Finite Group Databasebeta

← #2128×C167 #36320488

Subgroups — #2128×C167 #36320488

Order 21376.

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

normal subgroups only

This group has 270 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#410×C167C212yesmaximal, complemented
3#353×C167C214yesmaximal, complemented
4C2×#41×C167C2211yescharacteristic
5C22×D4×C167C411yescharacteristic, complemented
6#127×C167C2212yes—
7#124×C167C412yescomplemented
8#121×C167—22no—
9#120×C167—22no—
10#117×C167—22no—
11#117×C167C411yescharacteristic, complemented
12#99×C167—22no—
13#99×C167C2214yescomplemented
14C24×C167—21no—
15#41×C167—42no—
16C2×Q8×C167C2×C411yescharacteristic
17C2×D4×C167—24no—
18C2×D4×C167—21no—
19C2×D4×C167—42no—
20C2×D4×C167C2×C412yes—
21C2×D4×C167D412yes—
22C2×D4×C167C2311yescharacteristic
23C22×C4×C167C2×C411yescharacteristic
24C22×C4×C167D412yes—
25C4⋊C4×C167—21no—
26C4⋊C4×C167—42no—
27C22⋊C4×C167—21no—
28C22⋊C4×C167—24no—
29C22⋊C4×C167—24no—
30C42×C167—42no—
31C42×C167C2×C412yes—
32C23×C167—22no—
33C23×C167—42no—
34C23×C167—82no—
35C23×C167C22⋊C411yescharacteristic
36C23×C167C2×D412yes—
37Q8×C167—41no—
38D4×C167—42no—
39D4×C167—41no—
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×C167—44no—
48C2×C4×C167C22⋊C412yes—
49C2×C4×C167C22⋊C411yescharacteristic
50C2×C4×C167C22×C411yescharacteristic
51C22×C167—22no—
52C22×C167—21no—
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—84no—
63C4×C167#9912yes—
64C2×C167—21no—
65C2×C167—42no—
66C2×C167—41no—
67C2×C167—82no—
68C2×C167C2×#9911yescharacteristic
69C167#212811yescharacteristic, direct product factor, Sylow
70#2128C16711yesmaximal, characteristic, direct product factor, Sylow
71#535C2×C16711yescharacteristic
72#410C2×C16712yescomplemented
73#353C2×C16714yescomplemented
74C2×#41C22×C16711yescharacteristic
75C22×D4C4×C16711yescharacteristic, complemented
76#127C22×C16712yes—
77#124C4×C16712yescomplemented
78#121—22no—
79#120—22no—
80#117—22no—
81#117C4×C16711yescharacteristic, complemented
82#99—22no—
83#99C22×C16714yescomplemented
84C24—21no—
85#41—42no—
86C2×Q8C2×C4×C16711yescharacteristic
87C2×D4—24no—
88C2×D4—21no—
89C2×D4—42no—
90C2×D4C2×C4×C16712yes—
91C2×D4D4×C16712yes—
92C2×D4C23×C16711yescharacteristic
93C22×C4C2×C4×C16711yescharacteristic
94C22×C4D4×C16712yes—
95C4⋊C4—21no—
96C4⋊C4—42no—
97C22⋊C4—21no—
98C22⋊C4—24no—
99C22⋊C4—24no—
100C42—42no—
101C42C2×C4×C16712yes—
102C23—22no—
103C23—42no—
104C23—82no—
105C23C22⋊C4×C16711yescharacteristic
106C23C2×D4×C16712yes—
107Q8—41no—
108D4—42no—
109D4—41no—
110D4—42no—
111D4—42no—
112D4—84no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—44no—
118C2×C4C22⋊C4×C16712yes—
119C2×C4C22⋊C4×C16711yescharacteristic
120C2×C4C22×C4×C16711yescharacteristic
121C22—22no—
122C22—21no—
123C22—42no—
124C22—41no—
125C22—44no—
126C22—42no—
127C22—84no—
128C22C2×C22⋊C4×C16711yescharacteristic
129C4—22no—
130C4—21no—
131C4—42no—
132C4—84no—
133C4#99×C16712yes—
134C2—21no—
135C2—42no—
136C2—41no—
137C2—82no—
138C2C2×#99×C16711yescharacteristic
subgroupG/H
center, socleC2×C167C2×#99
commutatorC2×C4C22×C4×C167
Frattini, cosocleC2×D4C23×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. #410×C167C2semidirect
3. #353×C167C2semidirect
5. C22×D4×C167C4semidirect
7. #124×C167C4semidirect
11. #117×C167C4semidirect
13. #99×C167C22semidirect
69. C167#2128direct (N × H)
70. #2128C167direct (N × H)
72. #410C2×C167semidirect
73. #353C2×C167semidirect
75. C22×D4C4×C167semidirect
77. #124C4×C167semidirect
81. #117C4×C167semidirect
83. #99C22×C167semidirect

Maximal subgroups

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

subgroupindex
#535×C1672
#410×C1672
#353×C1672
#2128167