Finite Group Databasebeta

← #22550675

Subgroups — #22550675

Order 8128.

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

normal subgroups only

This group has 262 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
1C2×#14523249C211yesmaximal, characteristic, complemented
2#41×D127C211yesmaximal, characteristic, complemented
3#16861241C212yesmaximal, complemented
4#16861240C211yesmaximal, characteristic, complemented
5#16861237C211yesmaximal, characteristic, complemented
6#16861236C212yesmaximal, complemented
7#16861234C211yesmaximal, characteristic, complemented
8#16861232C212yesmaximal, complemented
9#16861230C212yesmaximal, complemented
10M16×D127C211yesmaximal, characteristic, complemented
11#137×C127C211yesmaximal, characteristic, complemented
12#14523249—21no—
13#14523249—21no—
14#14523249C2212yes—
15#14523249C2212yescomplemented
16#14523249C2211yescharacteristic, complemented
17#14523248—21no—
18#14523248C2211yescharacteristic
19#14523247—21no—
20#14523247C2211yescharacteristic, complemented
21Q8×D127C2211yescharacteristic, complemented
22C2×D508—21no—
23C2×D508C2211yescharacteristic, complemented
24D4×D127—21no—
25D4×D127C2211yescharacteristic, complemented
26C2×C4×D127—21no—
27C2×C4×D127—21no—
28C2×C4×D127C2211yescharacteristic
29#14523238C2212yescomplemented
30#14523237C2212yescomplemented
31#14523235C2212yescomplemented
32#14523235C2212yescomplemented
33D1016C2212yescomplemented
34#14523233C2212yescomplemented
35#14523232C2211yescharacteristic, complemented
36#14523231C2212yescomplemented
37C8×D127C2212yescomplemented
38#41×C127C2211yescharacteristic
39C2×Q8×C127C2211yescharacteristic, complemented
40Q16×C127C2212yescomplemented
41SD16×C127C2212yescomplemented
42M16×C127C2211yescharacteristic, complemented
43C22×D127—21no—
44C22×D127—21no—
45C22×D127D411yescharacteristic
46Dic254—21no—
47Dic254C2311yescharacteristic
48D508—21no—
49D508—22no—
50D508—22no—
51D508—21no—
52D508C2312yes—
53D508C2311yescharacteristic
54C22⋊D127—21no—
55C22⋊D127—21no—
56C22⋊D127—21no—
57C2×Dic127—21no—
58C2×Dic127D411yescharacteristic
59C4×D127—21no—
60C4×D127—21no—
61C4×D127—21no—
62C4×D127—22no—
63C4×D127—22no—
64C4×D127D411yescharacteristic
65C4×D127D411yescharacteristic
66C4×D127C2311yescharacteristic
67C4×D127C2311yescharacteristic
68C127⋊C8C2312yes—
69Q8×C127—21no—
70Q8×C127C2312yes—
71Q8×C127C2311yescharacteristic
72D4×C127—21no—
73D4×C127C2311yescharacteristic
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127C2311yescharacteristic
77C8×C127C2312yes—
78C22×C127—21no—
79C22×C127C2×D411yescharacteristic
80C2×D127—21no—
81C2×D127—22no—
82C2×D127—41no—
83C2×D127—42no—
84C2×D127—42no—
85C2×D127C2×D411yescharacteristic
86C2×D127C2×D411yescharacteristic
87C4×C127—21no—
88C4×C127—22no—
89C4×C127C2×D411yescharacteristic
90C4×C127C2411yescharacteristic
91Dic127—21no—
92Dic127C2×D411yescharacteristic
93Dic127C2×D411yescharacteristic
94C2×C127—21no—
95C2×C127—41no—
96C2×C127C22×D411yescharacteristic
97D127—21no—
98D127—21no—
99D127—41no—
100D127—42no—
101C127#57511yescharacteristic, complemented, Sylow
102#575—1271nomaximal, Sylow
103C2×#41—1271no—
104C2×#41—1271no—
105#137—1271no—
106#137—1272no—
107#137D12711yescharacteristic, complemented
108AGL(1,ℤ/8)—1271no—
109AGL(1,ℤ/8)—1272no—
110AGL(1,ℤ/8)—1271no—
111#135—1272no—
112#135—1272no—
113C2×M16—1271no—
114#41—1272no—
115#41—1271no—
116#41—1272no—
117#41—1271no—
118#41—1271no—
119#41—2541no—
120#41—2541no—
121#41—2541no—
122#41—2541no—
123#41C2×D12711yescharacteristic
124C2×Q8—1271no—
125C2×Q8C2×D12711yescharacteristic, complemented
126C2×D4—1271no—
127C2×D4—1271no—
128C2×D4—2541no—
129C2×D4—2541no—
130C22×C4—1271no—
131C22×C4—2541no—
132C22×C4—2541no—
133Q16—1272no—
134Q16C2×D12712yescomplemented
135SD16—1272no—
136SD16—1272no—
137SD16—1272no—
138SD16C2×D12712yescomplemented
139D8—1272no—
140D8—1272no—
141M16—1271no—
142M16—1272no—
143M16C2×D12711yescharacteristic, complemented
144C2×C8—1272no—
145C23—1271no—
146C23—2541no—
147C23—2541no—
148Q8—21no—
149Q8—1271no—
150Q8—2541no—
151Q8C22×D12711yescharacteristic
152Q8C22×D12712yes—
153D4—21no—
154D4—1272no—
155D4—1271no—
156D4—2541no—
157D4—2542no—
158D4—2542no—
159D4—2541no—
160D4—2541no—
161D4—2541no—
162D4—2541no—
163D4C22×D12711yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—1271no—
167C2×C4—1271no—
168C2×C4—1271no—
169C2×C4—1271no—
170C2×C4—1271no—
171C2×C4—2541no—
172C2×C4—2542no—
173C2×C4—2542no—
174C2×C4—2541no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4C22×D12711yescharacteristic
178C8—1272no—
179C8C22×D12712yes—
180C22—21no—
181C22—1271no—
182C22—1271no—
183C22—5082no—
184C22—5081no—
185C22—5082no—
186C22—2542no—
187C22—2541no—
188C22D4×D12711yescharacteristic
189C4—21no—
190C4—22no—
191C4—1271no—
192C4—1271no—
193C4—2541no—
194C4D4×D12711yescharacteristic
195C4C23×D12711yescharacteristic
196C2—21no—
197C2—41no—
198C2—5082no—
199C2—5081no—
200C2—2541no—
201C2—2541no—
202C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC2C2×D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
Fitting#137×C127C2
socleC2×C127C22×D4
hypercenter#137D127
hypocenterC127#575
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. C2×#14523249C2semidirect
2. #41×D127C2semidirect
3. #16861241C2semidirect
4. #16861240C2semidirect
5. #16861237C2semidirect
6. #16861236C2semidirect
7. #16861234C2semidirect
8. #16861232C2semidirect
9. #16861230C2semidirect
10. M16×D127C2semidirect
11. #137×C127C2semidirect
15. #14523249C22semidirect
16. #14523249C22semidirect
20. #14523247C22semidirect
21. Q8×D127C22semidirect
23. C2×D508C22semidirect
25. D4×D127C22semidirect
29. #14523238C22semidirect
30. #14523237C22semidirect
31. #14523235C22semidirect
32. #14523235C22semidirect
33. D1016C22semidirect
34. #14523233C22semidirect
35. #14523232C22semidirect
36. #14523231C22semidirect
37. C8×D127C22semidirect
39. C2×Q8×C127C22semidirect
40. Q16×C127C22semidirect
41. SD16×C127C22semidirect
42. M16×C127C22semidirect
101. C127#575semidirect
107. #137D127semidirect
125. C2×Q8C2×D127semidirect
134. Q16C2×D127semidirect
138. SD16C2×D127semidirect
143. M16C2×D127semidirect