Finite Group Databasebeta

← #22550458

Subgroups — #22550458

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#16861202C211yesmaximal, characteristic, complemented
3#16861196C211yesmaximal, characteristic, complemented
4#16861195C211yesmaximal, characteristic, complemented
5#16861192C211yesmaximal, characteristic, complemented
6#16861190C211yesmaximal, characteristic, complemented
7#16861187C211yesmaximal, characteristic, complemented
8#16861181C211yesmaximal, characteristic, complemented
9#16861180C211yesmaximal, characteristic, complemented
10#16861177C211yesmaximal, characteristic, complemented
11#16861177C211yesmaximal, characteristic, complemented
12C4×C22⋊D127C211yesmaximal, characteristic, complemented
13(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D127C211yesmaximal, characteristic, complemented
15#121×C127C211yesmaximal, characteristic, complemented
16C23×D127C2211yescharacteristic
17C2×Dic254C2211yescharacteristic, complemented
18C2×C22⋊D127C2211yescharacteristic
19C2×C22⋊D127C2211yescharacteristic
20C2×C22⋊D127C2211yescharacteristic
21C2×C22⋊D127C2211yescharacteristic
22C22×Dic127C2211yescharacteristic, complemented
23C2×C4×D127C2211yescharacteristic
24C2×C4×D127C2211yescharacteristic
25C4⋊Dic127C2211yescharacteristic, complemented
26Dic127⋊C4C2211yescharacteristic, complemented
27Dic127⋊C4C2211yescharacteristic, complemented
28Dic127⋊C4C2211yescharacteristic, complemented
29Dic127⋊C4C2211yescharacteristic, complemented
30C22⋊Dic127C2211yescharacteristic
31C22⋊Dic127C2211yescharacteristic
32C22⋊Dic127C2211yescharacteristic
33C22⋊Dic127C2211yescharacteristic, complemented
34C22⋊Dic127C2211yescharacteristic, complemented
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
42C4×Dic127C2211yescharacteristic, 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—22no—
52C22×D127—41no—
53C22×D127—41no—
54C22×D127C2311yescharacteristic
55C22×D127C2311yescharacteristic
56Dic254—41no—
57C22⋊D127—22no—
58C22⋊D127—22no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C2×Dic127—41no—
62C2×Dic127C2311yescharacteristic
63C2×Dic127C2311yescharacteristic
64C2×Dic127C2311yescharacteristic
65C2×Dic127C2311yescharacteristic
66C2×Dic127C2311yescharacteristic
67C2×Dic127C2311yescharacteristic
68C4×D127—41no—
69C4×D127—41no—
70C23×C127C2311yescharacteristic
71C23×C127C2311yescharacteristic
72C23×C127C2311yescharacteristic
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127C2311yescharacteristic
78C2×C4×C127C2311yescharacteristic
79C2×C4×C127C2311yescharacteristic
80C2×C4×C127C2311yescharacteristic
81C22×C127—21no—
82C22×C127—21no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127C2411yescharacteristic
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—
102Dic127—21no—
103Dic127—21no—
104Dic127—21no—
105Dic127—21no—
106Dic127—21no—
107Dic127—21no—
108C2×C127—41no—
109C2×C127—41no—
110C2×C127—41no—
111C2×C127C2×#4111yescharacteristic
112C2×C127H(5,𝔽2)11yescharacteristic
113C2×C127H(5,𝔽2)11yescharacteristic
114D127—22no—
115D127—41no—
116C127#53811yescharacteristic, complemented, Sylow
117#538—1271nomaximal, Sylow
118#126—1271no—
119#126—1271no—
120#124—1271no—
121#124—1271no—
122#123—1271no—
123#123—1271no—
124C22⋊Q8—1271no—
125#121—1271no—
126#121—1271no—
127#121D12711yescharacteristic, complemented
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×D4C2×D12711yescharacteristic
140C2×D4C2×D12711yescharacteristic
141C2×D4C2×D12711yescharacteristic, complemented
142C22×C4—1271no—
143C22×C4—1271no—
144C22×C4—1271no—
145C22×C4C2×D12711yescharacteristic, complemented
146C4⋊C4—1271no—
147C4⋊C4—1271no—
148C4⋊C4—1271no—
149C4⋊C4—1271no—
150C4⋊C4—1271no—
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⋊C4—1271no—
164C22⋊C4C2×D12711yescharacteristic
165C22⋊C4C2×D12711yescharacteristic, complemented
166C42—1271no—
167C42—1271no—
168C23—1271no—
169C23—1271no—
170C23—5081no—
171C23—5081no—
172C23—2542no—
173C23C22×D12711yescharacteristic
174C23C22×D12711yescharacteristic
175C23C22×D12711yescharacteristic
176Q8—5081no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—5081no—
181D4—5081no—
182D4—2542no—
183D4—2542no—
184C2×C4—41no—
185C2×C4—1271no—
186C2×C4—1271no—
187C2×C4—1271no—
188C2×C4—1271no—
189C2×C4—1271no—
190C2×C4—1271no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4C22×D12711yescharacteristic
195C2×C4C22×D12711yescharacteristic
196C2×C4C22×D12711yescharacteristic
197C2×C4C22×D12711yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—1272no—
208C22—5084no—
209C22—2541no—
210C22—2541no—
211C22—2541no—
212C22—2541no—
213C22—2541no—
214C22C23×D12711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—2541no—
220C4—2541no—
221C4—2541no—
222C4—2541no—
223C4—2541no—
224C4—2541no—
225C2—41no—
226C2—41no—
227C2—41no—
228C2—5081no—
229C2—2542no—
230C2#41×D12711yescharacteristic
231C2#1686125411yescharacteristic
232C2#1686125411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#121×C127C2
hypercenter#121D127
hypocenterC127#538
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #16861204C2semidirect
2. #16861202C2semidirect
3. #16861196C2semidirect
4. #16861195C2semidirect
5. #16861192C2semidirect
6. #16861190C2semidirect
7. #16861187C2semidirect
8. #16861181C2semidirect
9. #16861180C2semidirect
10. #16861177C2semidirect
11. #16861177C2semidirect
12. C4×C22⋊D127C2semidirect
13. (C22⋊D127)⋊C4C2semidirect
14. C22⋊C4×D127C2semidirect
15. #121×C127C2semidirect
17. C2×Dic254C22semidirect
22. C22×Dic127C22semidirect
25. C4⋊Dic127C22semidirect
26. Dic127⋊C4C22semidirect
27. Dic127⋊C4C22semidirect
28. Dic127⋊C4C22semidirect
29. Dic127⋊C4C22semidirect
33. C22⋊Dic127C22semidirect
34. C22⋊Dic127C22semidirect
35. C22⋊Dic127C22semidirect
42. C4×Dic127C22semidirect
43. C4×Dic127C22semidirect
46. C2×D4×C127C22semidirect
47. C22×C4×C127C22semidirect
48. C4⋊C4×C127C22semidirect
50. C22⋊C4×C127C22semidirect
116. C127#538semidirect
127. #121D127semidirect
141. C2×D4C2×D127semidirect
145. C22×C4C2×D127semidirect
151. C4⋊C4C2×D127semidirect
165. C22⋊C4C2×D127semidirect