Finite Group Databasebeta

← #551×C71 #17985682

Subgroups — #551×C71 #17985682

Order 4544.

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
1#126×C71C212yesmaximal, complemented
2#124×C71C211yesmaximal, characteristic, complemented
3#123×C71C211yesmaximal, characteristic, complemented
4#123×C71C212yesmaximal, complemented
5C22⋊Q8×C71C212yesmaximal, complemented
6#120×C71C211yesmaximal, characteristic
7C4×D4×C71C212yesmaximal, complemented
8#117×C71C212yesmaximal, complemented
9C2×C22⋊C4×C71C212yesmaximal, complemented
10C24×C71C2211yescharacteristic
11C2×Q8×C71C2211yescharacteristic, complemented
12C2×D4×C71C2212yes—
13C2×D4×C71C2211yescharacteristic
14C22×C4×C71C2212yes—
15C22×C4×C71C2212yes—
16C22×C4×C71C2211yescharacteristic, complemented
17C4⋊C4×C71C2212yescomplemented
18C4⋊C4×C71C2212yescomplemented
19C4⋊C4×C71C2212yescomplemented
20C4⋊C4×C71C2212yescomplemented
21C22⋊C4×C71C2212yes—
22C22⋊C4×C71C2212yes—
23C22⋊C4×C71C2212yes—
24C22⋊C4×C71C2212yes—
25C22⋊C4×C71C2212yes—
26C22⋊C4×C71C2211yescharacteristic
27C22⋊C4×C71C2212yescomplemented
28C22⋊C4×C71C2211yescharacteristic, complemented
29C42×C71C2211yescharacteristic, complemented
30C42×C71C2212yescomplemented
31C23×C71—24no—
32C23×C71—41no—
33C23×C71C2311yescharacteristic
34C23×C71C2311yescharacteristic
35C23×C71C2312yes—
36Q8×C71—41no—
37D4×C71—24no—
38D4×C71—41no—
39C2×C4×C71—24no—
40C2×C4×C71—42no—
41C2×C4×C71—41no—
42C2×C4×C71C2312yes—
43C2×C4×C71C2312yes—
44C2×C4×C71C2312yes—
45C2×C4×C71C2311yescharacteristic
46C2×C4×C71C2312yes—
47C2×C4×C71C2312yes—
48C22×C71—22no—
49C22×C71—22no—
50C22×C71—22no—
51C22×C71—22no—
52C22×C71—21no—
53C22×C71—21no—
54C22×C71—44no—
55C22×C71#4114yes—
56C22×C71C2411yescharacteristic
57C4×C71—22no—
58C4×C71—22no—
59C4×C71—22no—
60C4×C71—22no—
61C4×C71—22no—
62C4×C71—21no—
63C2×C71—24no—
64C2×C71—41no—
65C2×C71—41no—
66C2×C71C2×#4112yes—
67C2×C71H(5,𝔽2)11yescharacteristic
68C71#55111yescharacteristic, direct product factor, Sylow
69#551C7111yesmaximal, characteristic, direct product factor, Sylow
70#126C2×C7112yescomplemented
71#124C2×C7111yescharacteristic, complemented
72#123C2×C7111yescharacteristic, complemented
73#123C2×C7112yescomplemented
74C22⋊Q8C2×C7112yescomplemented
75#120C2×C7111yescharacteristic
76C4×D4C2×C7112yescomplemented
77#117C2×C7112yescomplemented
78C2×C22⋊C4C2×C7112yescomplemented
79C24C22×C7111yescharacteristic
80C2×Q8C22×C7111yescharacteristic, complemented
81C2×D4C22×C7112yes—
82C2×D4C22×C7111yescharacteristic
83C22×C4C22×C7112yes—
84C22×C4C22×C7112yes—
85C22×C4C22×C7111yescharacteristic, complemented
86C4⋊C4C22×C7112yescomplemented
87C4⋊C4C22×C7112yescomplemented
88C4⋊C4C22×C7112yescomplemented
89C4⋊C4C22×C7112yescomplemented
90C22⋊C4C22×C7112yes—
91C22⋊C4C22×C7112yes—
92C22⋊C4C22×C7112yes—
93C22⋊C4C22×C7112yes—
94C22⋊C4C22×C7112yes—
95C22⋊C4C22×C7111yescharacteristic
96C22⋊C4C22×C7112yescomplemented
97C22⋊C4C22×C7111yescharacteristic, complemented
98C42C22×C7111yescharacteristic, complemented
99C42C22×C7112yescomplemented
100C23—24no—
101C23—41no—
102C23C23×C7111yescharacteristic
103C23C23×C7111yescharacteristic
104C23C23×C7112yes—
105Q8—41no—
106D4—24no—
107D4—41no—
108C2×C4—24no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4C23×C7112yes—
112C2×C4C23×C7112yes—
113C2×C4C23×C7112yes—
114C2×C4C23×C7111yescharacteristic
115C2×C4C23×C7112yes—
116C2×C4C23×C7112yes—
117C22—22no—
118C22—22no—
119C22—22no—
120C22—22no—
121C22—21no—
122C22—21no—
123C22—44no—
124C22#41×C7114yes—
125C22C24×C7111yescharacteristic
126C4—22no—
127C4—22no—
128C4—22no—
129C4—22no—
130C4—22no—
131C4—21no—
132C2—24no—
133C2—41no—
134C2—41no—
135C2C2×#41×C7112yes—
136C2H(5,𝔽2)×C7111yescharacteristic
subgroupG/H
center, socleC22×C71C24
commutator, Frattini, cosocleC22C24×C71
prime radical#120×C71C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. #126×C71C2semidirect
2. #124×C71C2semidirect
3. #123×C71C2semidirect
4. #123×C71C2semidirect
5. C22⋊Q8×C71C2semidirect
7. C4×D4×C71C2semidirect
8. #117×C71C2semidirect
9. C2×C22⋊C4×C71C2semidirect
11. C2×Q8×C71C22semidirect
16. C22×C4×C71C22semidirect
17. C4⋊C4×C71C22semidirect
18. C4⋊C4×C71C22semidirect
19. C4⋊C4×C71C22semidirect
20. C4⋊C4×C71C22semidirect
27. C22⋊C4×C71C22semidirect
28. C22⋊C4×C71C22semidirect
29. C42×C71C22semidirect
30. C42×C71C22semidirect
68. C71#551direct (N × H)
69. #551C71direct (N × H)
70. #126C2×C71semidirect
71. #124C2×C71semidirect
72. #123C2×C71semidirect
73. #123C2×C71semidirect
74. C22⋊Q8C2×C71semidirect
76. C4×D4C2×C71semidirect
77. #117C2×C71semidirect
78. C2×C22⋊C4C2×C71semidirect
80. C2×Q8C22×C71semidirect
85. C22×C4C22×C71semidirect
86. C4⋊C4C22×C71semidirect
87. C4⋊C4C22×C71semidirect
88. C4⋊C4C22×C71semidirect
89. C4⋊C4C22×C71semidirect
96. C22⋊C4C22×C71semidirect
97. C22⋊C4C22×C71semidirect
98. C42C22×C71semidirect
99. C42C22×C71semidirect

Maximal subgroups

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

subgroupindex
#126×C712
#124×C712
#123×C712
C22⋊Q8×C712
#120×C712
C4×D4×C712
#117×C712
C2×C22⋊C4×C712
#55171