Finite Group Databasebeta

← #22550686

Subgroups — #22550686

Order 8128.

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

normal subgroups only

This group has 258 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#16861256C211yesmaximal, characteristic, complemented
2#16861254C211yesmaximal, characteristic, complemented
3#16861242C211yesmaximal, characteristic, complemented
4#16861238C212yesmaximal, complemented
5#16861233C212yesmaximal, complemented
6#16861229C211yesmaximal, characteristic, complemented
7#16861228C212yesmaximal, complemented
8C2×#14523236C211yesmaximal, characteristic, complemented
9SD16×D127C212yesmaximal, complemented
10#16861217C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12#14523248—21no—
13#14523248C2211yescharacteristic
14#14523248C2211yescharacteristic
15#14523247—21no—
16#14523247—22no—
17#14523247C2212yes—
18#14523247C2212yescomplemented
19C2×Dic254—21no—
20C2×Dic254C2211yescharacteristic, complemented
21Q8×D127C2212yescomplemented
22C2×C22⋊D127—22no—
23D4×D127—21no—
24D4×D127C2212yes—
25Dic508C2212yescomplemented
26#14523238C2211yescharacteristic, complemented
27#14523237C2212yescomplemented
28#14523236C2212yescomplemented
29#14523236C2212yescomplemented
30#14523236C2211yescharacteristic, complemented
31#14523235C2211yescharacteristic, complemented
32#14523233C2211yescharacteristic, complemented
33#14523231C2212yescomplemented
34C2×C127⋊C8C2211yescharacteristic, complemented
35C8×D127C2212yescomplemented
36#41×C127C2211yescharacteristic, complemented
37C2×D4×C127C2211yescharacteristic
38SD16×C127C2212yescomplemented
39D8×C127C2212yescomplemented
40M16×C127C2211yescharacteristic, complemented
41C22×D127—22no—
42Dic254—21no—
43Dic254—22no—
44Dic254D411yescharacteristic
45Dic254C2312yes—
46Dic254C2311yescharacteristic
47D508D411yescharacteristic
48D508C2311yescharacteristic
49C22⋊D127—21no—
50C22⋊D127—22no—
51C22⋊D127—22no—
52C22⋊D127—22no—
53C22⋊D127D412yes—
54C2×Dic127—21no—
55C2×Dic127—22no—
56C2×Dic127—22no—
57C4×D127—22no—
58C4×D127—21no—
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—82no—
77C2×D127C2×D412yes—
78C4×C127—21no—
79C4×C127C2×D411yescharacteristic
80C4×C127C2411yescharacteristic
81Dic127—21no—
82Dic127—22no—
83Dic127C2×D412yes—
84C2×C127—21no—
85C2×C127—42no—
86C2×C127—41no—
87C2×C127C22×D411yescharacteristic
88D127—22no—
89D127—41no—
90C127#57711yescharacteristic, complemented, Sylow
91#577—1271nomaximal, Sylow
922−1+4—1271no—
93H(5,𝔽2)—1271no—
94#137—1272no—
95#137—1271no—
96AGL(1,ℤ/8)—1272no—
97AGL(1,ℤ/8)D12711yescharacteristic, complemented
98#135—1271no—
99#135—1272no—
100C2×SD16—1271no—
101C2×SD16—1272no—
102#131—1271no—
103#41—1271no—
104#41—1272no—
105#41—1272no—
106#41—1271no—
107#41—2541no—
108#41—2542no—
109#41—2541no—
110#41C2×D12711yescharacteristic, complemented
111C2×Q8—1272no—
112C2×Q8—1271no—
113C2×Q8—2541no—
114C2×D4—1272no—
115C2×D4—2541no—
116C2×D4—2542no—
117C2×D4C2×D12711yescharacteristic
118Q16—1271no—
119Q16—1272no—
120SD16—1271no—
121SD16—1272no—
122SD16—1272no—
123SD16—1271no—
124SD16—1272no—
125SD16C2×D12712yescomplemented
126D8—1271no—
127D8C2×D12712yescomplemented
128M16—1272no—
129M16C2×D12711yescharacteristic, complemented
130C2×C8—1271no—
131C2×C8—1272no—
132C23—21no—
133C23—2542no—
134Q8—1271no—
135Q8—1272no—
136Q8—1271no—
137Q8—2542no—
138Q8—2541no—
139Q8C22×D12711yescharacteristic
140D4—21no—
141D4—21no—
142D4—1272no—
143D4—1271no—
144D4—1271no—
145D4—2542no—
146D4—2542no—
147D4—2542no—
148D4—2541no—
149D4C22×D12711yescharacteristic
150D4C22×D12712yes—
151C2×C4—21no—
152C2×C4—1272no—
153C2×C4—2542no—
154C2×C4—2542no—
155C2×C4—2542no—
156C2×C4—2541no—
157C2×C4—2541no—
158C2×C4C22×D12711yescharacteristic
159C8—1271no—
160C8—1271no—
161C8C22×D12712yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—1272no—
166C22—10162no—
167C22—2541no—
168C22D4×D12711yescharacteristic
169C4—21no—
170C4—1272no—
171C4—2541no—
172C4—2542no—
173C4D4×D12711yescharacteristic
174C4C23×D12711yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—5081no—
179C2—2542no—
180C2C2×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#577
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. #16861256C2semidirect
2. #16861254C2semidirect
3. #16861242C2semidirect
4. #16861238C2semidirect
5. #16861233C2semidirect
6. #16861229C2semidirect
7. #16861228C2semidirect
8. C2×#14523236C2semidirect
9. SD16×D127C2semidirect
10. #16861217C2semidirect
11. AGL(1,ℤ/8)×C127C2semidirect
18. #14523247C22semidirect
20. C2×Dic254C22semidirect
21. Q8×D127C22semidirect
25. Dic508C22semidirect
26. #14523238C22semidirect
27. #14523237C22semidirect
28. #14523236C22semidirect
29. #14523236C22semidirect
30. #14523236C22semidirect
31. #14523235C22semidirect
32. #14523233C22semidirect
33. #14523231C22semidirect
34. C2×C127⋊C8C22semidirect
35. C8×D127C22semidirect
36. #41×C127C22semidirect
38. SD16×C127C22semidirect
39. D8×C127C22semidirect
40. M16×C127C22semidirect
90. C127#577semidirect
97. AGL(1,ℤ/8)D127semidirect
110. #41C2×D127semidirect
125. SD16C2×D127semidirect
127. D8C2×D127semidirect
129. M16C2×D127semidirect