Finite Group Databasebeta

← #551×C239 #29504455

Subgroups — #551×C239 #29504455

Order 15296.

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×C239C212yesmaximal, complemented
2#124×C239C211yesmaximal, characteristic, complemented
3#123×C239C211yesmaximal, characteristic, complemented
4#123×C239C212yesmaximal, complemented
5C22⋊Q8×C239C212yesmaximal, complemented
6#120×C239C211yesmaximal, characteristic
7C4×D4×C239C212yesmaximal, complemented
8#117×C239C212yesmaximal, complemented
9C2×C22⋊C4×C239C212yesmaximal, complemented
10C24×C239C2211yescharacteristic
11C2×Q8×C239C2211yescharacteristic, complemented
12C2×D4×C239C2212yes—
13C2×D4×C239C2211yescharacteristic
14C22×C4×C239C2212yes—
15C22×C4×C239C2212yes—
16C22×C4×C239C2211yescharacteristic, complemented
17C4⋊C4×C239C2212yescomplemented
18C4⋊C4×C239C2212yescomplemented
19C4⋊C4×C239C2212yescomplemented
20C4⋊C4×C239C2212yescomplemented
21C22⋊C4×C239C2212yes—
22C22⋊C4×C239C2212yes—
23C22⋊C4×C239C2212yes—
24C22⋊C4×C239C2212yes—
25C22⋊C4×C239C2212yes—
26C22⋊C4×C239C2211yescharacteristic
27C22⋊C4×C239C2212yescomplemented
28C22⋊C4×C239C2211yescharacteristic, complemented
29C42×C239C2211yescharacteristic, complemented
30C42×C239C2212yescomplemented
31C23×C239—24no—
32C23×C239—41no—
33C23×C239C2311yescharacteristic
34C23×C239C2311yescharacteristic
35C23×C239C2312yes—
36Q8×C239—41no—
37D4×C239—24no—
38D4×C239—41no—
39C2×C4×C239—24no—
40C2×C4×C239—42no—
41C2×C4×C239—41no—
42C2×C4×C239C2312yes—
43C2×C4×C239C2312yes—
44C2×C4×C239C2312yes—
45C2×C4×C239C2311yescharacteristic
46C2×C4×C239C2312yes—
47C2×C4×C239C2312yes—
48C22×C239—22no—
49C22×C239—22no—
50C22×C239—22no—
51C22×C239—22no—
52C22×C239—21no—
53C22×C239—21no—
54C22×C239—44no—
55C22×C239#4114yes—
56C22×C239C2411yescharacteristic
57C4×C239—22no—
58C4×C239—22no—
59C4×C239—22no—
60C4×C239—22no—
61C4×C239—22no—
62C4×C239—21no—
63C2×C239—24no—
64C2×C239—41no—
65C2×C239—41no—
66C2×C239C2×#4112yes—
67C2×C239H(5,𝔽2)11yescharacteristic
68C239#55111yescharacteristic, direct product factor, Sylow
69#551C23911yesmaximal, characteristic, direct product factor, Sylow
70#126C2×C23912yescomplemented
71#124C2×C23911yescharacteristic, complemented
72#123C2×C23911yescharacteristic, complemented
73#123C2×C23912yescomplemented
74C22⋊Q8C2×C23912yescomplemented
75#120C2×C23911yescharacteristic
76C4×D4C2×C23912yescomplemented
77#117C2×C23912yescomplemented
78C2×C22⋊C4C2×C23912yescomplemented
79C24C22×C23911yescharacteristic
80C2×Q8C22×C23911yescharacteristic, complemented
81C2×D4C22×C23912yes—
82C2×D4C22×C23911yescharacteristic
83C22×C4C22×C23912yes—
84C22×C4C22×C23912yes—
85C22×C4C22×C23911yescharacteristic, complemented
86C4⋊C4C22×C23912yescomplemented
87C4⋊C4C22×C23912yescomplemented
88C4⋊C4C22×C23912yescomplemented
89C4⋊C4C22×C23912yescomplemented
90C22⋊C4C22×C23912yes—
91C22⋊C4C22×C23912yes—
92C22⋊C4C22×C23912yes—
93C22⋊C4C22×C23912yes—
94C22⋊C4C22×C23912yes—
95C22⋊C4C22×C23911yescharacteristic
96C22⋊C4C22×C23912yescomplemented
97C22⋊C4C22×C23911yescharacteristic, complemented
98C42C22×C23911yescharacteristic, complemented
99C42C22×C23912yescomplemented
100C23—24no—
101C23—41no—
102C23C23×C23911yescharacteristic
103C23C23×C23911yescharacteristic
104C23C23×C23912yes—
105Q8—41no—
106D4—24no—
107D4—41no—
108C2×C4—24no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4C23×C23912yes—
112C2×C4C23×C23912yes—
113C2×C4C23×C23912yes—
114C2×C4C23×C23911yescharacteristic
115C2×C4C23×C23912yes—
116C2×C4C23×C23912yes—
117C22—22no—
118C22—22no—
119C22—22no—
120C22—22no—
121C22—21no—
122C22—21no—
123C22—44no—
124C22#41×C23914yes—
125C22C24×C23911yescharacteristic
126C4—22no—
127C4—22no—
128C4—22no—
129C4—22no—
130C4—22no—
131C4—21no—
132C2—24no—
133C2—41no—
134C2—41no—
135C2C2×#41×C23912yes—
136C2H(5,𝔽2)×C23911yescharacteristic
subgroupG/H
center, socleC22×C239C24
commutator, Frattini, cosocleC22C24×C239
prime radical#120×C239C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C239 ◁ G

Complements

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

NH ≅ G/Ntype
1. #126×C239C2semidirect
2. #124×C239C2semidirect
3. #123×C239C2semidirect
4. #123×C239C2semidirect
5. C22⋊Q8×C239C2semidirect
7. C4×D4×C239C2semidirect
8. #117×C239C2semidirect
9. C2×C22⋊C4×C239C2semidirect
11. C2×Q8×C239C22semidirect
16. C22×C4×C239C22semidirect
17. C4⋊C4×C239C22semidirect
18. C4⋊C4×C239C22semidirect
19. C4⋊C4×C239C22semidirect
20. C4⋊C4×C239C22semidirect
27. C22⋊C4×C239C22semidirect
28. C22⋊C4×C239C22semidirect
29. C42×C239C22semidirect
30. C42×C239C22semidirect
68. C239#551direct (N × H)
69. #551C239direct (N × H)
70. #126C2×C239semidirect
71. #124C2×C239semidirect
72. #123C2×C239semidirect
73. #123C2×C239semidirect
74. C22⋊Q8C2×C239semidirect
76. C4×D4C2×C239semidirect
77. #117C2×C239semidirect
78. C2×C22⋊C4C2×C239semidirect
80. C2×Q8C22×C239semidirect
85. C22×C4C22×C239semidirect
86. C4⋊C4C22×C239semidirect
87. C4⋊C4C22×C239semidirect
88. C4⋊C4C22×C239semidirect
89. C4⋊C4C22×C239semidirect
96. C22⋊C4C22×C239semidirect
97. C22⋊C4C22×C239semidirect
98. C42C22×C239semidirect
99. C42C22×C239semidirect

Maximal subgroups

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

subgroupindex
#126×C2392
#124×C2392
#123×C2392
C22⋊Q8×C2392
#120×C2392
C4×D4×C2392
#117×C2392
C2×C22⋊C4×C2392
#551239