Finite Group Databasebeta

← C2×#6474 #90446

Subgroups — C2×#6474 #90446

Order 448.

298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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×#1159C211yesmaximal, characteristic, complemented
2C2×D4×D7C211yesmaximal, characteristic, complemented
3C2×C7×D8C211yesmaximal, characteristic, complemented
4C2×#1142C211yesmaximal, characteristic, complemented
5C2×#1141C211yesmaximal, characteristic, complemented
6#6474C218yesmaximal, direct product factor
7C2×#1132C211yesmaximal, characteristic, complemented
8C2×#1131C211yesmaximal, characteristic, complemented
9C23×D7—21no—
10C2×C7×D4C2211yescharacteristic
11C2×C7×D4C2211yescharacteristic, complemented
12C2×C22⋊D7—21no—
13C2×C22⋊D7—21no—
14C22×Dic7—21no—
15#1159—22no—
16#1159C2214yescomplemented
17D4×D7—22no—
18D4×D7C2214yescomplemented
19C2×D28C2211yescharacteristic
20C2×C4×D7C2211yescharacteristic
21C2×Dic14C2211yescharacteristic, complemented
22C7×D8C2214yescomplemented
23C2×C7×C8C2211yescharacteristic, complemented
24#1142C2214yescomplemented
25#1141C2214yescomplemented
26C2×C7⋊C8C2211yescharacteristic, complemented
27#1132C2214yescomplemented
28#1131C2214yescomplemented
29C2×AGL(1,ℤ/8)—71nomaximal, Sylow
30C23×C7—21no—
31C23×C7—21no—
32C22×D7—21no—
33C22×D7—24no—
34C22×D7—44no—
35C22×D7D411yescharacteristic
36C7×D4—21no—
37C7×D4—21no—
38C7×D4C2312yes—
39C7×D4C2312yescomplemented
40C2×C4×C7C2311yescharacteristic
41C22⋊D7—24no—
42C22⋊D7—24no—
43C2×Dic7—21no—
44C2×Dic7—24no—
45C2×Dic7D411yescharacteristic
46D28—21no—
47D28C2312yes—
48C4×D7D412yes—
49C4×D7C2312yes—
50Dic14—21no—
51Dic14C2312yescomplemented
52C7×C8C2312yescomplemented
53C7⋊C8C2312yescomplemented
54C2×#41—71no—
55C22×D4—71no—
56AGL(1,ℤ/8)—78no—
57C2×SD16—71no—
58C2×SD16—71no—
59C2×D8—71no—
60C2×D8D711yescharacteristic, complemented
61C2×M16—71no—
62C22×C7—22no—
63C22×C7—22no—
64C22×C7—42no—
65C22×C7—42no—
66C22×C7C2×D411yescharacteristic
67C2×D7—22no—
68C2×D7—22no—
69C2×D7—48no—
70C2×D7—42no—
71C2×D7C2×D412yes—
72C4×C7C2×D411yescharacteristic
73C4×C7C2411yescharacteristic
74Dic7—22no—
75Dic7C2×D412yes—
76C24—141no—
77#41—74no—
78#41—142no—
79C2×Q8—71no—
80C2×D4—74no—
81C2×D4—71no—
82C2×D4—142no—
83C2×D4—141no—
84C2×D4—141no—
85C2×D4C2×D711yescharacteristic
86C2×D4C2×D711yescharacteristic, complemented
87C22×C4—71no—
88C22×C4—141no—
89SD16—74no—
90SD16—74no—
91D8—74no—
92D8C2×D714yescomplemented
93M16—74no—
94C2×C8—71no—
95C2×C8C2×D711yescharacteristic, complemented
96C2×C7—42no—
97C2×C7—42no—
98C2×C7AGL(1,ℤ/8)12yescomplemented
99C2×C7C22×D411yescharacteristic
100D7—22no—
101D7—42no—
102C23—21no—
103C23—21no—
104C23—71no—
105C23—144no—
106C23—141no—
107C23—284no—
108Q8—72no—
109Q8—141no—
110D4—21no—
111D4—21no—
112D4—72no—
113D4—144no—
114D4—144no—
115D4—141no—
116D4C22×D712yes—
117D4C22×D712yescomplemented
118C2×C4—71no—
119C2×C4—72no—
120C2×C4—72no—
121C2×C4—144no—
122C2×C4—141no—
123C2×C4C22×D711yescharacteristic
124C8—72no—
125C8C22×D712yescomplemented
126C7C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
127C22—22no—
128C22—22no—
129C22—42no—
130C22—42no—
131C22—72no—
132C22—142no—
133C22—142no—
134C22—288no—
135C22—282no—
136C22D4×D711yescharacteristic
137C4—72no—
138C4—142no—
139C4D4×D711yescharacteristic
140C4C23×D711yescharacteristic
141C2—42no—
142C2—42no—
143C2—142no—
144C2—282no—
145C2#647412yesdirect product factor
146C2C2×D4×D711yescharacteristic
subgroupG/H
centerC22D4×D7
commutator, cosocleC4×C7C24
FrattiniC4C23×D7
FittingC2×C7×D8C2
socleC22×C7C2×D4
hypercenterC2×D8D7
hypocenterC7C2×AGL(1,ℤ/8)
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#1159C2semidirect
2. C2×D4×D7C2semidirect
3. C2×C7×D8C2semidirect
4. C2×#1142C2semidirect
5. C2×#1141C2semidirect
6. #6474C2direct (N × H)
7. C2×#1132C2semidirect
8. C2×#1131C2semidirect
11. C2×C7×D4C22semidirect
16. #1159C22semidirect
18. D4×D7C22semidirect
21. C2×Dic14C22semidirect
22. C7×D8C22semidirect
23. C2×C7×C8C22semidirect
24. #1142C22semidirect
25. #1141C22semidirect
26. C2×C7⋊C8C22semidirect
27. #1132C22semidirect
28. #1131C22semidirect
39. C7×D4C23semidirect
51. Dic14C23semidirect
52. C7×C8C23semidirect
53. C7⋊C8C23semidirect
60. C2×D8D7semidirect
86. C2×D4C2×D7semidirect
92. D8C2×D7semidirect
95. C2×C8C2×D7semidirect
98. C2×C7AGL(1,ℤ/8)semidirect
117. D4C22×D7semidirect
125. C8C22×D7semidirect
126. C7C2×AGL(1,ℤ/8)semidirect
145. C2#6474direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#11592
C2×D4×D72
C2×C7×D82
C2×#11422
C2×#11412
#64742
C2×#11322
C2×#11312
C2×AGL(1,ℤ/8)7