Finite Group Databasebeta

← #1391×C167 #36319751

Subgroups — #1391×C167 #36319751

Order 21376.

166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 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#410×C167C212yesmaximal
2#408×C167C211yesmaximal, characteristic
3C2×#41×C167C2211yescharacteristic
4C2×M16×C167C411yescharacteristic, complemented
5C22×C8×C167C411yescharacteristic, complemented
6#117×C167—22no—
7#99×C167—22no—
8#99×C167C414yescomplemented
9C22⋊C8×C167—22no—
10#41×C167—41no—
11#41×C167—41no—
12C2×Q8×C167Q811yescharacteristic
13C2×D4×C167C2×C412yes—
14C2×D4×C167D411yescharacteristic
15C22×C4×C167C2×C411yescharacteristic
16C22×C4×C167D412yes—
17M16×C167—41no—
18C2×C8×C167—21no—
19C2×C8×C167—21no—
20C2×C8×C167—41no—
21C4⋊C4×C167—42no—
22C22⋊C4×C167—24no—
23C42×C167—42no—
24C23×C167C4211yescharacteristic
25C23×C167C22⋊C412yes—
26Q8×C167—41no—
27D4×C167—42no—
28D4×C167—41no—
29C2×C4×C167—22no—
30C2×C4×C167—21no—
31C2×C4×C167—22no—
32C2×C4×C167—21no—
33C2×C4×C167—44no—
34C2×C4×C167C22⋊C411yescharacteristic
35C2×C4×C167C4⋊C411yescharacteristic
36C2×C4×C167C4⋊C412yes—
37C8×C167—41no—
38C8×C167—41no—
39C22×C167—22no—
40C22×C167—21no—
41C22×C167—41no—
42C22×C167—42no—
43C22×C167#9511yescharacteristic
44C4×C167—22no—
45C4×C167—21no—
46C4×C167—84no—
47C4×C167#9911yescharacteristic
48C4×C167#9911yescharacteristic
49C2×C167—21no—
50C2×C167—42no—
51C2×C167—41no—
52C2×C167#34211yescharacteristic
53C167#139111yescharacteristic, direct product factor, Sylow
54#1391C16711yesmaximal, characteristic, direct product factor, Sylow
55#410C2×C16712yes—
56#408C2×C16711yescharacteristic
57C2×#41C22×C16711yescharacteristic
58C2×M16C4×C16711yescharacteristic, complemented
59C22×C8C4×C16711yescharacteristic, complemented
60#117—22no—
61#99—22no—
62#99C4×C16714yescomplemented
63C22⋊C8—22no—
64#41—41no—
65#41—41no—
66C2×Q8Q8×C16711yescharacteristic
67C2×D4C2×C4×C16712yes—
68C2×D4D4×C16711yescharacteristic
69C22×C4C2×C4×C16711yescharacteristic
70C22×C4D4×C16712yes—
71M16—41no—
72C2×C8—21no—
73C2×C8—21no—
74C2×C8—41no—
75C4⋊C4—42no—
76C22⋊C4—24no—
77C42—42no—
78C23C42×C16711yescharacteristic
79C23C22⋊C4×C16712yes—
80Q8—41no—
81D4—42no—
82D4—41no—
83C2×C4—22no—
84C2×C4—22no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—44no—
88C2×C4C22⋊C4×C16711yescharacteristic
89C2×C4C4⋊C4×C16711yescharacteristic
90C2×C4C4⋊C4×C16712yes—
91C8—41no—
92C8—41no—
93C22—22no—
94C22—21no—
95C22—41no—
96C22—42no—
97C22#95×C16711yescharacteristic
98C4—22no—
99C4—21no—
100C4—84no—
101C4#99×C16711yescharacteristic
102C4#99×C16711yescharacteristic
103C2—21no—
104C2—42no—
105C2—41no—
106C2#342×C16711yescharacteristic
subgroupG/H
centerC4×C167#99
commutatorC23C42×C167
Frattini, cosocleC2×#41C22×C167
socleC2×C167#342
prime radicalC2×#41×C167C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C167 ◁ C2×C4×C167 ◁ C2×#41×C167 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×M16×C167C4semidirect
5. C22×C8×C167C4semidirect
8. #99×C167C4semidirect
53. C167#1391direct (N × H)
54. #1391C167direct (N × H)
58. C2×M16C4×C167semidirect
59. C22×C8C4×C167semidirect
62. #99C4×C167semidirect

Maximal subgroups

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

subgroupindex
#410×C1672
#408×C1672
#1391167