Finite Group Databasebeta

← C2×AGL(1,ℤ/8)×C269 #31770310

Subgroups — C2×AGL(1,ℤ/8)×C269 #31770310

Order 17216.

298 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 162 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×#41×C269C211yesmaximal, characteristic, complemented
2C22×D4×C269C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×C269C218yesmaximal, direct product factor
4C2×SD16×C269C212yesmaximal, complemented
5C2×D8×C269C212yesmaximal, complemented
6C2×M16×C269C211yesmaximal, characteristic, complemented
7C24×C269—21no—
8#41×C269—22no—
9#41×C269C2214yescomplemented
10C2×Q8×C269C2211yescharacteristic, complemented
11C2×D4×C269—21no—
12C2×D4×C269—22no—
13C2×D4×C269—21no—
14C2×D4×C269C2212yes—
15C2×D4×C269C2214yescomplemented
16C2×D4×C269C2211yescharacteristic, complemented
17C22×C4×C269—21no—
18C22×C4×C269C2211yescharacteristic
19SD16×C269C2218yescomplemented
20D8×C269C2218yescomplemented
21M16×C269C2214yescomplemented
22C2×C8×C269C2212yescomplemented
23C23×C269—24no—
24C23×C269—21no—
25C23×C269—22no—
26C23×C269—44no—
27C23×C269D411yescharacteristic
28Q8×C269—21no—
29Q8×C269C2312yescomplemented
30D4×C269—24no—
31D4×C269—22no—
32D4×C269—24no—
33D4×C269—21no—
34D4×C269C2314yes—
35D4×C269C2312yescomplemented
36C2×C4×C269—21no—
37C2×C4×C269—24no—
38C2×C4×C269D412yes—
39C2×C4×C269D411yescharacteristic
40C2×C4×C269C2312yes—
41C2×C4×C269C2311yescharacteristic
42C8×C269C2314yescomplemented
43C22×C269—22no—
44C22×C269—22no—
45C22×C269—24no—
46C22×C269—48no—
47C22×C269—42no—
48C22×C269—44no—
49C22×C269C2×D412yes—
50C22×C269C2×D411yescharacteristic
51C4×C269—22no—
52C4×C269C2×D412yes—
53C4×C269C2×D411yescharacteristic
54C4×C269C2411yescharacteristic
55C2×C269—22no—
56C2×C269—42no—
57C2×C269—44no—
58C2×C269AGL(1,ℤ/8)12yesdirect product factor
59C2×C269C22×D411yescharacteristic
60C269C2×AGL(1,ℤ/8)11yescharacteristic, direct product factor, Sylow
61C2×AGL(1,ℤ/8)C26911yesmaximal, characteristic, direct product factor, Sylow
62C2×#41C2×C26911yescharacteristic, complemented
63C22×D4C2×C26911yescharacteristic, complemented
64AGL(1,ℤ/8)C2×C26918yesdirect product factor
65C2×SD16C2×C26912yescomplemented
66C2×D8C2×C26912yescomplemented
67C2×M16C2×C26911yescharacteristic, complemented
68C24—21no—
69#41—22no—
70#41C22×C26914yescomplemented
71C2×Q8C22×C26911yescharacteristic, complemented
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—22no—
75C2×D4C22×C26912yes—
76C2×D4C22×C26914yescomplemented
77C2×D4C22×C26911yescharacteristic, complemented
78C22×C4—21no—
79C22×C4C22×C26911yescharacteristic
80SD16C22×C26918yescomplemented
81D8C22×C26918yescomplemented
82M16C22×C26914yescomplemented
83C2×C8C22×C26912yescomplemented
84C23—21no—
85C23—22no—
86C23—24no—
87C23—44no—
88C23D4×C26911yescharacteristic
89Q8—21no—
90Q8C23×C26912yescomplemented
91D4—24no—
92D4—22no—
93D4—24no—
94D4—21no—
95D4C23×C26914yes—
96D4C23×C26912yescomplemented
97C2×C4—21no—
98C2×C4—24no—
99C2×C4D4×C26911yescharacteristic
100C2×C4D4×C26912yes—
101C2×C4C23×C26911yescharacteristic
102C2×C4C23×C26912yes—
103C8C23×C26914yescomplemented
104C22—22no—
105C22—22no—
106C22—24no—
107C22—48no—
108C22—44no—
109C22—42no—
110C22C2×D4×C26912yes—
111C22C2×D4×C26911yescharacteristic
112C4—22no—
113C4C2×D4×C26912yes—
114C4C2×D4×C26911yescharacteristic
115C4C24×C26911yescharacteristic
116C2—22no—
117C2—44no—
118C2—42no—
119C2AGL(1,ℤ/8)×C26912yesdirect product factor
120C2C22×D4×C26911yescharacteristic
subgroupG/H
center, socleC22×C269C2×D4
commutator, Frattini, cosocleC4C24×C269
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C269 ◁ C22×C4×C269 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#41×C269C2semidirect
2. C22×D4×C269C2semidirect
3. AGL(1,ℤ/8)×C269C2direct (N × H)
4. C2×SD16×C269C2semidirect
5. C2×D8×C269C2semidirect
6. C2×M16×C269C2semidirect
9. #41×C269C22semidirect
10. C2×Q8×C269C22semidirect
15. C2×D4×C269C22semidirect
16. C2×D4×C269C22semidirect
19. SD16×C269C22semidirect
20. D8×C269C22semidirect
21. M16×C269C22semidirect
22. C2×C8×C269C22semidirect
29. Q8×C269C23semidirect
35. D4×C269C23semidirect
42. C8×C269C23semidirect
58. C2×C269AGL(1,ℤ/8)direct (N × H)
60. C269C2×AGL(1,ℤ/8)direct (N × H)
61. C2×AGL(1,ℤ/8)C269direct (N × H)
62. C2×#41C2×C269semidirect
63. C22×D4C2×C269semidirect
64. AGL(1,ℤ/8)C2×C269direct (N × H)
65. C2×SD16C2×C269semidirect
66. C2×D8C2×C269semidirect
67. C2×M16C2×C269semidirect
70. #41C22×C269semidirect
71. C2×Q8C22×C269semidirect
76. C2×D4C22×C269semidirect
77. C2×D4C22×C269semidirect
80. SD16C22×C269semidirect
81. D8C22×C269semidirect
82. M16C22×C269semidirect
83. C2×C8C22×C269semidirect
90. Q8C23×C269semidirect
96. D4C23×C269semidirect
103. C8C23×C269semidirect
119. C2AGL(1,ℤ/8)×C269direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#41×C2692
C22×D4×C2692
AGL(1,ℤ/8)×C2692
C2×SD16×C2692
C2×D8×C2692
C2×M16×C2692
C2×AGL(1,ℤ/8)269