Finite Group Databasebeta

← #135×D127 #22550654

Subgroups — #135×D127 #22550654

Order 8128.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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#41×D127C212yesmaximal, complemented
2#16861232C211yesmaximal, characteristic, complemented
3#16861231C211yesmaximal, characteristic, complemented
4#16861230C212yesmaximal, complemented
5#16861229C212yesmaximal, complemented
6#16861228C211yesmaximal, characteristic, complemented
7Q16×D127C211yesmaximal, characteristic, complemented
8SD16×D127C212yesmaximal, complemented
9D8×D127C211yesmaximal, characteristic, complemented
10C2×C8×D127C211yesmaximal, characteristic, complemented
11#135×C127C211yesmaximal, characteristic, complemented
12#14523249C2212yescomplemented
13#14523248—22no—
14#14523248C2212yescomplemented
15#14523247—22no—
16#14523247C2212yescomplemented
17Q8×D127C2212yes—
18D4×D127—22no—
19D4×D127C2212yes—
20C2×C4×D127—22no—
21C2×C4×D127C2211yescharacteristic
22Dic508C2211yescharacteristic, complemented
23#14523238C2212yescomplemented
24#14523237C2212yescomplemented
25#14523236C2212yescomplemented
26#14523235C2212yescomplemented
27D1016C2211yescharacteristic, complemented
28#14523233C2212yescomplemented
29C2×C127⋊C8C2211yescharacteristic, complemented
30C8×D127C2211yescharacteristic
31C8×D127C2211yescharacteristic
32C8×D127C2211yescharacteristic, complemented
33C8×D127C2211yescharacteristic, complemented
34#41×C127C2212yescomplemented
35Q16×C127C2211yescharacteristic, complemented
36SD16×C127C2212yescomplemented
37D8×C127C2211yescharacteristic, complemented
38C2×C8×C127C2211yescharacteristic, complemented
39C22×D127—22no—
40C22×D127D411yescharacteristic
41Dic254—22no—
42Dic254C2312yes—
43D508—22no—
44D508C2312yes—
45C22⋊D127—22no—
46C22⋊D127—22no—
47C22⋊D127—22no—
48C2×Dic127—22no—
49C2×Dic127D411yescharacteristic
50C4×D127—22no—
51C4×D127—22no—
52C4×D127—22no—
53C4×D127D411yescharacteristic
54C4×D127D411yescharacteristic
55C4×D127C2311yescharacteristic
56C4×D127C2311yescharacteristic
57C127⋊C8C2311yescharacteristic
58C127⋊C8C2311yescharacteristic
59Q8×C127C2312yes—
60D4×C127—22no—
61D4×C127C2312yes—
62C2×C4×C127—22no—
63C2×C4×C127C2311yescharacteristic
64C8×C127C2311yescharacteristic
65C8×C127C2311yescharacteristic
66C22×C127—22no—
67C22×C127C2×D411yescharacteristic
68C2×D127—22no—
69C2×D127—22no—
70C2×D127—44no—
71C2×D127C2×D411yescharacteristic
72C2×D127C2×D411yescharacteristic
73C4×C127—22no—
74C4×C127C2×D411yescharacteristic
75C4×C127C2411yescharacteristic
76Dic127—22no—
77Dic127C2×D411yescharacteristic
78Dic127C2×D411yescharacteristic
79C2×C127—21no—
80C2×C127—42no—
81C2×C127C22×D411yescharacteristic
82D127—21no—
83D127—42no—
84D127#13512yesdirect product factor
85C127C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—1271nomaximal, Sylow
87C2×#41—1272no—
88#135—1272no—
89#135—1271no—
90#135—1272no—
91#135—1271no—
92#135—1271no—
93#135D12711yescharacteristic, direct product factor
94C2×Q16—1271no—
95C2×SD16—1272no—
96C2×D8—1271no—
97C22×C8—1271no—
98#41—1272no—
99#41—1272no—
100#41—1272no—
101#41—2542no—
102#41—2542no—
103#41C2×D12712yescomplemented
104C2×Q8—1272no—
105C2×D4—1272no—
106C2×D4—2542no—
107C22×C4—1271no—
108C22×C4—2542no—
109Q16—1272no—
110Q16—1271no—
111Q16C2×D12711yescharacteristic, complemented
112SD16—1272no—
113SD16—1272no—
114SD16—1272no—
115SD16C2×D12712yescomplemented
116D8—1272no—
117D8—1271no—
118D8C2×D12711yescharacteristic, complemented
119C2×C8—1271no—
120C2×C8—1271no—
121C2×C8—1271no—
122C2×C8—1271no—
123C2×C8—1271no—
124C2×C8C2×D12711yescharacteristic, complemented
125C23—1271no—
126C23—2542no—
127Q8—1272no—
128Q8—2542no—
129Q8C22×D12712yes—
130D4—22no—
131D4—1272no—
132D4—2542no—
133D4—2542no—
134D4—2542no—
135D4—2542no—
136D4C22×D12712yes—
137C2×C4—22no—
138C2×C4—1271no—
139C2×C4—1271no—
140C2×C4—1271no—
141C2×C4—1271no—
142C2×C4—1271no—
143C2×C4—2542no—
144C2×C4—2542no—
145C2×C4—2542no—
146C2×C4—2542no—
147C2×C4C22×D12711yescharacteristic
148C8—1271no—
149C8—1271no—
150C8C22×D12711yescharacteristic
151C8C22×D12711yescharacteristic
152C22—22no—
153C22—1271no—
154C22—1271no—
155C22—5084no—
156C22—2542no—
157C22—2542no—
158C22D4×D12711yescharacteristic
159C4—22no—
160C4—1271no—
161C4—1271no—
162C4—2542no—
163C4D4×D12711yescharacteristic
164C4C23×D12711yescharacteristic
165C2—21no—
166C2—42no—
167C2—1272no—
168C2—5082no—
169C2—2541no—
170C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC4D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
Fitting#135×C127C2
socleC2×C127C22×D4
hypercenter#135D127
hypocenterC127C2×#135
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D127C2semidirect
2. #16861232C2semidirect
3. #16861231C2semidirect
4. #16861230C2semidirect
5. #16861229C2semidirect
6. #16861228C2semidirect
7. Q16×D127C2semidirect
8. SD16×D127C2semidirect
9. D8×D127C2semidirect
10. C2×C8×D127C2semidirect
11. #135×C127C2semidirect
12. #14523249C22semidirect
14. #14523248C22semidirect
16. #14523247C22semidirect
22. Dic508C22semidirect
23. #14523238C22semidirect
24. #14523237C22semidirect
25. #14523236C22semidirect
26. #14523235C22semidirect
27. D1016C22semidirect
28. #14523233C22semidirect
29. C2×C127⋊C8C22semidirect
32. C8×D127C22semidirect
33. C8×D127C22semidirect
34. #41×C127C22semidirect
35. Q16×C127C22semidirect
36. SD16×C127C22semidirect
37. D8×C127C22semidirect
38. C2×C8×C127C22semidirect
84. D127#135direct (N × H)
85. C127C2×#135semidirect
93. #135D127direct (N × H)
103. #41C2×D127semidirect
111. Q16C2×D127semidirect
115. SD16C2×D127semidirect
118. D8C2×D127semidirect
124. C2×C8C2×D127semidirect