Finite Group Databasebeta

← #22550680

Subgroups — #22550680

Order 8128.

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

normal subgroups only

This group has 268 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#16861255C211yesmaximal, characteristic, complemented
2#16861254C211yesmaximal, characteristic, complemented
3#16861237C211yesmaximal, characteristic, complemented
4#16861236C212yesmaximal, complemented
5#16861233C212yesmaximal, complemented
6#16861230C212yesmaximal, complemented
7#16861229C211yesmaximal, characteristic, complemented
8C2×#14523233C211yesmaximal, characteristic, complemented
9D8×D127C212yesmaximal, complemented
10#16861217C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12#14523249C2212yescomplemented
13#14523248—21no—
14#14523248C2211yescharacteristic
15#14523248C2211yescharacteristic, complemented
16#14523247—21no—
17#14523247C2212yescomplemented
18C2×D508—21no—
19C2×D508C2211yescharacteristic, complemented
20C2×C22⋊D127—22no—
21D4×D127—21no—
22D4×D127—22no—
23D4×D127C2212yescomplemented
24D4×D127C2212yescomplemented
25#14523238C2211yescharacteristic, complemented
26#14523237C2212yescomplemented
27#14523236C2211yescharacteristic, complemented
28#14523235C2211yescharacteristic, complemented
29D1016C2212yescomplemented
30#14523233C2211yescharacteristic, complemented
31#14523233C2212yescomplemented
32#14523233C2212yescomplemented
33#14523231C2212yescomplemented
34C2×C127⋊C8C2211yescharacteristic, complemented
35C8×D127C2212yescomplemented
36#41×C127C2211yescharacteristic, complemented
37C2×D4×C127C2211yescharacteristic, complemented
38SD16×C127C2212yescomplemented
39D8×C127C2212yescomplemented
40M16×C127C2211yescharacteristic, complemented
41C22×D127—21no—
42C22×D127—22no—
43C22×D127—22no—
44Dic254D411yescharacteristic
45Dic254C2311yescharacteristic
46D508—22no—
47D508—21no—
48D508D411yescharacteristic
49D508C2312yes—
50D508C2311yescharacteristic
51C22⋊D127—21no—
52C22⋊D127—22no—
53C22⋊D127—22no—
54C22⋊D127—22no—
55C22⋊D127D412yes—
56C2×Dic127—22no—
57C4×D127—21no—
58C4×D127—22no—
59C4×D127C2312yes—
60C127⋊C8C2311yescharacteristic
61C127⋊C8C2311yescharacteristic
62C23×C127—21no—
63Q8×C127C2311yescharacteristic
64D4×C127—21no—
65D4×C127—21no—
66D4×C127C2311yescharacteristic
67D4×C127C2312yes—
68C2×C4×C127—21no—
69C2×C4×C127C2311yescharacteristic
70C8×C127C2312yes—
71C22×C127—21no—
72C22×C127—22no—
73C22×C127—81no—
74C22×C127C2×D411yescharacteristic
75C2×D127—21no—
76C2×D127—22no—
77C2×D127—82no—
78C2×D127—82no—
79C2×D127—81no—
80C2×D127C2×D412yes—
81C4×C127—21no—
82C4×C127C2×D411yescharacteristic
83C4×C127C2411yescharacteristic
84Dic127—21no—
85Dic127C2×D412yes—
86C2×C127—21no—
87C2×C127—41no—
88C2×C127—42no—
89C2×C127C22×D411yescharacteristic
90D127—22no—
91D127—41no—
92D127—42no—
93C127#57611yescharacteristic, complemented, Sylow
94#576—1271nomaximal, Sylow
95H(5,𝔽2)—1271no—
96H(5,𝔽2)—1271no—
97AGL(1,ℤ/8)—1272no—
98AGL(1,ℤ/8)—1272no—
99AGL(1,ℤ/8)—1271no—
100AGL(1,ℤ/8)D12711yescharacteristic, complemented
101#135—1271no—
102#135—1272no—
103C2×D8—1271no—
104C2×D8—1272no—
105#131—1271no—
106#41—1271no—
107#41—1272no—
108#41—1272no—
109#41—1271no—
110#41—2541no—
111#41—2541no—
112#41C2×D12711yescharacteristic, complemented
113C2×D4—1272no—
114C2×D4—1272no—
115C2×D4—1271no—
116C2×D4—2541no—
117C2×D4—2542no—
118C2×D4—2541no—
119C2×D4—2542no—
120C2×D4C2×D12711yescharacteristic, complemented
121Q16—1271no—
122SD16—1271no—
123SD16—1271no—
124SD16—1272no—
125SD16C2×D12712yescomplemented
126D8—1272no—
127D8—1271no—
128D8—1272no—
129D8—1272no—
130D8C2×D12712yescomplemented
131M16—1272no—
132M16C2×D12711yescharacteristic, complemented
133C2×C8—1271no—
134C2×C8—1272no—
135C23—21no—
136C23—2542no—
137C23—2542no—
138C23—2541no—
139Q8—1271no—
140Q8—1271no—
141Q8C22×D12711yescharacteristic
142D4—21no—
143D4—21no—
144D4—1272no—
145D4—1271no—
146D4—1272no—
147D4—1271no—
148D4—2542no—
149D4—2542no—
150D4—2542no—
151D4—2542no—
152D4—2541no—
153D4—2541no—
154D4C22×D12712yes—
155D4C22×D12711yescharacteristic
156C2×C4—21no—
157C2×C4—1272no—
158C2×C4—2542no—
159C2×C4—2542no—
160C2×C4—2541no—
161C2×C4C22×D12711yescharacteristic
162C8—1271no—
163C8—1271no—
164C8C22×D12712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—1272no—
169C22—10161no—
170C22—10162no—
171C22—10162no—
172C22—2542no—
173C22—2541no—
174C22D4×D12711yescharacteristic
175C4—21no—
176C4—1272no—
177C4—2541no—
178C4D4×D12711yescharacteristic
179C4C23×D12711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—5082no—
184C2—5081no—
185C2—2542no—
186C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC2C2×D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
FittingAGL(1,ℤ/8)×C127C2
socleC2×C127C22×D4
hypercenterAGL(1,ℤ/8)D127
hypocenterC127#576
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16861255C2semidirect
2. #16861254C2semidirect
3. #16861237C2semidirect
4. #16861236C2semidirect
5. #16861233C2semidirect
6. #16861230C2semidirect
7. #16861229C2semidirect
8. C2×#14523233C2semidirect
9. D8×D127C2semidirect
10. #16861217C2semidirect
11. AGL(1,ℤ/8)×C127C2semidirect
12. #14523249C22semidirect
15. #14523248C22semidirect
17. #14523247C22semidirect
19. C2×D508C22semidirect
23. D4×D127C22semidirect
24. D4×D127C22semidirect
25. #14523238C22semidirect
26. #14523237C22semidirect
27. #14523236C22semidirect
28. #14523235C22semidirect
29. D1016C22semidirect
30. #14523233C22semidirect
31. #14523233C22semidirect
32. #14523233C22semidirect
33. #14523231C22semidirect
34. C2×C127⋊C8C22semidirect
35. C8×D127C22semidirect
36. #41×C127C22semidirect
37. C2×D4×C127C22semidirect
38. SD16×C127C22semidirect
39. D8×C127C22semidirect
40. M16×C127C22semidirect
93. C127#576semidirect
100. AGL(1,ℤ/8)D127semidirect
112. #41C2×D127semidirect
120. C2×D4C2×D127semidirect
125. SD16C2×D127semidirect
130. D8C2×D127semidirect
132. M16C2×D127semidirect