Finite Group Databasebeta

← #22550465

Subgroups — #22550465

Order 8128.

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#16861204C211yesmaximal, characteristic, complemented
2#16861197C211yesmaximal, characteristic, complemented
3#16861195C211yesmaximal, characteristic, complemented
4#16861194C211yesmaximal, characteristic, complemented
5#16861191C211yesmaximal, characteristic, complemented
6#16861187C211yesmaximal, characteristic, complemented
7#16861184C211yesmaximal, characteristic, complemented
8#16861184C211yesmaximal, characteristic, complemented
9#16861182C211yesmaximal, characteristic, complemented
10#16861179C211yesmaximal, characteristic, complemented
11#16861179C211yesmaximal, characteristic, complemented
12C4×D508C211yesmaximal, characteristic, complemented
13#16861171C211yesmaximal, characteristic, complemented
14C22⋊C4×D127C211yesmaximal, characteristic, complemented
15#126×C127C211yesmaximal, characteristic, complemented
16C23×D127C2211yescharacteristic
17C2×Dic254C2211yescharacteristic, complemented
18C2×D508C2211yescharacteristic
19C2×D508C2211yescharacteristic
20C2×D508C2211yescharacteristic
21C2×D508C2211yescharacteristic
22C2×D508C2211yescharacteristic, complemented
23C2×C22⋊D127C2211yescharacteristic
24C2×C22⋊D127C2211yescharacteristic
25C2×C4×D127C2211yescharacteristic
26C2×C4×D127C2211yescharacteristic
27C2×C4×D127C2211yescharacteristic, complemented
28C2×C4×D127C2211yescharacteristic, complemented
29C4⋊Dic127C2211yescharacteristic, complemented
30Dic127⋊C4C2211yescharacteristic, complemented
31Dic127⋊C4C2211yescharacteristic, complemented
32C22⋊Dic127C2211yescharacteristic
33(C2×D127)⋊C4C2211yescharacteristic
34(C2×D127)⋊C4C2211yescharacteristic
35(C2×D127)⋊C4C2211yescharacteristic
36(C2×D127)⋊C4C2211yescharacteristic
37(C2×D127)⋊C4C2211yescharacteristic
38(C2×D127)⋊C4C2211yescharacteristic
39(C2×D127)⋊C4C2211yescharacteristic, complemented
40(C2×D127)⋊C4C2211yescharacteristic, complemented
41(C2×D127)⋊C4C2211yescharacteristic, complemented
42(C2×D127)⋊C4C2211yescharacteristic, complemented
43C4×Dic127C2211yescharacteristic, complemented
44C4⋊C4×C127C2211yescharacteristic, complemented
45C4⋊C4×C127C2211yescharacteristic, complemented
46C4⋊C4×C127C2211yescharacteristic, complemented
47C22⋊C4×C127C2211yescharacteristic
48C22⋊C4×C127C2211yescharacteristic
49C22⋊C4×C127C2211yescharacteristic
50C42×C127C2211yescharacteristic, complemented
51C22×D127—22no—
52C22×D127—41no—
53C22×D127—41no—
54C22×D127C2311yescharacteristic
55C22×D127C2311yescharacteristic
56C22×D127C2311yescharacteristic
57C22×D127C2311yescharacteristic
58Dic254—41no—
59D508—22no—
60D508—22no—
61D508—41no—
62D508—41no—
63D508—41no—
64C22⋊D127—41no—
65C22⋊D127—41no—
66C2×Dic127C2311yescharacteristic
67C2×Dic127C2311yescharacteristic
68C2×Dic127C2311yescharacteristic
69C2×Dic127C2311yescharacteristic
70C4×D127—41no—
71C4×D127—41no—
72C4×D127—41no—
73C4×D127—41no—
74C23×C127C2311yescharacteristic
75C2×C4×C127C2311yescharacteristic
76C2×C4×C127C2311yescharacteristic
77C2×C4×C127C2311yescharacteristic
78C2×C4×C127C2311yescharacteristic
79C2×C4×C127C2311yescharacteristic
80C2×C4×C127C2311yescharacteristic
81C22×C127—21no—
82C22×C127—21no—
83C22×C127—21no—
84C22×C127C2411yescharacteristic
85C2×D127—21no—
86C2×D127—21no—
87C2×D127—21no—
88C2×D127—21no—
89C2×D127—21no—
90C2×D127—21no—
91C2×D127—21no—
92C2×D127—21no—
93C2×D127—21no—
94C2×D127—21no—
95C2×D127—21no—
96C2×D127—44no—
97C2×D127#4112yes—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—21no—
103C4×C127—21no—
104Dic127—21no—
105Dic127—21no—
106Dic127—21no—
107Dic127—21no—
108C2×C127—41no—
109C2×C127C2×#4111yescharacteristic
110C2×C127H(5,𝔽2)11yescharacteristic
111C2×C127H(5,𝔽2)11yescharacteristic
112D127—22no—
113D127—41no—
114D127—41no—
115D127—41no—
116C127#53811yescharacteristic, complemented, Sylow
117#538—1271nomaximal, Sylow
118#126—1271no—
119#126D12711yescharacteristic, complemented
120#124—1271no—
121#124—1271no—
122#123—1271no—
123#123—1271no—
124C22⋊Q8—1271no—
125#121—1271no—
126#121—1271no—
127#121—1271no—
128#120—1271no—
129#120—1271no—
130C4×D4—1271no—
131C4×D4—1271no—
132C2×C22⋊C4—1271no—
133C24—1271no—
134C2×Q8—1271no—
135C2×D4—1271no—
136C2×D4—1271no—
137C2×D4—1271no—
138C2×D4—1271no—
139C2×D4—1271no—
140C2×D4—1271no—
141C2×D4—1271no—
142C22×C4—1271no—
143C22×C4—1271no—
144C22×C4—1271no—
145C22×C4—1271no—
146C4⋊C4—1271no—
147C4⋊C4—1271no—
148C4⋊C4—1271no—
149C4⋊C4C2×D12711yescharacteristic, complemented
150C4⋊C4C2×D12711yescharacteristic, complemented
151C4⋊C4C2×D12711yescharacteristic, complemented
152C22⋊C4—1271no—
153C22⋊C4—1271no—
154C22⋊C4—1271no—
155C22⋊C4—1271no—
156C22⋊C4—1271no—
157C22⋊C4—1271no—
158C22⋊C4—1271no—
159C22⋊C4—1271no—
160C22⋊C4—1271no—
161C22⋊C4—1271no—
162C22⋊C4—1271no—
163C22⋊C4C2×D12711yescharacteristic
164C22⋊C4C2×D12711yescharacteristic
165C22⋊C4C2×D12711yescharacteristic
166C42—1271no—
167C42C2×D12711yescharacteristic, complemented
168C23—1271no—
169C23—1271no—
170C23—1271no—
171C23—1271no—
172C23—5081no—
173C23—5081no—
174C23—2542no—
175C23C22×D12711yescharacteristic
176Q8—5081no—
177D4—5081no—
178D4—5081no—
179D4—5081no—
180D4—5081no—
181D4—5081no—
182D4—2542no—
183D4—2542no—
184C2×C4—1271no—
185C2×C4—1271no—
186C2×C4—1271no—
187C2×C4—1271no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4C22×D12711yescharacteristic
193C2×C4C22×D12711yescharacteristic
194C2×C4C22×D12711yescharacteristic
195C2×C4C22×D12711yescharacteristic
196C2×C4C22×D12711yescharacteristic
197C2×C4C22×D12711yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—1272no—
202C22—5084no—
203C22—2541no—
204C22—2541no—
205C22—2541no—
206C22—2541no—
207C22—2541no—
208C22—2541no—
209C22—2541no—
210C22—2541no—
211C22—2541no—
212C22—2541no—
213C22—2541no—
214C22C23×D12711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—2541no—
222C4—2541no—
223C4—2541no—
224C4—2541no—
225C2—41no—
226C2—5081no—
227C2—5081no—
228C2—5081no—
229C2—2542no—
230C2#41×D12711yescharacteristic
231C2#1686125511yescharacteristic
232C2#1686125511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#126×C127C2
hypercenter#126D127
hypocenterC127#538
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #16861204C2semidirect
2. #16861197C2semidirect
3. #16861195C2semidirect
4. #16861194C2semidirect
5. #16861191C2semidirect
6. #16861187C2semidirect
7. #16861184C2semidirect
8. #16861184C2semidirect
9. #16861182C2semidirect
10. #16861179C2semidirect
11. #16861179C2semidirect
12. C4×D508C2semidirect
13. #16861171C2semidirect
14. C22⋊C4×D127C2semidirect
15. #126×C127C2semidirect
17. C2×Dic254C22semidirect
22. C2×D508C22semidirect
27. C2×C4×D127C22semidirect
28. C2×C4×D127C22semidirect
29. C4⋊Dic127C22semidirect
30. Dic127⋊C4C22semidirect
31. Dic127⋊C4C22semidirect
39. (C2×D127)⋊C4C22semidirect
40. (C2×D127)⋊C4C22semidirect
41. (C2×D127)⋊C4C22semidirect
42. (C2×D127)⋊C4C22semidirect
43. C4×Dic127C22semidirect
44. C4⋊C4×C127C22semidirect
45. C4⋊C4×C127C22semidirect
46. C4⋊C4×C127C22semidirect
50. C42×C127C22semidirect
116. C127#538semidirect
119. #126D127semidirect
149. C4⋊C4C2×D127semidirect
150. C4⋊C4C2×D127semidirect
151. C4⋊C4C2×D127semidirect
167. C42C2×D127semidirect