211yesmaximal, characteristic, complemented
9#16813595C211yesmaximal, characteristic, complemented
10#16813592C211yesmaximal, characteristic, complemented
11#16813592C211yesmaximal, characteristic, complemented
12C4×D452C211yesmaximal, characteristic, complemented
13#16813584C211yesmaximal, characteristic, complemented
14C22⋊C4×D113C211yesmaximal, characteristic, complemented
15#126×C113C211yesmaximal, characteristic, complemented
16C23×D113C2211yescharacteristic
17C2×Dic226C2211yescharacteristic, complemented
18C2×D452C2211yescharacteristic
19C2×D452C2211yescharacteristic
20C2×D452C2211yescharacteristic
21C2×D452C2211yescharacteristic
22C2×D452C2211yescharacteristic, complemented
23C2×C22⋊D113C2211yescharacteristic
24C2×C22⋊D113C2211yescharacteristic
25C2×C4×D113C2211yescharacteristic
26C2×C4×D113C2211yescharacteristic
27C2×C4×D113C2211yescharacteristic, complemented
28C2×C4×D113C2211yescharacteristic, complemented
29C4⋊Dic113C2211yescharacteristic, complemented
30#14268626C2211yescharacteristic, complemented
31#14268626C2211yescharacteristic, complemented
32C22⋊Dic113C2211yescharacteristic
33#14268624C2211yescharacteristic
34#14268624C2211yescharacteristic
35#14268624C2211yescharacteristic
36#14268624C2211yescharacteristic
37#14268624C2211yescharacteristic
38#14268624C2211yescharacteristic
39#14268624C2211yescharacteristic, complemented
40#14268624C2211yescharacteristic, complemented
41#14268624C2211yescharacteristic, complemented
42#14268624C2211yescharacteristic, complemented
43C4×Dic113C2211yescharacteristic, complemented
44C4⋊C4×C113C2211yescharacteristic, complemented
45C4⋊C4×C113C2211yescharacteristic, complemented
46C4⋊C4×C113C2211yescharacteristic, complemented
47C22⋊C4×C113C2211yescharacteristic
48C22⋊C4×C113C2211yescharacteristic
49C22⋊C4×C113C2211yescharacteristic
50C42×C113C2211yescharacteristic, complemented
51C23×C113C2311yescharacteristic
52C22×D113—22no—
53C22×D113—41no—
54C22×D113—41no—
55C22×D113C2311yescharacteristic
56C22×D113C2311yescharacteristic
57C22×D113C2311yescharacteristic
58C22×D113C2311yescharacteristic
59C2×C4×C113C2311yescharacteristic
60C2×C4×C113C2311yescharacteristic
61C2×C4×C113C2311yescharacteristic
62C2×C4×C113C2311yescharacteristic
63C2×C4×C113C2311yescharacteristic
64C2×C4×C113C2311yescharacteristic
65C22⋊D113—41no—
66C22⋊D113—41no—
67C2×Dic113C2311yescharacteristic
68C2×Dic113C2311yescharacteristic
69C2×Dic113C2311yescharacteristic
70C2×Dic113C2311yescharacteristic
71D452—22no—
72D452—22no—
73D452—41no—
74D452—41no—
75D452—41no—
76C4×D113—41no—
77C4×D113—41no—
78C4×D113—41no—
79C4×D113—41no—
80Dic226—41no—
81C22×C113—21no—
82C22×C113—21no—
83C22×C113—21no—
84C22×C113C2411yescharacteristic
85C2×D113—21no—
86C2×D113—21no—
87C2×D113—21no—
88C2×D113—21no—
89C2×D113—21no—
90C2×D113—21no—
91C2×D113—21no—
92C2×D113—21no—
93C2×D113—21no—
94C2×D113—21no—
95C2×D113—21no—
96C2×D113—44no—
97C2×D113#4112yes—
98C4×C113—21no—
99C4×C113—21no—
100C4×C113—21no—
101C4×C113—21no—
102C4×C113—21no—
103C4×C113—21no—
104Dic113—21no—
105Dic113—21no—
106Dic113—21no—
107Dic113—21no—
108C2×C113—41no—
109C2×C113C2×#4111yescharacteristic
110C2×C113H(5,𝔽2)11yescharacteristic
111C2×C113H(5,𝔽2)11yescharacteristic
112D113—22no—
113D113—41no—
114D113—41no—
115D113—41no—
116C113#53811yescharacteristic, complemented, Sylow
117#538—1131nomaximal, Sylow
118#126—1131no—
119#126D11311yescharacteristic, complemented
120#124—1131no—
121#124—1131no—
122#123—1131no—
123#123—1131no—
124C22⋊Q8—1131no—
125#121—1131no—
126#121—1131no—
127#121—1131no—
128#120—1131no—
129#120—1131no—
130C4×D4—1131no—
131C4×D4—1131no—
132C2×C22⋊C4—1131no—
133C24—1131no—
134C2×Q8—1131no—
135C2×D4—1131no—
136C2×D4—1131no—
137C2×D4—1131no—
138C2×D4—1131no—
139C2×D4—1131no—
140C2×D4—1131no—
141C2×D4—1131no—
142C22×C4—1131no—
143C22×C4—1131no—
144C22×C4—1131no—
145C22×C4—1131no—
146C4⋊C4—1131no—
147C4⋊C4—1131no—
148C4⋊C4—1131no—
149C4⋊C4C2×D11311yescharacteristic, complemented
150C4⋊C4C2×D11311yescharacteristic, complemented
151C4⋊C4C2×D11311yescharacteristic, complemented
152C22⋊C4—1131no—
153C22⋊C4—1131no—
154C22⋊C4—1131no—
155C22⋊C4—1131no—
156C22⋊C4—1131no—
157C22⋊C4—1131no—
158C22⋊C4—1131no—
159C22⋊C4—1131no—
160C22⋊C4—1131no—
161C22⋊C4—1131no—
162C22⋊C4—1131no—
163C22⋊C4C2×D11311yescharacteristic
164C22⋊C4C2×D11311yescharacteristic
165C22⋊C4C2×D11311yescharacteristic
166C42—1131no—
167C42C2×D11311yescharacteristic, complemented
168C23—1131no—
169C23—1131no—
170C23—1131no—
171C23—1131no—
172C23—4521no—
173C23—4521no—
174C23—2262no—
175C23C22×D11311yescharacteristic
176Q8—4521no—
177D4—4521no—
178D4—4521no—
179D4—4521no—
180D4—4521no—
181D4—4521no—
182D4—2262no—
183D4—2262no—
184C2×C4—1131no—
185C2×C4—1131no—
186C2×C4—1131no—
187C2×C4—1131no—
188C2×C4—4521no—
189C2×C4—4521no—
190C2×C4—4521no—
191C2×C4—4521no—
192C2×C4C22×D11311yescharacteristic
193C2×C4C22×D11311yescharacteristic
194C2×C4C22×D11311yescharacteristic
195C2×C4C22×D11311yescharacteristic
196C2×C4C22×D11311yescharacteristic
197C2×C4C22×D11311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—1132no—
202C22—4524no—
203C22—2261no—
204C22—2261no—
205C22—2261no—
206C22—2261no—
207C22—2261no—
208C22—2261no—
209C22—2261no—
210C22—2261no—
211C22—2261no—
212C22—2261no—
213C22—2261no—
214C22C23×D11311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—2261no—
222C4—2261no—
223C4—2261no—
224C4—2261no—
225C2—41no—
226C2—4521no—
227C2—4521no—
228C2—4521no—
229C2—2262no—
230C2#41×D11311yescharacteristic
231C2#1681366811yescharacteristic
232C2#1681366811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
Fitting#126×C113C2
hypercenter#126D113
hypocenterC113#538
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #16813617C2semidirect
2. #16813610C2semidirect
3. #16813608C2semidirect
4. #16813607C2semidirect
5. #16813604C2semidirect
6. #16813600C2semidirect
7. #16813597C2semidirect
8. #16813597C2semidirect
9. #16813595C2semidirect
10. #16813592C2semidirect
11. #16813592C2semidirect
12. C4×D452C2semidirect
13. #16813584C2semidirect
14. C22⋊C4×D113C2semidirect
15. #126×C113C2semidirect
17. C2×Dic226C22semidirect
22. C2×D452C22semidirect
27. C2×C4×D113C22semidirect
28. C2×C4×D113C22semidirect
29. C4⋊Dic113C22semidirect
30. #14268626C22semidirect
31. #14268626C22semidirect
39. #14268624C22semidirect
40. #14268624C22semidirect
41. #14268624C22semidirect
42. #14268624C22semidirect
43. C4×Dic113C22semidirect
44. C4⋊C4×C113C22semidirect
45. C4⋊C4×C113C22semidirect
46. C4⋊C4×C113C22semidirect
50. C42×C113C22semidirect
116. C113#538semidirect
119. #126D113semidirect
149. C4⋊C4C2×D113semidirect
150. C4⋊C4C2×D113semidirect
151. C4⋊C4C2×D113semidirect
167. C42C2×D113semidirect