Finite Group Databasebeta

← #36323175

Subgroups — #36323175

Order 21376.

246 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 57 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#24923585C212yesmaximal, complemented
2#24923208C211yesmaximal, characteristic, complemented
3#24923003C212yesmaximal, complemented
4#24922995C211yesmaximal, characteristic, complemented
5#537×C167C211yesmaximal, characteristic
6C2×C4⋊Dic167C2211yescharacteristic
7#19128763—22no—
8#19128753—22no—
9C22⋊(C167⋊C8)C2212yes—
10C167⋊#95—21no—
11C167⋊#95—21no—
12C167⋊#95C2211yescharacteristic
13C2×#41×C167—21no—
14C22⋊Q8×C167—22no—
15#121×C167—22no—
16#121×C167—21no—
17#121×C167C2212yes—
18C2×C4⋊C4×C167C2211yescharacteristic
19C22×Dic167—21no—
20C22×Dic167—21no—
21C2×C167⋊C8—22no—
22C4⋊Dic167—22no—
23#41×C167—42no—
24#41×C167—42no—
25C2×Q8×C167—21no—
26C2×D4×C167—21no—
27C2×D4×C167—22no—
28C2×D4×C167—21no—
29C2×D4×C167D412yes—
30C22×C4×C167—22no—
31C22×C4×C167D411yescharacteristic
32C22×C4×C167D411yescharacteristic
33C22×C4×C167C2311yescharacteristic
34C4⋊C4×C167—21no—
35C4⋊C4×C167—22no—
36C4⋊C4×C167D412yes—
37C22⋊C4×C167—22no—
38C22⋊C4×C167—22no—
39C2×Dic167—41no—
40C2×Dic167—41no—
41C2×Dic167—42no—
42C2×Dic167—41no—
43C2×Dic167—41no—
44C167⋊C8—42no—
45C23×C167—22no—
46C23×C167C2×D411yescharacteristic
47Q8×C167—42no—
48D4×C167—42no—
49D4×C167—42no—
50D4×C167—42no—
51D4×C167—42no—
52D4×C167—81no—
53C2×C4×C167—21no—
54C2×C4×C167—21no—
55C2×C4×C167—21no—
56C2×C4×C167—22no—
57C2×C4×C167—41no—
58C2×C4×C167—44no—
59C2×C4×C167—41no—
60C2×C4×C167SD1612yes—
61C2×C4×C167C2×D412yes—
62C22×C167—21no—
63C22×C167—21no—
64C22×C167—42no—
65C22×C167—42no—
66C22×C167—42no—
67C22×C167#12011yescharacteristic
68C22×C167C2×SD1611yescharacteristic
69C22×C167AGL(1,ℤ/8)11yescharacteristic
70C4×C167—22no—
71C4×C167—22no—
72C4×C167—42no—
73C4×C167—41no—
74Dic167—81no—
75Dic167—81no—
76C2×C167—21no—
77C2×C167—21no—
78C2×C167—82no—
79C2×C167#45011yescharacteristic
80C2×C167#45411yescharacteristic
81C2×C167#45811yescharacteristic
82C167#161611yescharacteristic, complemented, Sylow
83#1616—1671nomaximal, Sylow
84#537D16711yescharacteristic
85#481—1672no—
86#400—1671no—
87#327—1672no—
88#324—1671no—
89C2×#41—21no—
90C22⋊Q8—22no—
91#121—22no—
92#121—21no—
93#121C2×D16712yes—
94C2×C4⋊C4—1671no—
95C2×C4⋊C4C2×D16711yescharacteristic
96#106—3342no—
97#102—3342no—
98C22⋊C8—1672no—
99#95—3341no—
100#95—3341no—
101#95—1671no—
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4C22⋊D16712yes—
109C22×C4—22no—
110C22×C4—3341no—
111C22×C4—3341no—
112C22×C4C22⋊D16711yescharacteristic
113C22×C4C22⋊D16711yescharacteristic
114C22×C4C22×D16711yescharacteristic
115C2×C8—3342no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—3342no—
119C4⋊C4C22⋊D16712yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C23—22no—
123C23C2×C22⋊D16711yescharacteristic
124Q8—42no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—81no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—6681no—
138C2×C4—6681no—
139C2×C4—6682no—
140C2×C4—6681no—
141C2×C4—6681no—
142C2×C4#1458193812yes—
143C2×C4C2×C22⋊D16712yes—
144C8—6682no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22#1912880111yescharacteristic
151C22C2×#1458193811yescharacteristic
152C22#1912885811yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—41no—
157C4—13361no—
158C4—13361no—
159C2—21no—
160C2—21no—
161C2—82no—
162C2#2492342011yescharacteristic
163C2#2492344211yescharacteristic
164C2#2492345911yescharacteristic
subgroupG/H
centerC22#19128801
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting, prime radical#537×C167C2
socleC22×C167#120
hypercenter#537D167
hypocenterC167#1616
LCSG ▷ C22×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #24923585C2semidirect
2. #24923208C2semidirect
3. #24923003C2semidirect
4. #24922995C2semidirect
82. C167#1616semidirect

Maximal subgroups

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

subgroupindex
#249235852
#249232082
#249230032
#249229952
#537×C1672
#1616167