Finite Group Databasebeta

← #2134×C167 #36320494

Subgroups — #2134×C167 #36320494

Order 21376.

262 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 84 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#535×C167C211yesmaximal, characteristic
2#413×C167C212yesmaximal, complemented
3#355×C167C214yesmaximal, complemented
4C2×#41×C167C2211yescharacteristic
5C22×D4×C167C411yescharacteristic
6C2×M16×C167—22no—
7#127×C167C412yes—
8#124×C167C2212yes—
9#121×C167—22no—
10#120×C167—22no—
11#117×C167C411yescharacteristic
12#101×C167C2214yescomplemented
13#100×C167—22no—
14C24×C167—21no—
15#41×C167—42no—
16C2×Q8×C167C2311yescharacteristic
17C2×D4×C167—24no—
18C2×D4×C167—21no—
19C2×D4×C167—42no—
20C2×D4×C167C2×C412yes—
21C2×D4×C167C2×C411yescharacteristic
22C2×D4×C167D412yes—
23C22×C4×C167C2×C411yescharacteristic
24C22×C4×C167D412yes—
25M16×C167—24no—
26M16×C167—42no—
27C2×C8×C167—42no—
28C4⋊C4×C167—21no—
29C22⋊C4×C167—21no—
30C22⋊C4×C167—24no—
31C42×C167C2×C412yes—
32C23×C167—22no—
33C23×C167—42no—
34C23×C167—82no—
35C23×C167C22⋊C411yescharacteristic
36C23×C167C22⋊C412yes—
37Q8×C167—41no—
38D4×C167—41no—
39D4×C167—42no—
40D4×C167—42no—
41D4×C167—42no—
42D4×C167—84no—
43C2×C4×C167—22no—
44C2×C4×C167—22no—
45C2×C4×C167—22no—
46C2×C4×C167—22no—
47C2×C4×C167C22⋊C411yescharacteristic
48C2×C4×C167C22×C411yescharacteristic
49C2×C4×C167C2×D412yes—
50C8×C167—44no—
51C22×C167—21no—
52C22×C167—22no—
53C22×C167—42no—
54C22×C167—41no—
55C22×C167—44no—
56C22×C167—42no—
57C22×C167—84no—
58C22×C167C2×C22⋊C411yescharacteristic
59C4×C167—22no—
60C4×C167—21no—
61C4×C167—42no—
62C4×C167#9912yes—
63C2×C167—21no—
64C2×C167—42no—
65C2×C167—41no—
66C2×C167—82no—
67C2×C167C2×#9911yescharacteristic
68C167#213411yescharacteristic, direct product factor, Sylow
69#2134C16711yesmaximal, characteristic, direct product factor, Sylow
70#535C2×C16711yescharacteristic
71#413C2×C16712yescomplemented
72#355C2×C16714yescomplemented
73C2×#41C22×C16711yescharacteristic
74C22×D4C4×C16711yescharacteristic
75C2×M16—22no—
76#127C4×C16712yes—
77#124C22×C16712yes—
78#121—22no—
79#120—22no—
80#117C4×C16711yescharacteristic
81#101C22×C16714yescomplemented
82#100—22no—
83C24—21no—
84#41—42no—
85C2×Q8C23×C16711yescharacteristic
86C2×D4—24no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4C2×C4×C16712yes—
90C2×D4C2×C4×C16711yescharacteristic
91C2×D4D4×C16712yes—
92C22×C4C2×C4×C16711yescharacteristic
93C22×C4D4×C16712yes—
94M16—24no—
95M16—42no—
96C2×C8—42no—
97C4⋊C4—21no—
98C22⋊C4—21no—
99C22⋊C4—24no—
100C42C2×C4×C16712yes—
101C23—22no—
102C23—42no—
103C23—82no—
104C23C22⋊C4×C16711yescharacteristic
105C23C22⋊C4×C16712yes—
106Q8—41no—
107D4—41no—
108D4—42no—
109D4—42no—
110D4—42no—
111D4—84no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4C22⋊C4×C16711yescharacteristic
117C2×C4C22×C4×C16711yescharacteristic
118C2×C4C2×D4×C16712yes—
119C8—44no—
120C22—21no—
121C22—22no—
122C22—42no—
123C22—41no—
124C22—44no—
125C22—42no—
126C22—84no—
127C22C2×C22⋊C4×C16711yescharacteristic
128C4—22no—
129C4—21no—
130C4—42no—
131C4#99×C16712yes—
132C2—21no—
133C2—41no—
134C2—42no—
135C2—82no—
136C2C2×#99×C16711yescharacteristic
subgroupG/H
center, socleC2×C167C2×#99
commutatorC2×C4C22×C4×C167
Frattini, cosocleC2×Q8C23×C167
prime radical#535×C167C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C167 ◁ C2×C4×C167 ◁ C2×#41×C167 ◁ G

Complements

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

NH ≅ G/Ntype
2. #413×C167C2semidirect
3. #355×C167C2semidirect
12. #101×C167C22semidirect
68. C167#2134direct (N × H)
69. #2134C167direct (N × H)
71. #413C2×C167semidirect
72. #355C2×C167semidirect
81. #101C22×C167semidirect

Maximal subgroups

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

subgroupindex
#535×C1672
#413×C1672
#355×C1672
#2134167