Finite Group Databasebeta

← #22550614

Subgroups — #22550614

Order 8128.

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#16861205C211yesmaximal, characteristic, complemented
2#16861204C211yesmaximal, characteristic, complemented
3#16861202C211yesmaximal, characteristic, complemented
4#16861194C211yesmaximal, characteristic, complemented
5#16861193C211yesmaximal, characteristic, complemented
6#16861192C211yesmaximal, characteristic, complemented
7#16861183C211yesmaximal, characteristic, complemented
8#16861182C211yesmaximal, characteristic, complemented
9#16861181C211yesmaximal, characteristic, complemented
10#16861181C211yesmaximal, characteristic, complemented
11#16861180C211yesmaximal, characteristic, complemented
12#16861179C211yesmaximal, characteristic, complemented
13#16861177C211yesmaximal, characteristic, complemented
14#16861177C211yesmaximal, characteristic, complemented
15#121×C127C211yesmaximal, characteristic, complemented
16C23×D127C2211yescharacteristic
17C2×D508C2211yescharacteristic
18C2×D508C2211yescharacteristic
19C2×C22⋊D127C2211yescharacteristic
20C2×C22⋊D127C2211yescharacteristic
21C2×C22⋊D127C2211yescharacteristic
22C2×C22⋊D127C2211yescharacteristic
23C2×C22⋊D127C2211yescharacteristic
24C2×C22⋊D127C2211yescharacteristic, complemented
25C2×C22⋊D127C2211yescharacteristic, complemented
26C22×Dic127C2211yescharacteristic, complemented
27C2×C4×D127C2211yescharacteristic, complemented
28C4⋊Dic127C2211yescharacteristic, complemented
29C4⋊Dic127C2211yescharacteristic, complemented
30Dic127⋊C4C2211yescharacteristic, complemented
31Dic127⋊C4C2211yescharacteristic, complemented
32Dic127⋊C4C2211yescharacteristic, complemented
33C22⋊Dic127C2211yescharacteristic
34C22⋊Dic127C2211yescharacteristic
35C22⋊Dic127C2211yescharacteristic, complemented
36(C2×D127)⋊C4C2211yescharacteristic
37(C2×D127)⋊C4C2211yescharacteristic
38(C2×D127)⋊C4C2211yescharacteristic
39(C2×D127)⋊C4C2211yescharacteristic
40(C2×D127)⋊C4C2211yescharacteristic
41(C2×D127)⋊C4C2211yescharacteristic
42(C2×D127)⋊C4C2211yescharacteristic, complemented
43C4×Dic127C2211yescharacteristic, complemented
44C2×D4×C127C2211yescharacteristic
45C2×D4×C127C2211yescharacteristic
46C2×D4×C127C2211yescharacteristic, complemented
47C22×C4×C127C2211yescharacteristic, complemented
48C4⋊C4×C127C2211yescharacteristic, complemented
49C22⋊C4×C127C2211yescharacteristic
50C22⋊C4×C127C2211yescharacteristic, complemented
51C22×D127—41no—
52C22×D127—41no—
53C22×D127—41no—
54C22×D127C2311yescharacteristic
55C22×D127C2311yescharacteristic
56C22×D127C2311yescharacteristic
57D508—41no—
58D508—41no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C22⋊D127—41no—
62C22⋊D127—41no—
63C22⋊D127—41no—
64C22⋊D127—41no—
65C22⋊D127—41no—
66C2×Dic127—41no—
67C2×Dic127C2311yescharacteristic
68C2×Dic127C2311yescharacteristic
69C2×Dic127C2311yescharacteristic
70C2×Dic127C2311yescharacteristic
71C2×Dic127C2311yescharacteristic
72C4×D127—41no—
73C23×C127C2311yescharacteristic
74C23×C127C2311yescharacteristic
75C23×C127C2311yescharacteristic
76D4×C127—41no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127C2311yescharacteristic
81C2×C4×C127C2311yescharacteristic
82C2×C4×C127C2311yescharacteristic
83C2×C4×C127C2311yescharacteristic
84C22×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127C2411yescharacteristic
94C2×D127—21no—
95C2×D127—21no—
96C2×D127—21no—
97C2×D127—21no—
98C2×D127—21no—
99C2×D127—21no—
100C2×D127—21no—
101C2×D127—21no—
102C2×D127—21no—
103C2×D127—44no—
104C4×C127—21no—
105C4×C127—21no—
106C4×C127—21no—
107C4×C127—21no—
108Dic127—21no—
109Dic127—21no—
110Dic127—21no—
111Dic127—21no—
112Dic127—21no—
113C2×C127—41no—
114C2×C127—41no—
115C2×C127—41no—
116C2×C127H(5,𝔽2)11yescharacteristic
117C2×C127H(5,𝔽2)11yescharacteristic
118C2×C127H(5,𝔽2)11yescharacteristic
119D127—41no—
120D127—41no—
121D127—41no—
122C127#56011yescharacteristic, complemented, Sylow
123#560—1271nomaximal, Sylow
124#127—1271no—
125#126—1271no—
126#126—1271no—
127#123—1271no—
128#123—1271no—
129#123—1271no—
130#121—1271no—
131#121—1271no—
132#121—1271no—
133#121—1271no—
134#121—1271no—
135#121D12711yescharacteristic, complemented
136#120—1271no—
137#120—1271no—
138#120—1271no—
139C24—1271no—
140C2×D4—1271no—
141C2×D4—1271no—
142C2×D4—1271no—
143C2×D4—1271no—
144C2×D4—1271no—
145C2×D4—1271no—
146C2×D4—1271no—
147C2×D4—1271no—
148C2×D4—1271no—
149C2×D4C2×D12711yescharacteristic
150C2×D4C2×D12711yescharacteristic
151C2×D4C2×D12711yescharacteristic, complemented
152C22×C4—1271no—
153C22×C4—1271no—
154C22×C4C2×D12711yescharacteristic, complemented
155C4⋊C4—1271no—
156C4⋊C4—1271no—
157C4⋊C4—1271no—
158C4⋊C4—1271no—
159C4⋊C4—1271no—
160C4⋊C4C2×D12711yescharacteristic, complemented
161C22⋊C4—1271no—
162C22⋊C4—1271no—
163C22⋊C4—1271no—
164C22⋊C4—1271no—
165C22⋊C4—1271no—
166C22⋊C4—1271no—
167C22⋊C4—1271no—
168C22⋊C4—1271no—
169C22⋊C4—1271no—
170C22⋊C4—1271no—
171C22⋊C4C2×D12711yescharacteristic
172C22⋊C4C2×D12711yescharacteristic, complemented
173C42—1271no—
174C23—1271no—
175C23—1271no—
176C23—1271no—
177C23—5081no—
178C23—5081no—
179C23—5081no—
180C23C22×D12711yescharacteristic
181C23C22×D12711yescharacteristic
182C23C22×D12711yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—5081no—
187D4—5081no—
188D4—5081no—
189D4—5081no—
190D4—5081no—
191D4—5081no—
192D4—5081no—
193D4—5081no—
194D4—5081no—
195C2×C4—41no—
196C2×C4—1271no—
197C2×C4—1271no—
198C2×C4—1271no—
199C2×C4—1271no—
200C2×C4—1271no—
201C2×C4—5081no—
202C2×C4—5081no—
203C2×C4C22×D12711yescharacteristic
204C2×C4C22×D12711yescharacteristic
205C2×C4C22×D12711yescharacteristic
206C2×C4C22×D12711yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—5084no—
217C22—2541no—
218C22—2541no—
219C22—2541no—
220C22—2541no—
221C22—2541no—
222C22—2541no—
223C22—2541no—
224C22—2541no—
225C22—2541no—
226C22C23×D12711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—2541no—
232C4—2541no—
233C4—2541no—
234C4—2541no—
235C4—2541no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—5081no—
240C2—5081no—
241C2—5081no—
242C2#1686125411yescharacteristic
243C2#1686125411yescharacteristic
244C2#1686125511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#121×C127C2
hypercenter#121D127
hypocenterC127#560
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #16861205C2semidirect
2. #16861204C2semidirect
3. #16861202C2semidirect
4. #16861194C2semidirect
5. #16861193C2semidirect
6. #16861192C2semidirect
7. #16861183C2semidirect
8. #16861182C2semidirect
9. #16861181C2semidirect
10. #16861181C2semidirect
11. #16861180C2semidirect
12. #16861179C2semidirect
13. #16861177C2semidirect
14. #16861177C2semidirect
15. #121×C127C2semidirect
24. C2×C22⋊D127C22semidirect
25. C2×C22⋊D127C22semidirect
26. C22×Dic127C22semidirect
27. C2×C4×D127C22semidirect
28. C4⋊Dic127C22semidirect
29. C4⋊Dic127C22semidirect
30. Dic127⋊C4C22semidirect
31. Dic127⋊C4C22semidirect
32. Dic127⋊C4C22semidirect
35. C22⋊Dic127C22semidirect
42. (C2×D127)⋊C4C22semidirect
43. C4×Dic127C22semidirect
46. C2×D4×C127C22semidirect
47. C22×C4×C127C22semidirect
48. C4⋊C4×C127C22semidirect
50. C22⋊C4×C127C22semidirect
122. C127#560semidirect
135. #121D127semidirect
151. C2×D4C2×D127semidirect
154. C22×C4C2×D127semidirect
160. C4⋊C4C2×D127semidirect
172. C22⋊C4C2×D127semidirect