Finite Group Databasebeta

← C2×AGL(1,ℤ/8)×C71 #17985704

Subgroups — C2×AGL(1,ℤ/8)×C71 #17985704

Order 4544.

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×C71C211yesmaximal, characteristic, complemented
2C22×D4×C71C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×C71C218yesmaximal, direct product factor
4C2×SD16×C71C212yesmaximal, complemented
5C2×D8×C71C212yesmaximal, complemented
6C2×M16×C71C211yesmaximal, characteristic, complemented
7C24×C71—21no—
8#41×C71—22no—
9#41×C71C2214yescomplemented
10C2×Q8×C71C2211yescharacteristic, complemented
11C2×D4×C71—21no—
12C2×D4×C71—22no—
13C2×D4×C71—21no—
14C2×D4×C71C2212yes—
15C2×D4×C71C2214yescomplemented
16C2×D4×C71C2211yescharacteristic, complemented
17C22×C4×C71—21no—
18C22×C4×C71C2211yescharacteristic
19SD16×C71C2218yescomplemented
20D8×C71C2218yescomplemented
21M16×C71C2214yescomplemented
22C2×C8×C71C2212yescomplemented
23C23×C71—24no—
24C23×C71—21no—
25C23×C71—22no—
26C23×C71—44no—
27C23×C71D411yescharacteristic
28Q8×C71—21no—
29Q8×C71C2312yescomplemented
30D4×C71—24no—
31D4×C71—22no—
32D4×C71—24no—
33D4×C71—21no—
34D4×C71C2314yes—
35D4×C71C2312yescomplemented
36C2×C4×C71—21no—
37C2×C4×C71—24no—
38C2×C4×C71D412yes—
39C2×C4×C71D411yescharacteristic
40C2×C4×C71C2312yes—
41C2×C4×C71C2311yescharacteristic
42C8×C71C2314yescomplemented
43C22×C71—22no—
44C22×C71—22no—
45C22×C71—24no—
46C22×C71—48no—
47C22×C71—42no—
48C22×C71—44no—
49C22×C71C2×D412yes—
50C22×C71C2×D411yescharacteristic
51C4×C71—22no—
52C4×C71C2×D412yes—
53C4×C71C2×D411yescharacteristic
54C4×C71C2411yescharacteristic
55C2×C71—22no—
56C2×C71—42no—
57C2×C71—44no—
58C2×C71AGL(1,ℤ/8)12yesdirect product factor
59C2×C71C22×D411yescharacteristic
60C71C2×AGL(1,ℤ/8)11yescharacteristic, direct product factor, Sylow
61C2×AGL(1,ℤ/8)C7111yesmaximal, characteristic, direct product factor, Sylow
62C2×#41C2×C7111yescharacteristic, complemented
63C22×D4C2×C7111yescharacteristic, complemented
64AGL(1,ℤ/8)C2×C7118yesdirect product factor
65C2×SD16C2×C7112yescomplemented
66C2×D8C2×C7112yescomplemented
67C2×M16C2×C7111yescharacteristic, complemented
68C24—21no—
69#41—22no—
70#41C22×C7114yescomplemented
71C2×Q8C22×C7111yescharacteristic, complemented
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—22no—
75C2×D4C22×C7112yes—
76C2×D4C22×C7114yescomplemented
77C2×D4C22×C7111yescharacteristic, complemented
78C22×C4—21no—
79C22×C4C22×C7111yescharacteristic
80SD16C22×C7118yescomplemented
81D8C22×C7118yescomplemented
82M16C22×C7114yescomplemented
83C2×C8C22×C7112yescomplemented
84C23—21no—
85C23—22no—
86C23—24no—
87C23—44no—
88C23D4×C7111yescharacteristic
89Q8—21no—
90Q8C23×C7112yescomplemented
91D4—24no—
92D4—22no—
93D4—24no—
94D4—21no—
95D4C23×C7114yes—
96D4C23×C7112yescomplemented
97C2×C4—21no—
98C2×C4—24no—
99C2×C4D4×C7111yescharacteristic
100C2×C4D4×C7112yes—
101C2×C4C23×C7111yescharacteristic
102C2×C4C23×C7112yes—
103C8C23×C7114yescomplemented
104C22—22no—
105C22—22no—
106C22—24no—
107C22—48no—
108C22—44no—
109C22—42no—
110C22C2×D4×C7112yes—
111C22C2×D4×C7111yescharacteristic
112C4—22no—
113C4C2×D4×C7112yes—
114C4C2×D4×C7111yescharacteristic
115C4C24×C7111yescharacteristic
116C2—22no—
117C2—44no—
118C2—42no—
119C2AGL(1,ℤ/8)×C7112yesdirect product factor
120C2C22×D4×C7111yescharacteristic
subgroupG/H
center, socleC22×C71C2×D4
commutator, Frattini, cosocleC4C24×C71
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#41×C71C2semidirect
2. C22×D4×C71C2semidirect
3. AGL(1,ℤ/8)×C71C2direct (N × H)
4. C2×SD16×C71C2semidirect
5. C2×D8×C71C2semidirect
6. C2×M16×C71C2semidirect
9. #41×C71C22semidirect
10. C2×Q8×C71C22semidirect
15. C2×D4×C71C22semidirect
16. C2×D4×C71C22semidirect
19. SD16×C71C22semidirect
20. D8×C71C22semidirect
21. M16×C71C22semidirect
22. C2×C8×C71C22semidirect
29. Q8×C71C23semidirect
35. D4×C71C23semidirect
42. C8×C71C23semidirect
58. C2×C71AGL(1,ℤ/8)direct (N × H)
60. C71C2×AGL(1,ℤ/8)direct (N × H)
61. C2×AGL(1,ℤ/8)C71direct (N × H)
62. C2×#41C2×C71semidirect
63. C22×D4C2×C71semidirect
64. AGL(1,ℤ/8)C2×C71direct (N × H)
65. C2×SD16C2×C71semidirect
66. C2×D8C2×C71semidirect
67. C2×M16C2×C71semidirect
70. #41C22×C71semidirect
71. C2×Q8C22×C71semidirect
76. C2×D4C22×C71semidirect
77. C2×D4C22×C71semidirect
80. SD16C22×C71semidirect
81. D8C22×C71semidirect
82. M16C22×C71semidirect
83. C2×C8C22×C71semidirect
90. Q8C23×C71semidirect
96. D4C23×C71semidirect
103. C8C23×C71semidirect
119. C2AGL(1,ℤ/8)×C71direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41×C712
C22×D4×C712
AGL(1,ℤ/8)×C712
C2×SD16×C712
C2×D8×C712
C2×M16×C712
C2×AGL(1,ℤ/8)71