Finite Group Databasebeta

← AGL(1,ℤ/8)×D127 #22550660

Subgroups — AGL(1,ℤ/8)×D127 #22550660

Order 8128.

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

normal subgroups only

This group has 298 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×D127C211yesmaximal, characteristic, complemented
2C2×D4×D127C211yesmaximal, characteristic, complemented
3#16861237C211yesmaximal, characteristic, complemented
4#16861236C212yesmaximal, complemented
5#16861235C211yesmaximal, characteristic, complemented
6#16861234C211yesmaximal, characteristic, complemented
7#16861233C212yesmaximal, complemented
8SD16×D127C212yesmaximal, complemented
9D8×D127C212yesmaximal, complemented
10M16×D127C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C23×D127—21no—
13#14523249C2211yescharacteristic, complemented
14#14523248—21no—
15#14523248C2211yescharacteristic, complemented
16#14523247—21no—
17#14523247C2211yescharacteristic, complemented
18Q8×D127C2211yescharacteristic, complemented
19C2×D508C2211yescharacteristic, complemented
20C2×C22⋊D127—21no—
21D4×D127—21no—
22D4×D127—21no—
23D4×D127—21no—
24D4×D127C2212yes—
25D4×D127C2212yescomplemented
26D4×D127C2211yescharacteristic, complemented
27C2×C4×D127—21no—
28C2×C4×D127C2211yescharacteristic
29#14523237C2212yescomplemented
30#14523236C2212yescomplemented
31#14523235C2212yescomplemented
32D1016C2212yescomplemented
33#14523233C2212yescomplemented
34#14523233C2212yescomplemented
35#14523232C2211yescharacteristic, complemented
36#14523231C2212yescomplemented
37C8×D127C2212yescomplemented
38#41×C127C2211yescharacteristic, complemented
39C2×D4×C127C2211yescharacteristic, complemented
40SD16×C127C2212yescomplemented
41D8×C127C2212yescomplemented
42M16×C127C2211yescharacteristic, complemented
43C22×D127—21no—
44C22×D127—21no—
45C22×D127—22no—
46C22×D127—22no—
47C22×D127—44no—
48C22×D127D411yescharacteristic
49Dic254—21no—
50Dic254C2311yescharacteristic, complemented
51D508—21no—
52D508—21no—
53D508C2312yes—
54D508C2311yescharacteristic, complemented
55C22⋊D127—21no—
56C22⋊D127—21no—
57C22⋊D127—22no—
58C22⋊D127—22no—
59C22⋊D127—21no—
60C2×Dic127—21no—
61C2×Dic127D411yescharacteristic
62C4×D127—21no—
63C4×D127—21no—
64C4×D127—21no—
65C4×D127D411yescharacteristic
66C4×D127D411yescharacteristic
67C4×D127C2311yescharacteristic
68C4×D127C2311yescharacteristic
69C127⋊C8C2312yescomplemented
70C23×C127—21no—
71Q8×C127C2311yescharacteristic, complemented
72D4×C127—21no—
73D4×C127—21no—
74D4×C127C2312yes—
75D4×C127C2311yescharacteristic, complemented
76C2×C4×C127—21no—
77C2×C4×C127C2311yescharacteristic
78C8×C127C2312yescomplemented
79C22×C127—21no—
80C22×C127—22no—
81C22×C127—42no—
82C22×C127C2×D411yescharacteristic
83C2×D127—22no—
84C2×D127—21no—
85C2×D127—22no—
86C2×D127—44no—
87C2×D127—42no—
88C2×D127—42no—
89C2×D127—44no—
90C2×D127C2×D411yescharacteristic
91C2×D127C2×D411yescharacteristic
92C4×C127—21no—
93C4×C127C2×D411yescharacteristic
94C4×C127C2411yescharacteristic
95Dic127—21no—
96Dic127C2×D411yescharacteristic
97Dic127C2×D411yescharacteristic
98C2×C127—21no—
99C2×C127—42no—
100C2×C127—41no—
101C2×C127C22×D411yescharacteristic
102D127—21no—
103D127—41no—
104D127—42no—
105D127AGL(1,ℤ/8)12yesdirect product factor
106C127C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—1271nomaximal, Sylow
108C2×#41—1271no—
109C22×D4—1271no—
110AGL(1,ℤ/8)—1271no—
111AGL(1,ℤ/8)—1271no—
112AGL(1,ℤ/8)—1272no—
113AGL(1,ℤ/8)—1272no—
114AGL(1,ℤ/8)—1271no—
115AGL(1,ℤ/8)D12711yescharacteristic, direct product factor
116C2×SD16—1272no—
117C2×D8—1272no—
118C2×M16—1271no—
119C24—2541no—
120#41—1271no—
121#41—1271no—
122#41—1271no—
123#41—2541no—
124#41—2541no—
125#41C2×D12711yescharacteristic, complemented
126C2×Q8—1271no—
127C2×D4—1272no—
128C2×D4—1272no—
129C2×D4—1271no—
130C2×D4—1271no—
131C2×D4—2541no—
132C2×D4—2541no—
133C2×D4—2541no—
134C2×D4—2541no—
135C2×D4C2×D12711yescharacteristic, complemented
136C22×C4—1271no—
137C22×C4—2541no—
138SD16—1272no—
139SD16—1272no—
140SD16—1272no—
141SD16C2×D12712yescomplemented
142D8—1272no—
143D8—1272no—
144D8—1272no—
145D8C2×D12712yescomplemented
146M16—1271no—
147M16—1272no—
148M16C2×D12711yescharacteristic, complemented
149C2×C8—1272no—
150C23—21no—
151C23—1271no—
152C23—5084no—
153C23—2541no—
154C23—2542no—
155C23—2542no—
156C23—2541no—
157Q8—1271no—
158Q8—2541no—
159Q8C22×D12711yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—1271no—
163D4—1272no—
164D4—2541no—
165D4—2542no—
166D4—2542no—
167D4—2541no—
168D4—2541no—
169D4—2541no—
170D4—2541no—
171D4C22×D12712yes—
172D4C22×D12711yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—1271no—
175C2×C4—1271no—
176C2×C4—1271no—
177C2×C4—1271no—
178C2×C4—1271no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4—2541no—
182C2×C4—2541no—
183C2×C4C22×D12711yescharacteristic
184C8—1272no—
185C8C22×D12712yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—1271no—
190C22—1271no—
191C22—5082no—
192C22—5084no—
193C22—5082no—
194C22—5084no—
195C22—2542no—
196C22—2541no—
197C22—2542no—
198C22D4×D12711yescharacteristic
199C4—21no—
200C4—1271no—
201C4—1271no—
202C4—2541no—
203C4D4×D12711yescharacteristic
204C4C23×D12711yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—1272no—
209C2—5082no—
210C2—5081no—
211C2—2541no—
212C2C2×D4×D12711yescharacteristic
subgroupG/H
centerC2C2×D4×D127
commutatorC4×C127C24
FrattiniC4C23×D127
FittingAGL(1,ℤ/8)×C127C2
socleC2×C127C22×D4
hypercenterAGL(1,ℤ/8)D127
hypocenterC127C2×AGL(1,ℤ/8)
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. #41×D127C2semidirect
2. C2×D4×D127C2semidirect
3. #16861237C2semidirect
4. #16861236C2semidirect
5. #16861235C2semidirect
6. #16861234C2semidirect
7. #16861233C2semidirect
8. SD16×D127C2semidirect
9. D8×D127C2semidirect
10. M16×D127C2semidirect
11. AGL(1,ℤ/8)×C127C2semidirect
13. #14523249C22semidirect
15. #14523248C22semidirect
17. #14523247C22semidirect
18. Q8×D127C22semidirect
19. C2×D508C22semidirect
25. D4×D127C22semidirect
26. D4×D127C22semidirect
29. #14523237C22semidirect
30. #14523236C22semidirect
31. #14523235C22semidirect
32. D1016C22semidirect
33. #14523233C22semidirect
34. #14523233C22semidirect
35. #14523232C22semidirect
36. #14523231C22semidirect
37. C8×D127C22semidirect
38. #41×C127C22semidirect
39. C2×D4×C127C22semidirect
40. SD16×C127C22semidirect
41. D8×C127C22semidirect
42. M16×C127C22semidirect
50. Dic254C23semidirect
54. D508C23semidirect
69. C127⋊C8C23semidirect
71. Q8×C127C23semidirect
75. D4×C127C23semidirect
78. C8×C127C23semidirect
105. D127AGL(1,ℤ/8)direct (N × H)
106. C127C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D127direct (N × H)
125. #41C2×D127semidirect
135. C2×D4C2×D127semidirect
141. SD16C2×D127semidirect
145. D8C2×D127semidirect
148. M16C2×D127semidirect
159. Q8C22×D127semidirect
172. D4C22×D127semidirect
185. C8C22×D127semidirect