Finite Group Databasebeta

← #538×C167 #24922939

Subgroups — #538×C167 #24922939

Order 10688.

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
1#126×C167C212yesmaximal, complemented
2#124×C167C212yesmaximal, complemented
3#123×C167C212yesmaximal, complemented
4C22⋊Q8×C167C211yesmaximal, characteristic, complemented
5#121×C167C212yesmaximal, complemented
6#121×C167C211yesmaximal, characteristic, complemented
7#120×C167C212yesmaximal, complemented
8C4×D4×C167C212yesmaximal, complemented
9C2×C22⋊C4×C167C211yesmaximal, characteristic, complemented
10C24×C167C2211yescharacteristic
11C2×Q8×C167C2211yescharacteristic, complemented
12C2×D4×C167C2212yes—
13C2×D4×C167C2212yes—
14C2×D4×C167C2212yes—
15C2×D4×C167C2211yescharacteristic, complemented
16C22×C4×C167C2211yescharacteristic
17C22×C4×C167C2211yescharacteristic
18C22×C4×C167C2212yescomplemented
19C4⋊C4×C167C2212yescomplemented
20C4⋊C4×C167C2212yescomplemented
21C4⋊C4×C167C2211yescharacteristic, complemented
22C4⋊C4×C167C2211yescharacteristic, complemented
23C22⋊C4×C167C2212yes—
24C22⋊C4×C167C2212yes—
25C22⋊C4×C167C2212yes—
26C22⋊C4×C167C2212yes—
27C22⋊C4×C167C2212yes—
28C22⋊C4×C167C2212yescomplemented
29C22⋊C4×C167C2212yescomplemented
30C42×C167C2212yescomplemented
31C23×C167—22no—
32C23×C167—41no—
33C23×C167—41no—
34C23×C167C2312yes—
35C23×C167C2312yes—
36C23×C167C2311yescharacteristic
37Q8×C167—41no—
38D4×C167—24no—
39D4×C167—42no—
40D4×C167—42no—
41D4×C167—41no—
42C2×C4×C167—41no—
43C2×C4×C167—41no—
44C2×C4×C167—42no—
45C2×C4×C167C2312yes—
46C2×C4×C167C2312yes—
47C2×C4×C167C2312yes—
48C2×C4×C167C2311yescharacteristic
49C2×C4×C167C2311yescharacteristic
50C2×C4×C167C2312yes—
51C22×C167—21no—
52C22×C167—21no—
53C22×C167—22no—
54C22×C167—22no—
55C22×C167—22no—
56C22×C167—22no—
57C22×C167—22no—
58C22×C167—22no—
59C22×C167—44no—
60C22×C167#4112yes—
61C22×C167C2411yescharacteristic
62C4×C167—22no—
63C4×C167—22no—
64C4×C167—21no—
65C4×C167—22no—
66C4×C167—21no—
67C4×C167—22no—
68C2×C167—22no—
69C2×C167—42no—
70C2×C167—42no—
71C2×C167C2×#4111yescharacteristic
72C2×C167H(5,𝔽2)11yescharacteristic
73C2×C167H(5,𝔽2)11yescharacteristic
74C167#53811yescharacteristic, direct product factor, Sylow
75#538C16711yesmaximal, characteristic, direct product factor, Sylow
76#126C2×C16712yescomplemented
77#124C2×C16712yescomplemented
78#123C2×C16712yescomplemented
79C22⋊Q8C2×C16711yescharacteristic, complemented
80#121C2×C16712yescomplemented
81#121C2×C16711yescharacteristic, complemented
82#120C2×C16712yescomplemented
83C4×D4C2×C16712yescomplemented
84C2×C22⋊C4C2×C16711yescharacteristic, complemented
85C24C22×C16711yescharacteristic
86C2×Q8C22×C16711yescharacteristic, complemented
87C2×D4C22×C16712yes—
88C2×D4C22×C16712yes—
89C2×D4C22×C16712yes—
90C2×D4C22×C16711yescharacteristic, complemented
91C22×C4C22×C16711yescharacteristic
92C22×C4C22×C16711yescharacteristic
93C22×C4C22×C16712yescomplemented
94C4⋊C4C22×C16712yescomplemented
95C4⋊C4C22×C16712yescomplemented
96C4⋊C4C22×C16711yescharacteristic, complemented
97C4⋊C4C22×C16711yescharacteristic, complemented
98C22⋊C4C22×C16712yes—
99C22⋊C4C22×C16712yes—
100C22⋊C4C22×C16712yes—
101C22⋊C4C22×C16712yes—
102C22⋊C4C22×C16712yes—
103C22⋊C4C22×C16712yescomplemented
104C22⋊C4C22×C16712yescomplemented
105C42C22×C16712yescomplemented
106C23—22no—
107C23—41no—
108C23—41no—
109C23C23×C16712yes—
110C23C23×C16712yes—
111C23C23×C16711yescharacteristic
112Q8—41no—
113D4—24no—
114D4—42no—
115D4—42no—
116D4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4C23×C16712yes—
121C2×C4C23×C16712yes—
122C2×C4C23×C16712yes—
123C2×C4C23×C16711yescharacteristic
124C2×C4C23×C16711yescharacteristic
125C2×C4C23×C16712yes—
126C22—21no—
127C22—21no—
128C22—22no—
129C22—22no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—22no—
134C22—44no—
135C22#41×C16712yes—
136C22C24×C16711yescharacteristic
137C4—22no—
138C4—22no—
139C4—21no—
140C4—22no—
141C4—21no—
142C4—22no—
143C2—22no—
144C2—42no—
145C2—42no—
146C2C2×#41×C16711yescharacteristic
147C2H(5,𝔽2)×C16711yescharacteristic
148C2H(5,𝔽2)×C16711yescharacteristic
subgroupG/H
center, socleC22×C167C24
commutator, Frattini, cosocleC22C24×C167
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C167 ◁ G

Complements

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

NH ≅ G/Ntype
1. #126×C167C2semidirect
2. #124×C167C2semidirect
3. #123×C167C2semidirect
4. C22⋊Q8×C167C2semidirect
5. #121×C167C2semidirect
6. #121×C167C2semidirect
7. #120×C167C2semidirect
8. C4×D4×C167C2semidirect
9. C2×C22⋊C4×C167C2semidirect
11. C2×Q8×C167C22semidirect
15. C2×D4×C167C22semidirect
18. C22×C4×C167C22semidirect
19. C4⋊C4×C167C22semidirect
20. C4⋊C4×C167C22semidirect
21. C4⋊C4×C167C22semidirect
22. C4⋊C4×C167C22semidirect
28. C22⋊C4×C167C22semidirect
29. C22⋊C4×C167C22semidirect
30. C42×C167C22semidirect
74. C167#538direct (N × H)
75. #538C167direct (N × H)
76. #126C2×C167semidirect
77. #124C2×C167semidirect
78. #123C2×C167semidirect
79. C22⋊Q8C2×C167semidirect
80. #121C2×C167semidirect
81. #121C2×C167semidirect
82. #120C2×C167semidirect
83. C4×D4C2×C167semidirect
84. C2×C22⋊C4C2×C167semidirect
86. C2×Q8C22×C167semidirect
90. C2×D4C22×C167semidirect
93. C22×C4C22×C167semidirect
94. C4⋊C4C22×C167semidirect
95. C4⋊C4C22×C167semidirect
96. C4⋊C4C22×C167semidirect
97. C4⋊C4C22×C167semidirect
103. C22⋊C4C22×C167semidirect
104. C22⋊C4C22×C167semidirect
105. C42C22×C167semidirect

Maximal subgroups

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

subgroupindex
#126×C1672
#124×C1672
#123×C1672
C22⋊Q8×C1672
#121×C1672
#120×C1672
C4×D4×C1672
C2×C22⋊C4×C1672
#538167