Finite Group Databasebeta

← C7×#538 #90559

Subgroups — C7×#538 #90559

Order 448.

250 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 146 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×#124C212yesmaximal, complemented
3C7×#123C212yesmaximal, complemented
4C7×C22⋊Q8C211yesmaximal, characteristic, complemented
5C7×#121C212yesmaximal, complemented
6C7×#121C211yesmaximal, characteristic, complemented
7C7×#120C212yesmaximal, complemented
8C4×C7×D4C212yesmaximal, complemented
9C2×C7×C22⋊C4C211yesmaximal, characteristic, complemented
10C24×C7C2211yescharacteristic
11C2×C7×Q8C2211yescharacteristic, complemented
12C2×C7×D4C2212yes—
13C2×C7×D4C2212yes—
14C2×C7×D4C2212yes—
15C2×C7×D4C2211yescharacteristic, complemented
16C22×C4×C7C2211yescharacteristic
17C22×C4×C7C2211yescharacteristic
18C22×C4×C7C2212yescomplemented
19C7×C4⋊C4C2212yescomplemented
20C7×C4⋊C4C2212yescomplemented
21C7×C4⋊C4C2211yescharacteristic, complemented
22C7×C4⋊C4C2211yescharacteristic, complemented
23C7×C22⋊C4C2212yes—
24C7×C22⋊C4C2212yes—
25C7×C22⋊C4C2212yes—
26C7×C22⋊C4C2212yes—
27C7×C22⋊C4C2212yes—
28C7×C22⋊C4C2212yescomplemented
29C7×C22⋊C4C2212yescomplemented
30C42×C7C2212yescomplemented
31#538C711yesmaximal, characteristic, direct product factor, Sylow
32C23×C7—22no—
33C23×C7—41no—
34C23×C7—41no—
35C23×C7C2312yes—
36C23×C7C2312yes—
37C23×C7C2311yescharacteristic
38C7×Q8—41no—
39C7×D4—24no—
40C7×D4—42no—
41C7×D4—42no—
42C7×D4—41no—
43C2×C4×C7—41no—
44C2×C4×C7—41no—
45C2×C4×C7—42no—
46C2×C4×C7C2312yes—
47C2×C4×C7C2312yes—
48C2×C4×C7C2312yes—
49C2×C4×C7C2311yescharacteristic
50C2×C4×C7C2311yescharacteristic
51C2×C4×C7C2312yes—
52#126C2×C712yescomplemented
53#124C2×C712yescomplemented
54#123C2×C712yescomplemented
55C22⋊Q8C2×C711yescharacteristic, complemented
56#121C2×C712yescomplemented
57#121C2×C711yescharacteristic, complemented
58#120C2×C712yescomplemented
59C4×D4C2×C712yescomplemented
60C2×C22⋊C4C2×C711yescharacteristic, complemented
61C22×C7—21no—
62C22×C7—21no—
63C22×C7—22no—
64C22×C7—22no—
65C22×C7—22no—
66C22×C7—22no—
67C22×C7—22no—
68C22×C7—22no—
69C22×C7—44no—
70C22×C7#4112yes—
71C22×C7C2411yescharacteristic
72C4×C7—22no—
73C4×C7—22no—
74C4×C7—21no—
75C4×C7—22no—
76C4×C7—21no—
77C4×C7—22no—
78C24C22×C711yescharacteristic
79C2×Q8C22×C711yescharacteristic, complemented
80C2×D4C22×C712yes—
81C2×D4C22×C712yes—
82C2×D4C22×C712yes—
83C2×D4C22×C711yescharacteristic, complemented
84C22×C4C22×C711yescharacteristic
85C22×C4C22×C711yescharacteristic
86C22×C4C22×C712yescomplemented
87C4⋊C4C22×C712yescomplemented
88C4⋊C4C22×C712yescomplemented
89C4⋊C4C22×C711yescharacteristic, complemented
90C4⋊C4C22×C711yescharacteristic, complemented
91C22⋊C4C22×C712yes—
92C22⋊C4C22×C712yes—
93C22⋊C4C22×C712yes—
94C22⋊C4C22×C712yes—
95C22⋊C4C22×C712yes—
96C22⋊C4C22×C712yescomplemented
97C22⋊C4C22×C712yescomplemented
98C42C22×C712yescomplemented
99C2×C7—22no—
100C2×C7—42no—
101C2×C7—42no—
102C2×C7C2×#4111yescharacteristic
103C2×C7H(5,𝔽2)11yescharacteristic
104C2×C7H(5,𝔽2)11yescharacteristic
105C23—22no—
106C23—41no—
107C23—41no—
108C23C23×C712yes—
109C23C23×C712yes—
110C23C23×C711yescharacteristic
111Q8—41no—
112D4—24no—
113D4—42no—
114D4—42no—
115D4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4C23×C712yes—
120C2×C4C23×C712yes—
121C2×C4C23×C712yes—
122C2×C4C23×C711yescharacteristic
123C2×C4C23×C711yescharacteristic
124C2×C4C23×C712yes—
125C7#53811yescharacteristic, direct product factor, Sylow
126C22—21no—
127C22—21no—
128C22—22no—
129C22—22no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—22no—
134C22—44no—
135C22C7×#4112yes—
136C22C24×C711yescharacteristic
137C4—22no—
138C4—22no—
139C4—21no—
140C4—22no—
141C4—21no—
142C4—22no—
143C2—22no—
144C2—42no—
145C2—42no—
146C2C2×C7×#4111yescharacteristic
147C2C7×H(5,𝔽2)11yescharacteristic
148C2C7×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C7C24
commutator, Frattini, cosocleC22C24×C7
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×C22⋊Q8C2semidirect
5. C7×#121C2semidirect
6. C7×#121C2semidirect
7. C7×#120C2semidirect
8. C4×C7×D4C2semidirect
9. C2×C7×C22⋊C4C2semidirect
11. C2×C7×Q8C22semidirect
15. C2×C7×D4C22semidirect
18. C22×C4×C7C22semidirect
19. C7×C4⋊C4C22semidirect
20. C7×C4⋊C4C22semidirect
21. C7×C4⋊C4C22semidirect
22. C7×C4⋊C4C22semidirect
28. C7×C22⋊C4C22semidirect
29. C7×C22⋊C4C22semidirect
30. C42×C7C22semidirect
31. #538C7direct (N × H)
52. #126C2×C7semidirect
53. #124C2×C7semidirect
54. #123C2×C7semidirect
55. C22⋊Q8C2×C7semidirect
56. #121C2×C7semidirect
57. #121C2×C7semidirect
58. #120C2×C7semidirect
59. C4×D4C2×C7semidirect
60. C2×C22⋊C4C2×C7semidirect
79. C2×Q8C22×C7semidirect
83. C2×D4C22×C7semidirect
86. C22×C4C22×C7semidirect
87. C4⋊C4C22×C7semidirect
88. C4⋊C4C22×C7semidirect
89. C4⋊C4C22×C7semidirect
90. C4⋊C4C22×C7semidirect
96. C22⋊C4C22×C7semidirect
97. C22⋊C4C22×C7semidirect
98. C42C22×C7semidirect
125. C7#538direct (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×#1212
C7×#1202
C4×C7×D42
C2×C7×C22⋊C42
#5387