Finite Group Databasebeta

← #22550461

Subgroups — #22550461

Order 8128.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#16861202C211yesmaximal, characteristic, complemented
2#16861202C211yesmaximal, characteristic, complemented
3#16861196C211yesmaximal, characteristic, complemented
4#16861195C211yesmaximal, characteristic, complemented
5#16861191C211yesmaximal, characteristic, complemented
6#16861190C211yesmaximal, characteristic, complemented
7#16861186C211yesmaximal, characteristic, complemented
8#16861181C211yesmaximal, characteristic, complemented
9#16861181C211yesmaximal, characteristic, complemented
10#16861180C211yesmaximal, characteristic, complemented
11#16861178C211yesmaximal, characteristic, complemented
12D4×Dic127C211yesmaximal, characteristic, complemented
13(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic127C211yesmaximal, characteristic, complemented
15#120×C127C211yesmaximal, characteristic, complemented
16C2×Dic254C2211yescharacteristic, complemented
17C2×C22⋊D127C2211yescharacteristic
18C2×C22⋊D127C2211yescharacteristic
19C2×C22⋊D127C2211yescharacteristic
20C2×C22⋊D127C2211yescharacteristic, complemented
21C22×Dic127C2211yescharacteristic
22C22×Dic127C2211yescharacteristic
23C22×Dic127C2211yescharacteristic, complemented
24C2×C4×D127C2211yescharacteristic, complemented
25C4⋊Dic127C2211yescharacteristic, complemented
26C4⋊Dic127C2211yescharacteristic, complemented
27Dic127⋊C4C2211yescharacteristic, complemented
28Dic127⋊C4C2211yescharacteristic, complemented
29Dic127⋊C4C2211yescharacteristic, complemented
30Dic127⋊C4C2211yescharacteristic, complemented
31C22⋊Dic127C2211yescharacteristic
32C22⋊Dic127C2211yescharacteristic
33C22⋊Dic127C2211yescharacteristic
34C22⋊Dic127C2211yescharacteristic
35C22⋊Dic127C2211yescharacteristic
36C22⋊Dic127C2211yescharacteristic
37C22⋊Dic127C2211yescharacteristic
38C22⋊Dic127C2211yescharacteristic, complemented
39C22⋊Dic127C2211yescharacteristic, complemented
40(C2×D127)⋊C4C2211yescharacteristic, complemented
41(C2×D127)⋊C4C2211yescharacteristic, complemented
42C4×Dic127C2211yescharacteristic, complemented
43C4×Dic127C2211yescharacteristic, complemented
44C24×C127C2211yescharacteristic
45C2×D4×C127C2211yescharacteristic
46C2×D4×C127C2211yescharacteristic
47C2×D4×C127C2211yescharacteristic
48C22⋊C4×C127C2211yescharacteristic
49C22⋊C4×C127C2211yescharacteristic
50C22⋊C4×C127C2211yescharacteristic
51C22×D127C2311yescharacteristic
52Dic254—41no—
53C22⋊D127—22no—
54C22⋊D127—41no—
55C22⋊D127—41no—
56C22⋊D127—41no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C2×Dic127C2311yescharacteristic
61C2×Dic127C2311yescharacteristic
62C2×Dic127C2311yescharacteristic
63C2×Dic127C2311yescharacteristic
64C2×Dic127C2311yescharacteristic
65C2×Dic127C2311yescharacteristic
66C2×Dic127C2311yescharacteristic
67C4×D127—41no—
68C23×C127—22no—
69C23×C127—41no—
70C23×C127—41no—
71C23×C127C2311yescharacteristic
72C23×C127C2311yescharacteristic
73C23×C127C2311yescharacteristic
74C23×C127C2311yescharacteristic
75D4×C127—22no—
76D4×C127—41no—
77D4×C127—41no—
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×C127—21no—
91C22×C127—21no—
92C22×C127—44no—
93C22×C127#4112yes—
94C22×C127C2411yescharacteristic
95C2×D127—21no—
96C2×D127—21no—
97C2×D127—21no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—21no—
101Dic127—21no—
102Dic127—21no—
103Dic127—21no—
104Dic127—21no—
105Dic127—21no—
106Dic127—21no—
107Dic127—21no—
108C2×C127—22no—
109C2×C127—41no—
110C2×C127—41no—
111C2×C127—41no—
112C2×C127C2×#4111yescharacteristic
113C2×C127H(5,𝔽2)11yescharacteristic
114C2×C127H(5,𝔽2)11yescharacteristic
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#121—1271no—
128#120—1271no—
129#120D12711yescharacteristic, complemented
130C4×D4—1271no—
131C4×D4—1271no—
132C2×C22⋊C4—1271no—
133C24C2×D12711yescharacteristic
134C2×Q8—1271no—
135C2×D4—1271no—
136C2×D4—1271no—
137C2×D4—1271no—
138C2×D4—1271no—
139C2×D4C2×D12711yescharacteristic
140C2×D4C2×D12711yescharacteristic
141C2×D4C2×D12711yescharacteristic
142C22×C4—1271no—
143C22×C4—1271no—
144C22×C4—1271no—
145C22×C4—1271no—
146C4⋊C4—1271no—
147C4⋊C4—1271no—
148C4⋊C4—1271no—
149C4⋊C4—1271no—
150C4⋊C4—1271no—
151C4⋊C4—1271no—
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—
167C42—1271no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—1271no—
172C23C22×D12711yescharacteristic
173C23C22×D12711yescharacteristic
174C23C22×D12711yescharacteristic
175C23C22×D12711yescharacteristic
176Q8—5081no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—5081no—
181D4—5081no—
182D4—5081no—
183D4—2542no—
184C2×C4—1271no—
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×C4—5081no—
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—21no—
208C22—21no—
209C22—44no—
210C22—2541no—
211C22—2541no—
212C22—2541no—
213C22#1452324712yes—
214C22C23×D12711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—2541no—
219C4—2541no—
220C4—2541no—
221C4—2541no—
222C4—2541no—
223C4—2541no—
224C4—2541no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—5081no—
230C2C2×#1452324711yescharacteristic
231C2#1686125411yescharacteristic
232C2#1686125411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#120×C127C2
hypercenter#120D127
hypocenterC127#538
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #16861202C2semidirect
2. #16861202C2semidirect
3. #16861196C2semidirect
4. #16861195C2semidirect
5. #16861191C2semidirect
6. #16861190C2semidirect
7. #16861186C2semidirect
8. #16861181C2semidirect
9. #16861181C2semidirect
10. #16861180C2semidirect
11. #16861178C2semidirect
12. D4×Dic127C2semidirect
13. (C22⋊D127)⋊C4C2semidirect
14. C2×C22⋊Dic127C2semidirect
15. #120×C127C2semidirect
16. C2×Dic254C22semidirect
20. C2×C22⋊D127C22semidirect
23. C22×Dic127C22semidirect
24. C2×C4×D127C22semidirect
25. C4⋊Dic127C22semidirect
26. C4⋊Dic127C22semidirect
27. Dic127⋊C4C22semidirect
28. Dic127⋊C4C22semidirect
29. Dic127⋊C4C22semidirect
30. Dic127⋊C4C22semidirect
38. C22⋊Dic127C22semidirect
39. C22⋊Dic127C22semidirect
40. (C2×D127)⋊C4C22semidirect
41. (C2×D127)⋊C4C22semidirect
42. C4×Dic127C22semidirect
43. C4×Dic127C22semidirect
116. C127#538semidirect
129. #120D127semidirect