Finite Group Databasebeta

← #22550552

Subgroups — #22550552

Order 8128.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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#16861195C211yesmaximal, characteristic, complemented
4#16861193C211yesmaximal, characteristic, complemented
5#16861192C211yesmaximal, characteristic, complemented
6#16861191C211yesmaximal, characteristic, complemented
7#16861188C211yesmaximal, characteristic, complemented
8#16861186C211yesmaximal, characteristic, complemented
9#16861178C211yesmaximal, characteristic
10C4×C22⋊D127C211yesmaximal, characteristic, complemented
11(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
12#16861168C211yesmaximal, characteristic, complemented
13#16861165C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic127C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C127C211yesmaximal, characteristic, complemented
16C2×Dic254C2211yescharacteristic, complemented
17C2×C22⋊D127C2211yescharacteristic
18C2×C22⋊D127C2211yescharacteristic
19C2×C22⋊D127C2211yescharacteristic
20C22×Dic127C2211yescharacteristic
21C22×Dic127C2211yescharacteristic
22C2×C4×D127C2211yescharacteristic, complemented
23C4⋊Dic127C2211yescharacteristic, complemented
24C4⋊Dic127C2211yescharacteristic, complemented
25C4⋊Dic127C2211yescharacteristic, complemented
26Dic127⋊C4C2211yescharacteristic, 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
38(C2×D127)⋊C4C2211yescharacteristic, complemented
39(C2×D127)⋊C4C2211yescharacteristic, complemented
40(C2×D127)⋊C4C2211yescharacteristic, complemented
41C4×Dic127C2211yescharacteristic, complemented
42C4×Dic127C2211yescharacteristic, complemented
43C4×Dic127C2211yescharacteristic, complemented
44C24×C127C2211yescharacteristic
45C22×C4×C127C2211yescharacteristic
46C22×C4×C127C2211yescharacteristic
47C22⋊C4×C127C2211yescharacteristic
48C22⋊C4×C127C2211yescharacteristic
49C22⋊C4×C127C2211yescharacteristic
50C22⋊C4×C127C2211yescharacteristic
51C22×D127C2311yescharacteristic
52Dic254—41no—
53C22⋊D127—22no—
54C22⋊D127—22no—
55C22⋊D127—41no—
56C2×Dic127—22no—
57C2×Dic127—41no—
58C2×Dic127C2311yescharacteristic
59C2×Dic127C2311yescharacteristic
60C2×Dic127C2311yescharacteristic
61C2×Dic127C2311yescharacteristic
62C2×Dic127C2311yescharacteristic
63C2×Dic127C2311yescharacteristic
64C2×Dic127C2311yescharacteristic
65C4×D127—41no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127—41no—
69C23×C127C2311yescharacteristic
70C23×C127C2311yescharacteristic
71C23×C127C2311yescharacteristic
72C2×C4×C127—22no—
73C2×C4×C127—41no—
74C2×C4×C127C2311yescharacteristic
75C2×C4×C127C2311yescharacteristic
76C2×C4×C127C2311yescharacteristic
77C2×C4×C127C2311yescharacteristic
78C22×C127—21no—
79C22×C127—21no—
80C22×C127—21no—
81C22×C127—21no—
82C22×C127—21no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—44no—
86C22×C127#4112yes—
87C22×C127#4112yes—
88C22×C127C2411yescharacteristic
89C2×D127—21no—
90C2×D127—21no—
91C2×D127—21no—
92C4×C127—21no—
93C4×C127—21no—
94C4×C127—21no—
95C4×C127—21no—
96Dic127—21no—
97Dic127—21no—
98Dic127—21no—
99Dic127—21no—
100Dic127—21no—
101Dic127—21no—
102Dic127—21no—
103C2×C127—22no—
104C2×C127—22no—
105C2×C127—41no—
106C2×C127C2×#4111yescharacteristic
107C2×C127C2×#4111yescharacteristic
108C2×C127H(5,𝔽2)11yescharacteristic
109D127—41no—
110C127#55111yescharacteristic, complemented, Sylow
111#551—1271nomaximal, Sylow
112#126—1271no—
113#126—1271no—
114#124—1271no—
115#123—1271no—
116#123—1271no—
117#123—1271no—
118C22⋊Q8—1271no—
119C22⋊Q8—1271no—
120#120—1271no—
121C4×D4—1271no—
122C4×D4—1271no—
123#117—1271no—
124#117—1271no—
125C2×C22⋊C4—1271no—
126C2×C22⋊C4D12711yescharacteristic, complemented
127C24C2×D12711yescharacteristic
128C2×Q8—1271no—
129C2×D4—1271no—
130C2×D4—1271no—
131C2×D4—1271no—
132C22×C4—1271no—
133C22×C4—1271no—
134C22×C4—1271no—
135C22×C4C2×D12711yescharacteristic
136C22×C4C2×D12711yescharacteristic
137C4⋊C4—1271no—
138C4⋊C4—1271no—
139C4⋊C4—1271no—
140C4⋊C4—1271no—
141C4⋊C4—1271no—
142C4⋊C4—1271no—
143C4⋊C4—1271no—
144C4⋊C4—1271no—
145C22⋊C4—1271no—
146C22⋊C4—1271no—
147C22⋊C4—1271no—
148C22⋊C4—1271no—
149C22⋊C4—1271no—
150C22⋊C4—1271no—
151C22⋊C4—1271no—
152C22⋊C4—1271no—
153C22⋊C4—1271no—
154C22⋊C4—1271no—
155C22⋊C4C2×D12711yescharacteristic
156C22⋊C4C2×D12711yescharacteristic
157C22⋊C4C2×D12711yescharacteristic
158C22⋊C4C2×D12711yescharacteristic
159C42—1271no—
160C42—1271no—
161C42—1271no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—1271no—
166C23C22×D12711yescharacteristic
167C23C22×D12711yescharacteristic
168C23C22×D12711yescharacteristic
169Q8—5081no—
170D4—5081no—
171D4—2542no—
172D4—2542no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—1271no—
176C2×C4—1271no—
177C2×C4—1271no—
178C2×C4—1271no—
179C2×C4—1271no—
180C2×C4—1271no—
181C2×C4—1271no—
182C2×C4—5081no—
183C2×C4—5081no—
184C2×C4—2542no—
185C2×C4C22×D12711yescharacteristic
186C2×C4C22×D12711yescharacteristic
187C2×C4C22×D12711yescharacteristic
188C2×C4C22×D12711yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—44no—
197C22—2541no—
198C22—2541no—
199C22—2541no—
200C22#1452324712yes—
201C22#1452324812yes—
202C22C23×D12711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—2541no—
208C4—2541no—
209C4—2541no—
210C4—2541no—
211C4—2541no—
212C4—2541no—
213C4—2541no—
214C2—22no—
215C2—22no—
216C2—41no—
217C2—5081no—
218C2C2×#1452324711yescharacteristic
219C2C2×#1452324811yescharacteristic
220C2#1686125411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
FittingC2×C22⋊C4×C127C2
hypercenterC2×C22⋊C4D127
hypocenterC127#551
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #16861202C2semidirect
2. #16861202C2semidirect
3. #16861195C2semidirect
4. #16861193C2semidirect
5. #16861192C2semidirect
6. #16861191C2semidirect
7. #16861188C2semidirect
8. #16861186C2semidirect
10. C4×C22⋊D127C2semidirect
11. (C22⋊D127)⋊C4C2semidirect
12. #16861168C2semidirect
13. #16861165C2semidirect
14. C2×C22⋊Dic127C2semidirect
15. C2×C22⋊C4×C127C2semidirect
16. C2×Dic254C22semidirect
22. C2×C4×D127C22semidirect
23. C4⋊Dic127C22semidirect
24. C4⋊Dic127C22semidirect
25. C4⋊Dic127C22semidirect
26. Dic127⋊C4C22semidirect
27. Dic127⋊C4C22semidirect
28. Dic127⋊C4C22semidirect
29. Dic127⋊C4C22semidirect
30. Dic127⋊C4C22semidirect
38. (C2×D127)⋊C4C22semidirect
39. (C2×D127)⋊C4C22semidirect
40. (C2×D127)⋊C4C22semidirect
41. C4×Dic127C22semidirect
42. C4×Dic127C22semidirect
43. C4×Dic127C22semidirect
110. C127#551semidirect
126. C2×C22⋊C4D127semidirect