Finite Group Databasebeta

← #22550672

Subgroups — #22550672

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×#14523247C211yesmaximal, characteristic, complemented
2#41×D127C211yesmaximal, characteristic, complemented
3#16861242C211yesmaximal, characteristic, complemented
4#16861239C211yesmaximal, characteristic, complemented
5#16861238C212yesmaximal, complemented
6#16861235C211yesmaximal, characteristic, complemented
7#16861233C212yesmaximal, complemented
8#16861230C212yesmaximal, complemented
9#16861228C212yesmaximal, complemented
10M16×D127C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12#14523249C2211yescharacteristic, complemented
13#14523248—21no—
14#14523248C2211yescharacteristic
15#14523247—21no—
16#14523247—21no—
17#14523247—21no—
18#14523247C2212yes—
19#14523247C2211yescharacteristic, complemented
20#14523247C2212yescomplemented
21C2×Dic254C2211yescharacteristic, complemented
22Q8×D127C2211yescharacteristic, complemented
23C2×C22⋊D127—21no—
24D4×D127—21no—
25D4×D127C2211yescharacteristic, complemented
26C22×Dic127—21no—
27C2×C4×D127—21no—
28C2×C4×D127C2211yescharacteristic
29Dic508C2212yescomplemented
30#14523238C2212yescomplemented
31#14523237C2212yescomplemented
32#14523236C2212yescomplemented
33#14523236C2212yescomplemented
34#14523233C2212yescomplemented
35#14523232C2211yescharacteristic, complemented
36#14523231C2212yescomplemented
37C8×D127C2212yescomplemented
38#41×C127C2211yescharacteristic
39C2×D4×C127C2211yescharacteristic, complemented
40SD16×C127C2212yescomplemented
41D8×C127C2212yescomplemented
42M16×C127C2211yescharacteristic, complemented
43C22×D127—21no—
44C22×D127D411yescharacteristic
45Dic254—21no—
46Dic254—21no—
47Dic254C2311yescharacteristic
48Dic254C2312yes—
49D508—21no—
50D508C2311yescharacteristic
51C22⋊D127—21no—
52C22⋊D127—21no—
53C22⋊D127—22no—
54C22⋊D127—22no—
55C22⋊D127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127D411yescharacteristic
61C4×D127—21no—
62C4×D127—21no—
63C4×D127—21no—
64C4×D127D411yescharacteristic
65C4×D127D411yescharacteristic
66C4×D127C2311yescharacteristic
67C4×D127C2311yescharacteristic
68C127⋊C8C2312yes—
69C23×C127—21no—
70Q8×C127C2311yescharacteristic
71D4×C127—21no—
72D4×C127—21no—
73D4×C127C2311yescharacteristic
74D4×C127C2312yes—
75C2×C4×C127—21no—
76C2×C4×C127C2311yescharacteristic
77C8×C127C2312yes—
78C22×C127—21no—
79C22×C127—22no—
80C22×C127—42no—
81C22×C127C2×D411yescharacteristic
82C2×D127—21no—
83C2×D127—41no—
84C2×D127—42no—
85C2×D127C2×D411yescharacteristic
86C2×D127C2×D411yescharacteristic
87C4×C127—21no—
88C4×C127C2×D411yescharacteristic
89C4×C127C2411yescharacteristic
90Dic127—21no—
91Dic127—22no—
92Dic127C2×D411yescharacteristic
93Dic127C2×D411yescharacteristic
94C2×C127—21no—
95C2×C127—41no—
96C2×C127—42no—
97C2×C127C22×D411yescharacteristic
98D127—21no—
99D127—21no—
100D127—41no—
101C127#57511yescharacteristic, complemented, Sylow
102#575—1271nomaximal, Sylow
103C2×#41—1271no—
104C2×#41—1271no—
105#137—1271no—
106#137—1272no—
107#137—1271no—
108AGL(1,ℤ/8)—1271no—
109AGL(1,ℤ/8)—1272no—
110AGL(1,ℤ/8)D12711yescharacteristic, complemented
111#135—1272no—
112#135—1272no—
113C2×M16—1271no—
114#41—1271no—
115#41—1272no—
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×Q8—1271no—
126C2×D4—1271no—
127C2×D4—2541no—
128C2×D4—2541no—
129C2×D4C2×D12711yescharacteristic, complemented
130C22×C4—1271no—
131C22×C4—2541no—
132C22×C4—2541no—
133Q16—1272no—
134Q16—1272no—
135SD16—1272no—
136SD16—1272no—
137SD16—1272no—
138SD16C2×D12712yescomplemented
139D8—1272no—
140D8C2×D12712yescomplemented
141M16—1271no—
142M16—1272no—
143M16C2×D12711yescharacteristic, complemented
144C2×C8—1272no—
145C23—21no—
146C23—1271no—
147C23—2541no—
148Q8—1271no—
149Q8—1272no—
150Q8—2541no—
151Q8—2541no—
152Q8C22×D12711yescharacteristic
153D4—21no—
154D4—21no—
155D4—1271no—
156D4—2541no—
157D4—2541no—
158D4—2541no—
159D4—2542no—
160D4—2542no—
161D4—2541no—
162D4C22×D12712yes—
163D4C22×D12711yescharacteristic
164C2×C4—21no—
165C2×C4—1271no—
166C2×C4—1271no—
167C2×C4—1271no—
168C2×C4—1271no—
169C2×C4—1271no—
170C2×C4—2541no—
171C2×C4—2541no—
172C2×C4—2541no—
173C2×C4—2542no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4C22×D12711yescharacteristic
178C8—1272no—
179C8C22×D12712yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—1271no—
184C22—1271no—
185C22—5081no—
186C22—5082no—
187C22—2541no—
188C22D4×D12711yescharacteristic
189C4—21no—
190C4—1271no—
191C4—1271no—
192C4—2541no—
193C4—2542no—
194C4D4×D12711yescharacteristic
195C4C23×D12711yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—5081no—
200C2—2541no—
201C2—2541no—
202C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC2C2×D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
FittingAGL(1,ℤ/8)×C127C2
socleC2×C127C22×D4
hypercenterAGL(1,ℤ/8)D127
hypocenterC127#575
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#14523247C2semidirect
2. #41×D127C2semidirect
3. #16861242C2semidirect
4. #16861239C2semidirect
5. #16861238C2semidirect
6. #16861235C2semidirect
7. #16861233C2semidirect
8. #16861230C2semidirect
9. #16861228C2semidirect
10. M16×D127C2semidirect
11. AGL(1,ℤ/8)×C127C2semidirect
12. #14523249C22semidirect
19. #14523247C22semidirect
20. #14523247C22semidirect
21. C2×Dic254C22semidirect
22. Q8×D127C22semidirect
25. D4×D127C22semidirect
29. Dic508C22semidirect
30. #14523238C22semidirect
31. #14523237C22semidirect
32. #14523236C22semidirect
33. #14523236C22semidirect
34. #14523233C22semidirect
35. #14523232C22semidirect
36. #14523231C22semidirect
37. C8×D127C22semidirect
39. C2×D4×C127C22semidirect
40. SD16×C127C22semidirect
41. D8×C127C22semidirect
42. M16×C127C22semidirect
101. C127#575semidirect
110. AGL(1,ℤ/8)D127semidirect
129. C2×D4C2×D127semidirect
138. SD16C2×D127semidirect
140. D8C2×D127semidirect
143. M16C2×D127semidirect