Finite Group Databasebeta

← #22550682

Subgroups — #22550682

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#16861254C211yesmaximal, characteristic, complemented
3#16861240C211yesmaximal, characteristic, complemented
4#16861238C212yesmaximal, complemented
5#16861236C212yesmaximal, complemented
6#16861235C211yesmaximal, characteristic, complemented
7#16861231C211yesmaximal, characteristic, complemented
8#16861230C212yesmaximal, complemented
9SD16×D127C212yesmaximal, complemented
10#16861215C211yesmaximal, characteristic, complemented
11C2×SD16×C127C211yesmaximal, characteristic, complemented
12#14523249—21no—
13#14523249C2212yescomplemented
14#14523248—22no—
15#14523248C2211yescharacteristic
16#14523248C2211yescharacteristic
17#14523248C2211yescharacteristic, complemented
18#14523247—21no—
19#14523247C2212yes—
20Q8×D127—21no—
21Q8×D127C2212yescomplemented
22C2×C22⋊D127—22no—
23D4×D127—21no—
24D4×D127C2212yes—
25Dic508C2211yescharacteristic, complemented
26#14523238C2212yescomplemented
27#14523237C2211yescharacteristic, complemented
28#14523237C2211yescharacteristic, complemented
29#14523236C2212yescomplemented
30#14523235C2212yescomplemented
31D1016C2211yescharacteristic, complemented
32#14523233C2212yescomplemented
33#14523232C2211yescharacteristic, complemented
34#14523231C2212yescomplemented
35C8×D127C2212yescomplemented
36C2×Q8×C127C2211yescharacteristic, complemented
37C2×D4×C127C2211yescharacteristic
38SD16×C127C2212yescomplemented
39SD16×C127C2212yescomplemented
40C2×C8×C127C2211yescharacteristic, complemented
41C22×D127—22no—
42Dic254—22no—
43Dic254D411yescharacteristic
44Dic254C2311yescharacteristic
45Dic254C2311yescharacteristic
46D508—22no—
47D508D411yescharacteristic
48D508C2311yescharacteristic
49D508C2311yescharacteristic
50C22⋊D127—21no—
51C22⋊D127—21no—
52C22⋊D127—22no—
53C22⋊D127—22no—
54C22⋊D127D412yes—
55C2×Dic127—22no—
56C4×D127—22no—
57C4×D127—21no—
58C4×D127—21no—
59C4×D127—22no—
60C4×D127C2312yes—
61C127⋊C8C2312yes—
62C23×C127—21no—
63Q8×C127—21no—
64Q8×C127C2312yes—
65D4×C127—21no—
66D4×C127C2312yes—
67C2×C4×C127—21no—
68C2×C4×C127C2311yescharacteristic
69C8×C127C2311yescharacteristic
70C8×C127C2311yescharacteristic
71C22×C127—22no—
72C22×C127—81no—
73C22×C127C2×D411yescharacteristic
74C2×D127—21no—
75C2×D127—21no—
76C2×D127—82no—
77C2×D127C2×D412yes—
78C4×C127—22no—
79C4×C127C2×D411yescharacteristic
80C4×C127C2411yescharacteristic
81Dic127—21no—
82Dic127—21no—
83Dic127C2×D412yes—
84C2×C127—21no—
85C2×C127—42no—
86C2×C127C22×D411yescharacteristic
87D127—22no—
88D127—41no—
89D127—41no—
90C127#57711yescharacteristic, complemented, Sylow
91#577—1271nomaximal, Sylow
922−1+4—1271no—
93H(5,𝔽2)—1271no—
94#137—1271no—
95#137—1272no—
96AGL(1,ℤ/8)—1271no—
97AGL(1,ℤ/8)—1272no—
98#135—1272no—
99#135—1271no—
100C2×SD16—1272no—
101C2×SD16D12711yescharacteristic, complemented
102#131—1271no—
103#41—1271no—
104#41—1272no—
105#41—1272no—
106#41—1271no—
107#41—1271no—
108#41—2542no—
109#41—2541no—
110#41—2541no—
111C2×Q8—1272no—
112C2×Q8—2541no—
113C2×Q8C2×D12711yescharacteristic, complemented
114C2×D4—1272no—
115C2×D4—2541no—
116C2×D4—2542no—
117C2×D4C2×D12711yescharacteristic
118Q16—1272no—
119Q16—1271no—
120SD16—1272no—
121SD16—1272no—
122SD16—1271no—
123SD16—1271no—
124SD16C2×D12712yescomplemented
125SD16C2×D12712yescomplemented
126D8—1272no—
127D8—1271no—
128M16—1271no—
129M16—1272no—
130C2×C8—1272no—
131C2×C8C2×D12711yescharacteristic, complemented
132C23—21no—
133C23—2542no—
134Q8—21no—
135Q8—1271no—
136Q8—1271no—
137Q8—1271no—
138Q8—2542no—
139Q8C22×D12712yes—
140D4—21no—
141D4—1272no—
142D4—1271no—
143D4—1271no—
144D4—1271no—
145D4—2542no—
146D4—2542no—
147D4—2542no—
148D4—2541no—
149D4—2541no—
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—1272no—
160C8C22×D12711yescharacteristic
161C8C22×D12711yescharacteristic
162C22—22no—
163C22—81no—
164C22—1272no—
165C22—10162no—
166C22—2541no—
167C22—2541no—
168C22D4×D12711yescharacteristic
169C4—22no—
170C4—1272no—
171C4—2541no—
172C4—2541no—
173C4D4×D12711yescharacteristic
174C4C23×D12711yescharacteristic
175C2—21no—
176C2—42no—
177C2—5081no—
178C2—5081no—
179C2—2542no—
180C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC2C2×D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
FittingC2×SD16×C127C2
socleC2×C127C22×D4
hypercenterC2×SD16D127
hypocenterC127#577
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #16861257C2semidirect
2. #16861254C2semidirect
3. #16861240C2semidirect
4. #16861238C2semidirect
5. #16861236C2semidirect
6. #16861235C2semidirect
7. #16861231C2semidirect
8. #16861230C2semidirect
9. SD16×D127C2semidirect
10. #16861215C2semidirect
11. C2×SD16×C127C2semidirect
13. #14523249C22semidirect
17. #14523248C22semidirect
21. Q8×D127C22semidirect
25. Dic508C22semidirect
26. #14523238C22semidirect
27. #14523237C22semidirect
28. #14523237C22semidirect
29. #14523236C22semidirect
30. #14523235C22semidirect
31. D1016C22semidirect
32. #14523233C22semidirect
33. #14523232C22semidirect
34. #14523231C22semidirect
35. C8×D127C22semidirect
36. C2×Q8×C127C22semidirect
38. SD16×C127C22semidirect
39. SD16×C127C22semidirect
40. C2×C8×C127C22semidirect
90. C127#577semidirect
101. C2×SD16D127semidirect
113. C2×Q8C2×D127semidirect
124. SD16C2×D127semidirect
125. SD16C2×D127semidirect
131. C2×C8C2×D127semidirect