Finite Group Databasebeta

← #20299107

Subgroups — #20299107

Order 7232.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14268647C211yesmaximal, characteristic, complemented
2C22×C4×D113C211yesmaximal, characteristic, complemented
3#16813607C211yesmaximal, characteristic, complemented
4#16813604C211yesmaximal, characteristic, complemented
5#16813600C211yesmaximal, characteristic, complemented
6#16813599C211yesmaximal, characteristic, complemented
7#16813595C211yesmaximal, characteristic, complemented
8#16813594C211yesmaximal, characteristic, complemented
9#16813592C211yesmaximal, characteristic, complemented
10#16813590C211yesmaximal, characteristic, complemented
11C4×D452C211yesmaximal, characteristic, complemented
12C4×C22⋊D113C211yesmaximal, characteristic, complemented
13C4×C22⋊D113C211yesmaximal, characteristic, complemented
14#16813579C211yesmaximal, characteristic, complemented
15C4×D4×C113C211yesmaximal, characteristic, complemented
16C23×D113C2211yescharacteristic
17#14268647—24no—
18C2×Dic226C2211yescharacteristic, complemented
19C2×D452C2211yescharacteristic
20C2×D452C2211yescharacteristic
21C2×C22⋊D113C2211yescharacteristic
22C2×C22⋊D113C2211yescharacteristic
23C2×C22⋊D113C2211yescharacteristic
24C2×C22⋊D113C2211yescharacteristic, complemented
25C22×Dic113C2211yescharacteristic
26C2×C4×D113—24no—
27C2×C4×D113C2211yescharacteristic
28C2×C4×D113C2211yescharacteristic
29C2×C4×D113C2211yescharacteristic
30C2×C4×D113C2211yescharacteristic
31C2×C4×D113C2211yescharacteristic, complemented
32C4⋊Dic113C2211yescharacteristic, complemented
33#14268626C2211yescharacteristic, complemented
34#14268626C2211yescharacteristic, complemented
35#14268626C2211yescharacteristic, complemented
36#14268626C2211yescharacteristic, complemented
37C22⋊Dic113C2211yescharacteristic
38C22⋊Dic113C2211yescharacteristic, complemented
39#14268624C2211yescharacteristic
40#14268624C2211yescharacteristic
41#14268624C2211yescharacteristic
42#14268624C2211yescharacteristic
43#14268624C2211yescharacteristic, complemented
44#14268624C2211yescharacteristic, complemented
45C4×Dic113C2211yescharacteristic, complemented
46C2×D4×C113C2211yescharacteristic
47C22×C4×C113C2211yescharacteristic
48C22×C4×C113C2211yescharacteristic, complemented
49C4⋊C4×C113C2211yescharacteristic, complemented
50C22⋊C4×C113C2211yescharacteristic
51C22⋊C4×C113C2211yescharacteristic, complemented
52C42×C113C2211yescharacteristic, complemented
53C23×C113C2311yescharacteristic
54C23×C113C2311yescharacteristic
55C22×D113—22no—
56C22×D113—22no—
57C22×D113—22no—
58C22×D113C2311yescharacteristic
59C22×D113C2311yescharacteristic
60C22×D113C2311yescharacteristic
61D4×C113—22no—
62C2×C4×C113—22no—
63C2×C4×C113—22no—
64C2×C4×C113C2311yescharacteristic
65C2×C4×C113C2311yescharacteristic
66C2×C4×C113C2311yescharacteristic
67C2×C4×C113C2311yescharacteristic
68C2×C4×C113C2311yescharacteristic
69C22⋊D113—22no—
70C22⋊D113—22no—
71C22⋊D113—22no—
72C22⋊D113—22no—
73C2×Dic113—22no—
74C2×Dic113C2311yescharacteristic
75C2×Dic113C2311yescharacteristic
76C2×Dic113C2311yescharacteristic
77C2×Dic113C2311yescharacteristic
78C2×Dic113C2311yescharacteristic
79D452—22no—
80D452—22no—
81C4×D113—22no—
82C4×D113—22no—
83C4×D113—22no—
84C4×D113—22no—
85C4×D113D414yescomplemented
86Dic226—22no—
87C22×C113—21no—
88C22×C113—21no—
89C22×C113—21no—
90C22×C113—21no—
91C22×C113—21no—
92C22×C113#4112yescomplemented
93C22×C113C2411yescharacteristic
94C2×D113—28no—
95C2×D113—21no—
96C2×D113—21no—
97C2×D113—21no—
98C2×D113—21no—
99C2×D113—21no—
100C2×D113—21no—
101C2×D113—21no—
102C2×D113C2×D412yes—
103C2×D113#4112yescomplemented
104C4×C113—21no—
105C4×C113—21no—
106C4×C113—21no—
107C4×C113—21no—
108C4×C113C2×D412yes—
109Dic113—21no—
110Dic113—21no—
111Dic113—21no—
112Dic113—21no—
113Dic113C2×D412yes—
114C2×C113—22no—
115C2×C113—41no—
116C2×C113C22×D411yescharacteristic
117C2×C113C2×#4111yescharacteristic
118C2×C113C2×#4111yescharacteristic
119D113—22no—
120D113—22no—
121D113—41no—
122C113#52511yescharacteristic, complemented, Sylow
123#525—1131nomaximal, Sylow
124C2×#41—1131no—
125C23×C4—1131no—
126#123—1131no—
127#123—1131no—
128C22⋊Q8—1131no—
129C22⋊Q8—1131no—
130#121—1131no—
131#121—1131no—
132#120—1131no—
133#120—1131no—
134C4×D4—1131no—
135C4×D4—1131no—
136C4×D4—1131no—
137C4×D4D11311yescharacteristic, complemented
138#117—1131no—
139C24—1131no—
140#41—2264no—
141C2×Q8—1131no—
142C2×D4—1131no—
143C2×D4—1131no—
144C2×D4—1131no—
145C2×D4—1131no—
146C2×D4—1131no—
147C2×D4—1131no—
148C2×D4C2×D11311yescharacteristic
149C22×C4—1131no—
150C22×C4—1131no—
151C22×C4—1131no—
152C22×C4—1131no—
153C22×C4—1131no—
154C22×C4—1131no—
155C22×C4—2264no—
156C22×C4C2×D11311yescharacteristic
157C22×C4C2×D11311yescharacteristic, complemented
158C4⋊C4—1131no—
159C4⋊C4—1131no—
160C4⋊C4—1131no—
161C4⋊C4—1131no—
162C4⋊C4—1131no—
163C4⋊C4C2×D11311yescharacteristic, complemented
164C22⋊C4—1131no—
165C22⋊C4—1131no—
166C22⋊C4—1131no—
167C22⋊C4—1131no—
168C22⋊C4—1131no—
169C22⋊C4—1131no—
170C22⋊C4—1131no—
171C22⋊C4—1131no—
172C22⋊C4C2×D11311yescharacteristic
173C22⋊C4C2×D11311yescharacteristic, complemented
174C42—1131no—
175C42C2×D11311yescharacteristic, complemented
176C23—1131no—
177C23—1131no—
178C23—1131no—
179C23—2262no—
180C23—2262no—
181C23—2262no—
182C23C22×D11311yescharacteristic
183C23C22×D11311yescharacteristic
184Q8—2262no—
185D4—22no—
186D4—2262no—
187D4—2262no—
188D4—2262no—
189D4—2262no—
190D4—2262no—
191D4—2262no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—1131no—
195C2×C4—1134no—
196C2×C4—1131no—
197C2×C4—1131no—
198C2×C4—1131no—
199C2×C4—1131no—
200C2×C4—2262no—
201C2×C4—2262no—
202C2×C4—2262no—
203C2×C4—2262no—
204C2×C4—2262no—
205C2×C4C22×D11311yescharacteristic
206C2×C4C22×D11311yescharacteristic
207C2×C4C22×D11311yescharacteristic
208C2×C4C22×D11311yescharacteristic
209C2×C4C22×D11311yescharacteristic
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—1132no—
216C22—1132no—
217C22—2268no—
218C22—2261no—
219C22—2261no—
220C22—2261no—
221C22—2261no—
222C22—2261no—
223C22—2261no—
224C22—2261no—
225C22#1426864712yescomplemented
226C22C23×D11311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—1132no—
232C4—2261no—
233C4—2261no—
234C4—2261no—
235C4—2261no—
236C4D4×D11312yes—
237C2—22no—
238C2—41no—
239C2—4521no—
240C2—2262no—
241C2—2262no—
242C2C2×D4×D11311yescharacteristic
243C2#41×D11311yescharacteristic
244C2C2×#1426864711yescharacteristic
subgroupG/H
centerC2×C4C22×D113
commutator, socleC22×C113C24
FrattiniC22C23×D113
FittingC4×D4×C113C2
hypercenterC4×D4D113
hypocenterC113#525
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14268647C2semidirect
2. C22×C4×D113C2semidirect
3. #16813607C2semidirect
4. #16813604C2semidirect
5. #16813600C2semidirect
6. #16813599C2semidirect
7. #16813595C2semidirect
8. #16813594C2semidirect
9. #16813592C2semidirect
10. #16813590C2semidirect
11. C4×D452C2semidirect
12. C4×C22⋊D113C2semidirect
13. C4×C22⋊D113C2semidirect
14. #16813579C2semidirect
15. C4×D4×C113C2semidirect
18. C2×Dic226C22semidirect
24. C2×C22⋊D113C22semidirect
31. C2×C4×D113C22semidirect
32. C4⋊Dic113C22semidirect
33. #14268626C22semidirect
34. #14268626C22semidirect
35. #14268626C22semidirect
36. #14268626C22semidirect
38. C22⋊Dic113C22semidirect
43. #14268624C22semidirect
44. #14268624C22semidirect
45. C4×Dic113C22semidirect
48. C22×C4×C113C22semidirect
49. C4⋊C4×C113C22semidirect
51. C22⋊C4×C113C22semidirect
52. C42×C113C22semidirect
85. C4×D113D4semidirect
92. C22×C113#41semidirect
103. C2×D113#41semidirect
122. C113#525semidirect
137. C4×D4D113semidirect
157. C22×C4C2×D113semidirect
163. C4⋊C4C2×D113semidirect
173. C22⋊C4C2×D113semidirect
175. C42C2×D113semidirect
225. C22#14268647semidirect