Finite Group Databasebeta

← C31×#538 #14520719

Subgroups — C31×#538 #14520719

Order 1984.

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
1C31×#126C212yesmaximal, complemented
2C31×#124C212yesmaximal, complemented
3C31×#123C212yesmaximal, complemented
4C31×C22⋊Q8C211yesmaximal, characteristic, complemented
5C31×#121C212yesmaximal, complemented
6C31×#121C211yesmaximal, characteristic, complemented
7C31×#120C212yesmaximal, complemented
8C4×D4×C31C212yesmaximal, complemented
9C2×C22⋊C4×C31C211yesmaximal, characteristic, complemented
10C24×C31C2211yescharacteristic
11C2×Q8×C31C2211yescharacteristic, complemented
12C2×D4×C31C2212yes—
13C2×D4×C31C2212yes—
14C2×D4×C31C2212yes—
15C2×D4×C31C2211yescharacteristic, complemented
16C22×C4×C31C2211yescharacteristic
17C22×C4×C31C2211yescharacteristic
18C22×C4×C31C2212yescomplemented
19C4⋊C4×C31C2212yescomplemented
20C4⋊C4×C31C2212yescomplemented
21C4⋊C4×C31C2211yescharacteristic, complemented
22C4⋊C4×C31C2211yescharacteristic, complemented
23C22⋊C4×C31C2212yes—
24C22⋊C4×C31C2212yes—
25C22⋊C4×C31C2212yes—
26C22⋊C4×C31C2212yes—
27C22⋊C4×C31C2212yes—
28C22⋊C4×C31C2212yescomplemented
29C22⋊C4×C31C2212yescomplemented
30C42×C31C2212yescomplemented
31C23×C31—22no—
32C23×C31—41no—
33C23×C31—41no—
34C23×C31C2312yes—
35C23×C31C2312yes—
36C23×C31C2311yescharacteristic
37Q8×C31—41no—
38D4×C31—24no—
39D4×C31—42no—
40D4×C31—42no—
41D4×C31—41no—
42C2×C4×C31—41no—
43C2×C4×C31—41no—
44C2×C4×C31—42no—
45C2×C4×C31C2312yes—
46C2×C4×C31C2312yes—
47C2×C4×C31C2312yes—
48C2×C4×C31C2311yescharacteristic
49C2×C4×C31C2311yescharacteristic
50C2×C4×C31C2312yes—
51C22×C31—21no—
52C22×C31—21no—
53C22×C31—22no—
54C22×C31—22no—
55C22×C31—22no—
56C22×C31—22no—
57C22×C31—22no—
58C22×C31—22no—
59C22×C31—44no—
60C22×C31#4112yes—
61C22×C31C2411yescharacteristic
62C4×C31—22no—
63C4×C31—22no—
64C4×C31—21no—
65C4×C31—22no—
66C4×C31—21no—
67C4×C31—22no—
68#538C3111yesmaximal, characteristic, direct product factor, Sylow
69C2×C31—22no—
70C2×C31—42no—
71C2×C31—42no—
72C2×C31C2×#4111yescharacteristic
73C2×C31H(5,𝔽2)11yescharacteristic
74C2×C31H(5,𝔽2)11yescharacteristic
75#126C2×C3112yescomplemented
76#124C2×C3112yescomplemented
77#123C2×C3112yescomplemented
78C22⋊Q8C2×C3111yescharacteristic, complemented
79#121C2×C3112yescomplemented
80#121C2×C3111yescharacteristic, complemented
81#120C2×C3112yescomplemented
82C4×D4C2×C3112yescomplemented
83C2×C22⋊C4C2×C3111yescharacteristic, complemented
84C31#53811yescharacteristic, direct product factor, Sylow
85C24C22×C3111yescharacteristic
86C2×Q8C22×C3111yescharacteristic, complemented
87C2×D4C22×C3112yes—
88C2×D4C22×C3112yes—
89C2×D4C22×C3112yes—
90C2×D4C22×C3111yescharacteristic, complemented
91C22×C4C22×C3111yescharacteristic
92C22×C4C22×C3111yescharacteristic
93C22×C4C22×C3112yescomplemented
94C4⋊C4C22×C3112yescomplemented
95C4⋊C4C22×C3112yescomplemented
96C4⋊C4C22×C3111yescharacteristic, complemented
97C4⋊C4C22×C3111yescharacteristic, complemented
98C22⋊C4C22×C3112yes—
99C22⋊C4C22×C3112yes—
100C22⋊C4C22×C3112yes—
101C22⋊C4C22×C3112yes—
102C22⋊C4C22×C3112yes—
103C22⋊C4C22×C3112yescomplemented
104C22⋊C4C22×C3112yescomplemented
105C42C22×C3112yescomplemented
106C23—22no—
107C23—41no—
108C23—41no—
109C23C23×C3112yes—
110C23C23×C3112yes—
111C23C23×C3111yescharacteristic
112Q8—41no—
113D4—24no—
114D4—42no—
115D4—42no—
116D4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4C23×C3112yes—
121C2×C4C23×C3112yes—
122C2×C4C23×C3112yes—
123C2×C4C23×C3111yescharacteristic
124C2×C4C23×C3111yescharacteristic
125C2×C4C23×C3112yes—
126C22—21no—
127C22—21no—
128C22—22no—
129C22—22no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—22no—
134C22—44no—
135C22#41×C3112yes—
136C22C24×C3111yescharacteristic
137C4—22no—
138C4—22no—
139C4—21no—
140C4—22no—
141C4—21no—
142C4—22no—
143C2—22no—
144C2—42no—
145C2—42no—
146C2C2×#41×C3111yescharacteristic
147C2C31×H(5,𝔽2)11yescharacteristic
148C2C31×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C24
commutator, Frattini, cosocleC22C24×C31
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#126C2semidirect
2. C31×#124C2semidirect
3. C31×#123C2semidirect
4. C31×C22⋊Q8C2semidirect
5. C31×#121C2semidirect
6. C31×#121C2semidirect
7. C31×#120C2semidirect
8. C4×D4×C31C2semidirect
9. C2×C22⋊C4×C31C2semidirect
11. C2×Q8×C31C22semidirect
15. C2×D4×C31C22semidirect
18. C22×C4×C31C22semidirect
19. C4⋊C4×C31C22semidirect
20. C4⋊C4×C31C22semidirect
21. C4⋊C4×C31C22semidirect
22. C4⋊C4×C31C22semidirect
28. C22⋊C4×C31C22semidirect
29. C22⋊C4×C31C22semidirect
30. C42×C31C22semidirect
68. #538C31direct (N × H)
75. #126C2×C31semidirect
76. #124C2×C31semidirect
77. #123C2×C31semidirect
78. C22⋊Q8C2×C31semidirect
79. #121C2×C31semidirect
80. #121C2×C31semidirect
81. #120C2×C31semidirect
82. C4×D4C2×C31semidirect
83. C2×C22⋊C4C2×C31semidirect
84. C31#538direct (N × H)
86. C2×Q8C22×C31semidirect
90. C2×D4C22×C31semidirect
93. C22×C4C22×C31semidirect
94. C4⋊C4C22×C31semidirect
95. C4⋊C4C22×C31semidirect
96. C4⋊C4C22×C31semidirect
97. C4⋊C4C22×C31semidirect
103. C22⋊C4C22×C31semidirect
104. C22⋊C4C22×C31semidirect
105. C42C22×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#1262
C31×#1242
C31×#1232
C31×C22⋊Q82
C31×#1212
C31×#1202
C4×D4×C312
C2×C22⋊C4×C312
#53831