Finite Group Databasebeta

← #1624×C167 #36319984

Subgroups — #1624×C167 #36319984

Order 21376.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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#555×C167C211yesmaximal, characteristic, complemented
2#492×C167C211yesmaximal, characteristic, complemented
3#459×C167C212yesmaximal, complemented
4#333×C167C211yesmaximal, characteristic, complemented
5D4⋊C8×C167C212yesmaximal, complemented
6C2×SD16×C167—22no—
7C2×D8×C167—22no—
8C4⋊Q8×C167C2211yescharacteristic
9#127×C167C2211yescharacteristic
10#124×C167—22no—
11#121×C167—22no—
12C4×D4×C167C2212yes—
13#117×C167—22no—
14C4⋊C8×C167C2212yes—
15#102×C167—22no—
16C4×C8×C167C2211yescharacteristic
17C2×Q8×C167D412yes—
18C2×D4×C167—22no—
19C2×D4×C167—22no—
20C2×D4×C167—21no—
21C2×D4×C167D412yes—
22C22×C4×C167—22no—
23SD16×C167—44no—
24D8×C167—44no—
25C2×C8×C167—21no—
26C2×C8×C167—22no—
27C4⋊C4×C167—22no—
28C4⋊C4×C167D412yes—
29C22⋊C4×C167—24no—
30C22⋊C4×C167—22no—
31C42×C167—22no—
32C42×C167C2311yescharacteristic
33C23×C167—22no—
34C23×C167—41no—
35Q8×C167—42no—
36D4×C167—22no—
37D4×C167—22no—
38D4×C167—42no—
39D4×C167—42no—
40D4×C167—42no—
41D4×C167—82no—
42C2×C4×C167—22no—
43C2×C4×C167—22no—
44C2×C4×C167—42no—
45C2×C4×C167—42no—
46C2×C4×C167C2×D411yescharacteristic
47C2×C4×C167C2×D412yes—
48C8×C167—22no—
49C8×C167—42no—
50C22×C167—42no—
51C22×C167—42no—
52C22×C167—42no—
53C22×C167—82no—
54C22×C167—81no—
55C22×C167#12011yescharacteristic
56C4×C167—21no—
57C4×C167—42no—
58C4×C167—42no—
59C4×C167#13512yes—
60C4×C167AGL(1,ℤ/8)12yes—
61C2×C167—82no—
62C2×C167—161no—
63C2×C167#44912yes—
64C2×C167#45311yescharacteristic
65C167#162411yescharacteristic, direct product factor, Sylow
66#1624C16711yesmaximal, characteristic, direct product factor, Sylow
67#555C2×C16711yescharacteristic, complemented
68#492C2×C16711yescharacteristic, complemented
69#459C2×C16712yescomplemented
70#333C2×C16711yescharacteristic, complemented
71D4⋊C8C2×C16712yescomplemented
72C2×SD16—22no—
73C2×D8—22no—
74C4⋊Q8C22×C16711yescharacteristic
75#127C22×C16711yescharacteristic
76#124—22no—
77#121—22no—
78C4×D4C22×C16712yes—
79#117—22no—
80C4⋊C8C22×C16712yes—
81#102—22no—
82C4×C8C22×C16711yescharacteristic
83C2×Q8D4×C16712yes—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—21no—
87C2×D4D4×C16712yes—
88C22×C4—22no—
89SD16—44no—
90D8—44no—
91C2×C8—21no—
92C2×C8—22no—
93C4⋊C4—22no—
94C4⋊C4D4×C16712yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C23×C16711yescharacteristic
99C23—22no—
100C23—41no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—42no—
105D4—42no—
106D4—42no—
107D4—82no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×D4×C16711yescharacteristic
113C2×C4C2×D4×C16712yes—
114C8—22no—
115C8—42no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—82no—
120C22—81no—
121C22#120×C16711yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#135×C16712yes—
126C4AGL(1,ℤ/8)×C16712yes—
127C2—82no—
128C2—161no—
129C2#449×C16712yes—
130C2#453×C16711yescharacteristic
subgroupG/H
center, socleC22×C167#120
commutator, Frattini, cosocleC42C23×C167
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C167 ◁ C42×C167 ◁ G

Complements

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

NH ≅ G/Ntype
1. #555×C167C2semidirect
2. #492×C167C2semidirect
3. #459×C167C2semidirect
4. #333×C167C2semidirect
5. D4⋊C8×C167C2semidirect
65. C167#1624direct (N × H)
66. #1624C167direct (N × H)
67. #555C2×C167semidirect
68. #492C2×C167semidirect
69. #459C2×C167semidirect
70. #333C2×C167semidirect
71. D4⋊C8C2×C167semidirect

Maximal subgroups

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

subgroupindex
#555×C1672
#492×C1672
#459×C1672
#333×C1672
D4⋊C8×C1672
#1624167