Finite Group Databasebeta

← C7×#551 #90572

Subgroups — C7×#551 #90572

Order 448.

248 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C7×#126C212yesmaximal, complemented
2C7×#124C211yesmaximal, characteristic, complemented
3C7×#123C211yesmaximal, characteristic, complemented
4C7×#123C212yesmaximal, complemented
5C7×C22⋊Q8C212yesmaximal, complemented
6C7×#120C211yesmaximal, characteristic
7C4×C7×D4C212yesmaximal, complemented
8C7×#117C212yesmaximal, complemented
9C2×C7×C22⋊C4C212yesmaximal, complemented
10C24×C7C2211yescharacteristic
11C2×C7×Q8C2211yescharacteristic, complemented
12C2×C7×D4C2212yes—
13C2×C7×D4C2211yescharacteristic
14C22×C4×C7C2212yes—
15C22×C4×C7C2212yes—
16C22×C4×C7C2211yescharacteristic, complemented
17C7×C4⋊C4C2212yescomplemented
18C7×C4⋊C4C2212yescomplemented
19C7×C4⋊C4C2212yescomplemented
20C7×C4⋊C4C2212yescomplemented
21C7×C22⋊C4C2212yes—
22C7×C22⋊C4C2212yes—
23C7×C22⋊C4C2212yes—
24C7×C22⋊C4C2212yes—
25C7×C22⋊C4C2212yes—
26C7×C22⋊C4C2211yescharacteristic
27C7×C22⋊C4C2212yescomplemented
28C7×C22⋊C4C2211yescharacteristic, complemented
29C42×C7C2211yescharacteristic, complemented
30C42×C7C2212yescomplemented
31#551C711yesmaximal, characteristic, direct product factor, Sylow
32C23×C7—24no—
33C23×C7—41no—
34C23×C7C2311yescharacteristic
35C23×C7C2311yescharacteristic
36C23×C7C2312yes—
37C7×Q8—41no—
38C7×D4—24no—
39C7×D4—41no—
40C2×C4×C7—24no—
41C2×C4×C7—42no—
42C2×C4×C7—41no—
43C2×C4×C7C2312yes—
44C2×C4×C7C2312yes—
45C2×C4×C7C2312yes—
46C2×C4×C7C2311yescharacteristic
47C2×C4×C7C2312yes—
48C2×C4×C7C2312yes—
49#126C2×C712yescomplemented
50#124C2×C711yescharacteristic, complemented
51#123C2×C711yescharacteristic, complemented
52#123C2×C712yescomplemented
53C22⋊Q8C2×C712yescomplemented
54#120C2×C711yescharacteristic
55C4×D4C2×C712yescomplemented
56#117C2×C712yescomplemented
57C2×C22⋊C4C2×C712yescomplemented
58C22×C7—22no—
59C22×C7—22no—
60C22×C7—22no—
61C22×C7—22no—
62C22×C7—21no—
63C22×C7—21no—
64C22×C7—44no—
65C22×C7#4114yes—
66C22×C7C2411yescharacteristic
67C4×C7—22no—
68C4×C7—22no—
69C4×C7—22no—
70C4×C7—22no—
71C4×C7—22no—
72C4×C7—21no—
73C24C22×C711yescharacteristic
74C2×Q8C22×C711yescharacteristic, complemented
75C2×D4C22×C712yes—
76C2×D4C22×C711yescharacteristic
77C22×C4C22×C712yes—
78C22×C4C22×C712yes—
79C22×C4C22×C711yescharacteristic, complemented
80C4⋊C4C22×C712yescomplemented
81C4⋊C4C22×C712yescomplemented
82C4⋊C4C22×C712yescomplemented
83C4⋊C4C22×C712yescomplemented
84C22⋊C4C22×C712yes—
85C22⋊C4C22×C712yes—
86C22⋊C4C22×C712yes—
87C22⋊C4C22×C712yes—
88C22⋊C4C22×C712yes—
89C22⋊C4C22×C711yescharacteristic
90C22⋊C4C22×C712yescomplemented
91C22⋊C4C22×C711yescharacteristic, complemented
92C42C22×C711yescharacteristic, complemented
93C42C22×C712yescomplemented
94C2×C7—24no—
95C2×C7—41no—
96C2×C7—41no—
97C2×C7C2×#4112yes—
98C2×C7H(5,𝔽2)11yescharacteristic
99C23—24no—
100C23—41no—
101C23C23×C711yescharacteristic
102C23C23×C711yescharacteristic
103C23C23×C712yes—
104Q8—41no—
105D4—24no—
106D4—41no—
107C2×C4—24no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4C23×C712yes—
111C2×C4C23×C712yes—
112C2×C4C23×C712yes—
113C2×C4C23×C711yescharacteristic
114C2×C4C23×C712yes—
115C2×C4C23×C712yes—
116C7#55111yescharacteristic, direct product factor, Sylow
117C22—22no—
118C22—22no—
119C22—22no—
120C22—22no—
121C22—21no—
122C22—21no—
123C22—44no—
124C22C7×#4114yes—
125C22C24×C711yescharacteristic
126C4—22no—
127C4—22no—
128C4—22no—
129C4—22no—
130C4—22no—
131C4—21no—
132C2—24no—
133C2—41no—
134C2—41no—
135C2C2×C7×#4112yes—
136C2C7×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C7C24
commutator, Frattini, cosocleC22C24×C7
prime radicalC7×#120C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#126C2semidirect
2. C7×#124C2semidirect
3. C7×#123C2semidirect
4. C7×#123C2semidirect
5. C7×C22⋊Q8C2semidirect
7. C4×C7×D4C2semidirect
8. C7×#117C2semidirect
9. C2×C7×C22⋊C4C2semidirect
11. C2×C7×Q8C22semidirect
16. C22×C4×C7C22semidirect
17. C7×C4⋊C4C22semidirect
18. C7×C4⋊C4C22semidirect
19. C7×C4⋊C4C22semidirect
20. C7×C4⋊C4C22semidirect
27. C7×C22⋊C4C22semidirect
28. C7×C22⋊C4C22semidirect
29. C42×C7C22semidirect
30. C42×C7C22semidirect
31. #551C7direct (N × H)
49. #126C2×C7semidirect
50. #124C2×C7semidirect
51. #123C2×C7semidirect
52. #123C2×C7semidirect
53. C22⋊Q8C2×C7semidirect
55. C4×D4C2×C7semidirect
56. #117C2×C7semidirect
57. C2×C22⋊C4C2×C7semidirect
74. C2×Q8C22×C7semidirect
79. C22×C4C22×C7semidirect
80. C4⋊C4C22×C7semidirect
81. C4⋊C4C22×C7semidirect
82. C4⋊C4C22×C7semidirect
83. C4⋊C4C22×C7semidirect
90. C22⋊C4C22×C7semidirect
91. C22⋊C4C22×C7semidirect
92. C42C22×C7semidirect
93. C42C22×C7semidirect
116. C7#551direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#1262
C7×#1242
C7×#1232
C7×C22⋊Q82
C7×#1202
C4×C7×D42
C7×#1172
C2×C7×C22⋊C42
#5517