Finite Group Databasebeta

← #22550683

Subgroups — #22550683

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#16861255C211yesmaximal, characteristic, complemented
3#16861241C212yesmaximal, complemented
4#16861239C211yesmaximal, characteristic, complemented
5#16861234C211yesmaximal, characteristic, complemented
6#16861233C212yesmaximal, complemented
7#16861230C212yesmaximal, complemented
8C2×#14523237C211yesmaximal, characteristic, complemented
9SD16×D127C212yesmaximal, complemented
10#16861215C211yesmaximal, characteristic, complemented
11#135×C127C211yesmaximal, characteristic, complemented
12#14523249C2212yes—
13#14523248—21no—
14#14523248—21no—
15#14523248C2211yescharacteristic
16#14523247—22no—
17#14523247C2212yescomplemented
18C2×Dic254—21no—
19C2×Dic254C2211yescharacteristic, complemented
20Q8×D127C2212yescomplemented
21C2×D508—21no—
22C2×D508C2211yescharacteristic
23D4×D127—22no—
24D4×D127C2212yes—
25#14523238C2212yescomplemented
26#14523237C2212yescomplemented
27#14523237C2212yescomplemented
28#14523236C2212yescomplemented
29#14523235C2212yescomplemented
30#14523233C2212yescomplemented
31#14523232C2211yescharacteristic, complemented
32#14523231C2212yescomplemented
33C8×D127C2212yescomplemented
34#41×C127C2211yescharacteristic
35#41×C127C2211yescharacteristic, complemented
36Q16×C127C2211yescharacteristic, complemented
37SD16×C127C2211yescharacteristic, complemented
38SD16×C127C2211yescharacteristic, complemented
39D8×C127C2211yescharacteristic, complemented
40C2×C8×C127C2211yescharacteristic, complemented
41C22×D127—21no—
42C22×D127—22no—
43Dic254—21no—
44Dic254—22no—
45Dic254D411yescharacteristic
46Dic254C2312yes—
47D508—22no—
48D508—21no—
49D508D411yescharacteristic
50D508C2312yes—
51C22⋊D127—22no—
52C22⋊D127—22no—
53C22⋊D127D412yes—
54C2×Dic127—21no—
55C2×Dic127—22no—
56C4×D127—22no—
57C4×D127—22no—
58C4×D127C2312yes—
59C127⋊C8C2312yes—
60Q8×C127C2311yescharacteristic
61Q8×C127C2311yescharacteristic
62D4×C127—21no—
63D4×C127—21no—
64D4×C127C2311yescharacteristic
65D4×C127C2311yescharacteristic
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127C2311yescharacteristic
69C8×C127C2311yescharacteristic
70C8×C127C2311yescharacteristic
71C22×C127—21no—
72C22×C127—21no—
73C22×C127C2×D411yescharacteristic
74C2×D127—22no—
75C2×D127—82no—
76C2×D127—81no—
77C2×D127C2×D412yes—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127C2×D411yescharacteristic
81C4×C127C2411yescharacteristic
82Dic127—22no—
83Dic127C2×D412yes—
84C2×C127—21no—
85C2×C127—41no—
86C2×C127—41no—
87C2×C127C22×D411yescharacteristic
88D127—22no—
89D127—42no—
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#135D12711yescharacteristic, complemented
100C2×SD16—1272no—
101C2×SD16—1271no—
102#131—1271no—
103#41—1271no—
104#41—1272no—
105#41—1272no—
106#41—2541no—
107#41—2541no—
108#41—2542no—
109#41C2×D12711yescharacteristic
110#41C2×D12711yescharacteristic, complemented
111C2×Q8—1272no—
112C2×Q8—1271no—
113C2×Q8—2541no—
114C2×D4—1272no—
115C2×D4—1271no—
116C2×D4—2541no—
117C2×D4—2542no—
118Q16—1272no—
119Q16C2×D12711yescharacteristic, complemented
120SD16—1272no—
121SD16—1272no—
122SD16—1272no—
123SD16—1272no—
124SD16C2×D12711yescharacteristic, complemented
125SD16C2×D12711yescharacteristic, complemented
126D8—1272no—
127D8C2×D12711yescharacteristic, complemented
128M16—1271no—
129M16—1272no—
130C2×C8—1272no—
131C2×C8C2×D12711yescharacteristic, complemented
132C23—2542no—
133C23—2541no—
134Q8—1271no—
135Q8—1272no—
136Q8—2542no—
137Q8—2541no—
138Q8C22×D12711yescharacteristic
139Q8C22×D12711yescharacteristic
140D4—21no—
141D4—21no—
142D4—1272no—
143D4—1271no—
144D4—1272no—
145D4—2542no—
146D4—2542no—
147D4—2542no—
148D4—2541no—
149D4C22×D12711yescharacteristic
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—1272no—
160C8C22×D12711yescharacteristic
161C8C22×D12711yescharacteristic
162C22—21no—
163C22—21no—
164C22—1272no—
165C22—10161no—
166C22—10162no—
167C22—2542no—
168C22D4×D12711yescharacteristic
169C4—21no—
170C4—21no—
171C4—1272no—
172C4—2542no—
173C4D4×D12711yescharacteristic
174C4C23×D12711yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—5082no—
179C2—2542no—
180C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC2C2×D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
Fitting#135×C127C2
socleC2×C127C22×D4
hypercenter#135D127
hypocenterC127#577
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #16861256C2semidirect
2. #16861255C2semidirect
3. #16861241C2semidirect
4. #16861239C2semidirect
5. #16861234C2semidirect
6. #16861233C2semidirect
7. #16861230C2semidirect
8. C2×#14523237C2semidirect
9. SD16×D127C2semidirect
10. #16861215C2semidirect
11. #135×C127C2semidirect
17. #14523247C22semidirect
19. C2×Dic254C22semidirect
20. Q8×D127C22semidirect
25. #14523238C22semidirect
26. #14523237C22semidirect
27. #14523237C22semidirect
28. #14523236C22semidirect
29. #14523235C22semidirect
30. #14523233C22semidirect
31. #14523232C22semidirect
32. #14523231C22semidirect
33. C8×D127C22semidirect
35. #41×C127C22semidirect
36. Q16×C127C22semidirect
37. SD16×C127C22semidirect
38. SD16×C127C22semidirect
39. D8×C127C22semidirect
40. C2×C8×C127C22semidirect
90. C127#577semidirect
99. #135D127semidirect
110. #41C2×D127semidirect
119. Q16C2×D127semidirect
124. SD16C2×D127semidirect
125. SD16C2×D127semidirect
127. D8C2×D127semidirect
131. C2×C8C2×D127semidirect