Finite Group Databasebeta

← #24923458

Subgroups — #24923458

Order 10688.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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#19128877C211yesmaximal, characteristic
2#19128815C212yesmaximal, complemented
3#19128747C211yesmaximal, characteristic, complemented
4#19128746C212yesmaximal, complemented
5#123×C167C211yesmaximal, characteristic, complemented
6#14581950—21no—
7#14581949—22no—
8C2×C22⋊D167—21no—
9C2×C22⋊D167C2212yes—
10D4×D167—22no—
11Dic167⋊C4—22no—
12C22⋊Dic167C2212yescomplemented
13(C2×D167)⋊C4—22no—
14C2×D4×C167C2211yescharacteristic
15C22×C4×C167C2211yescharacteristic, complemented
16C4⋊C4×C167—21no—
17C22⋊C4×C167—21no—
18C22⋊C4×C167C2211yescharacteristic, complemented
19C22×D167—21no—
20C22×D167D412yes—
21Dic334—21no—
22D668—21no—
23C22⋊D167—22no—
24C22⋊D167—22no—
25C22⋊D167—22no—
26C2×Dic167—21no—
27C2×Dic167—22no—
28C2×Dic167D412yes—
29C4×D167—22no—
30C23×C167D411yescharacteristic
31C23×C167C2311yescharacteristic
32D4×C167—22no—
33C2×C4×C167—21no—
34C2×C4×C167—21no—
35C2×C4×C167—22no—
36C2×C4×C167D411yescharacteristic
37C22×C167—21no—
38C22×C167—41no—
39C22×C167—41no—
40C22×C167C2×D412yes—
41C22×C167C2×D411yescharacteristic
42C2×D167—22no—
43C2×D167—42no—
44C2×D167—81no—
45C4×C167—21no—
46C4×C167—41no—
47C4×C167—41no—
48Dic167—22no—
49Dic167—42no—
50C2×C167—22no—
51C2×C167—21no—
52C2×C167—41no—
53C2×C167#12011yescharacteristic
54D167—42no—
55C167#45811yescharacteristic, complemented, Sylow
56#458—1671nomaximal, Sylow
57H(5,𝔽2)—1671no—
58#123—1672no—
59#123D16711yescharacteristic, complemented
60#99—1671no—
61#99—1672no—
62#41—3342no—
63#41—3341no—
64C2×D4—3342no—
65C2×D4—3341no—
66C2×D4—1672no—
67C2×D4C2×D16711yescharacteristic
68C22×C4C2×D16711yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—3342no—
71C22⋊C4—21no—
72C22⋊C4—3342no—
73C22⋊C4—1672no—
74C22⋊C4C2×D16711yescharacteristic, complemented
75C23—3341no—
76C23—1672no—
77C23C22⋊D16711yescharacteristic
78C23C22×D16711yescharacteristic
79Q8—3341no—
80D4—22no—
81D4—3341no—
82D4—3342no—
83D4—3342no—
84D4—3342no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3342no—
89C2×C4—3341no—
90C2×C4—3342no—
91C2×C4—1672no—
92C2×C4C22⋊D16711yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—13361no—
97C22—3342no—
98C22—6682no—
99C22D4×D16712yes—
100C22C2×C22⋊D16711yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—3342no—
105C4—6682no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—6682no—
110C2#1912880011yescharacteristic
subgroupG/H
centerC2#19128800
commutator, cosocleC23×C167C23
FrattiniC23C22×D167
Fitting#123×C167C2
socleC2×C167#120
prime radical#19128877C2
hypercenter#123D167
hypocenterC167#458
LCSG ▷ C23×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #19128815C2semidirect
3. #19128747C2semidirect
4. #19128746C2semidirect
5. #123×C167C2semidirect
12. C22⋊Dic167C22semidirect
15. C22×C4×C167C22semidirect
18. C22⋊C4×C167C22semidirect
55. C167#458semidirect
59. #123D167semidirect
68. C22×C4C2×D167semidirect
74. C22⋊C4C2×D167semidirect