Finite Group Databasebeta

← #22550685

Subgroups — #22550685

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#16861257C211yesmaximal, characteristic, complemented
2#16861255C211yesmaximal, characteristic, complemented
3#16861241C212yesmaximal, complemented
4#16861237C211yesmaximal, characteristic, complemented
5#16861236C212yesmaximal, complemented
6#16861232C212yesmaximal, complemented
7#16861229C211yesmaximal, characteristic, complemented
8C2×#14523235C211yesmaximal, characteristic, complemented
9SD16×D127C212yesmaximal, complemented
10#16861217C211yesmaximal, characteristic, complemented
11#137×C127C211yesmaximal, characteristic, complemented
12#14523249—21no—
13#14523249C2212yes—
14#14523249C2212yescomplemented
15#14523248—21no—
16#14523248—22no—
17#14523248C2211yescharacteristic
18#14523248C2211yescharacteristic, complemented
19Q8×D127—21no—
20Q8×D127C2212yescomplemented
21C2×D508—21no—
22C2×D508C2211yescharacteristic
23D4×D127—22no—
24D4×D127C2212yes—
25#14523238C2211yescharacteristic, complemented
26#14523237C2212yescomplemented
27#14523236C2211yescharacteristic, complemented
28#14523235C2212yescomplemented
29#14523235C2212yescomplemented
30#14523235C2211yescharacteristic, complemented
31D1016C2212yescomplemented
32#14523233C2211yescharacteristic, complemented
33#14523231C2212yescomplemented
34C2×C127⋊C8C2211yescharacteristic, complemented
35C8×D127C2212yescomplemented
36#41×C127C2211yescharacteristic
37C2×Q8×C127C2211yescharacteristic, complemented
38Q16×C127C2212yescomplemented
39SD16×C127C2212yescomplemented
40M16×C127C2211yescharacteristic, complemented
41C22×D127—21no—
42C22×D127—22no—
43Dic254—22no—
44Dic254D411yescharacteristic
45Dic254C2311yescharacteristic
46D508—22no—
47D508—22no—
48D508—21no—
49D508D411yescharacteristic
50D508C2311yescharacteristic
51D508C2312yes—
52C22⋊D127—21no—
53C22⋊D127—22no—
54C22⋊D127D412yes—
55C4×D127—22no—
56C4×D127—21no—
57C4×D127—22no—
58C4×D127—22no—
59C4×D127C2312yes—
60C127⋊C8C2311yescharacteristic
61C127⋊C8C2311yescharacteristic
62Q8×C127—21no—
63Q8×C127C2312yes—
64Q8×C127C2311yescharacteristic
65D4×C127—21no—
66D4×C127C2311yescharacteristic
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127C2311yescharacteristic
70C8×C127C2312yes—
71C22×C127—21no—
72C22×C127C2×D411yescharacteristic
73C2×D127—21no—
74C2×D127—22no—
75C2×D127—82no—
76C2×D127—81no—
77C2×D127C2×D412yes—
78C4×C127—21no—
79C4×C127—22no—
80C4×C127C2×D411yescharacteristic
81C4×C127C2411yescharacteristic
82Dic127—21no—
83Dic127C2×D412yes—
84C2×C127—21no—
85C2×C127—41no—
86C2×C127C22×D411yescharacteristic
87D127—22no—
88D127—41no—
89D127—42no—
90C127#57711yescharacteristic, complemented, Sylow
91#577—1271nomaximal, Sylow
922−1+4—1271no—
93H(5,𝔽2)—1271no—
94#137—1272no—
95#137D12711yescharacteristic, complemented
96AGL(1,ℤ/8)—1272no—
97AGL(1,ℤ/8)—1271no—
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—2542no—
108#41—2541no—
109#41—2541no—
110#41C2×D12711yescharacteristic
111C2×Q8—1272no—
112C2×Q8—2541no—
113C2×Q8C2×D12711yescharacteristic, complemented
114C2×D4—1272no—
115C2×D4—1271no—
116C2×D4—2541no—
117C2×D4—2542no—
118Q16—1271no—
119Q16C2×D12712yescomplemented
120SD16—1272no—
121SD16—1272no—
122SD16—1271no—
123SD16—1271no—
124SD16—1272no—
125SD16C2×D12712yescomplemented
126D8—1271no—
127D8—1272no—
128M16—1272no—
129M16C2×D12711yescharacteristic, complemented
130C2×C8—1271no—
131C2×C8—1272no—
132C23—2542no—
133C23—2541no—
134Q8—21no—
135Q8—1271no—
136Q8—1271no—
137Q8—2542no—
138Q8C22×D12712yes—
139Q8C22×D12711yescharacteristic
140D4—21no—
141D4—1272no—
142D4—1271no—
143D4—1271no—
144D4—1272no—
145D4—2542no—
146D4—2542no—
147D4—2542no—
148D4—2541no—
149D4—2541no—
150D4C22×D12711yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—1272no—
154C2×C4—2542no—
155C2×C4—2542no—
156C2×C4—2542no—
157C2×C4—2541no—
158C2×C4C22×D12711yescharacteristic
159C8—1271no—
160C8—1271no—
161C8C22×D12712yes—
162C22—21no—
163C22—1272no—
164C22—10161no—
165C22—10162no—
166C22—2541no—
167C22—2542no—
168C22D4×D12711yescharacteristic
169C4—21no—
170C4—22no—
171C4—1272no—
172C4—2541no—
173C4D4×D12711yescharacteristic
174C4C23×D12711yescharacteristic
175C2—21no—
176C2—41no—
177C2—5082no—
178C2—5081no—
179C2—2542no—
180C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC2C2×D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
Fitting#137×C127C2
socleC2×C127C22×D4
hypercenter#137D127
hypocenterC127#577
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #16861257C2semidirect
2. #16861255C2semidirect
3. #16861241C2semidirect
4. #16861237C2semidirect
5. #16861236C2semidirect
6. #16861232C2semidirect
7. #16861229C2semidirect
8. C2×#14523235C2semidirect
9. SD16×D127C2semidirect
10. #16861217C2semidirect
11. #137×C127C2semidirect
14. #14523249C22semidirect
18. #14523248C22semidirect
20. Q8×D127C22semidirect
25. #14523238C22semidirect
26. #14523237C22semidirect
27. #14523236C22semidirect
28. #14523235C22semidirect
29. #14523235C22semidirect
30. #14523235C22semidirect
31. D1016C22semidirect
32. #14523233C22semidirect
33. #14523231C22semidirect
34. C2×C127⋊C8C22semidirect
35. C8×D127C22semidirect
37. C2×Q8×C127C22semidirect
38. Q16×C127C22semidirect
39. SD16×C127C22semidirect
40. M16×C127C22semidirect
90. C127#577semidirect
95. #137D127semidirect
113. C2×Q8C2×D127semidirect
119. Q16C2×D127semidirect
125. SD16C2×D127semidirect
129. M16C2×D127semidirect