Finite Group Databasebeta

← #36323129

Subgroups — #36323129

Order 21376.

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

normal subgroups only

This group has 228 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#24923597C212yesmaximal, complemented
2#24923194C211yesmaximal, characteristic
3#24923006C212yesmaximal, complemented
4#24922993C211yesmaximal, characteristic
5#543×C167C211yesmaximal, characteristic
6C2×C4⋊Dic167C2211yescharacteristic
7C2×C22⋊Dic167—21no—
8#19128763—22no—
9#19128756—22no—
10C22⋊(C167⋊C8)C2212yes—
11C167⋊#95—21no—
12C167⋊#95C2211yescharacteristic
13C22⋊Q8×C167—22no—
14C22⋊Q8×C167—21no—
15C22⋊Q8×C167—22no—
16C22⋊Q8×C167C2212yes—
17C2×C22⋊C4×C167—21no—
18C2×C22⋊C4×C167C2211yescharacteristic
19C22×Dic167—21no—
20C22×Dic167—21no—
21C2×C167⋊C8—22no—
22C4⋊Dic167—22no—
23C22⋊Dic167—42no—
24C24×C167D411yescharacteristic
25C2×Q8×C167—21no—
26C2×Q8×C167D412yes—
27C22×C4×C167—21no—
28C22×C4×C167—21no—
29C22×C4×C167D411yescharacteristic
30C22×C4×C167C2311yescharacteristic
31C4⋊C4×C167—21no—
32C4⋊C4×C167—22no—
33C4⋊C4×C167—22no—
34C4⋊C4×C167D412yes—
35C22⋊C4×C167—22no—
36C22⋊C4×C167—22no—
37C22⋊C4×C167—22no—
38C2×Dic167—41no—
39C2×Dic167—41no—
40C2×Dic167—41no—
41C2×Dic167—41no—
42C2×Dic167—41no—
43C2×Dic167—41no—
44C167⋊C8—42no—
45C23×C167—21no—
46C23×C167—21no—
47C23×C167—41no—
48C23×C167—41no—
49C23×C167SD1612yes—
50C23×C167C2×D411yescharacteristic
51Q8×C167—42no—
52Q8×C167—81no—
53C2×C4×C167—22no—
54C2×C4×C167—21no—
55C2×C4×C167—22no—
56C2×C4×C167—41no—
57C2×C4×C167—41no—
58C2×C4×C167—81no—
59C2×C4×C167—81no—
60C2×C4×C167C2×D412yes—
61C22×C167—21no—
62C22×C167—21no—
63C22×C167—22no—
64C22×C167—42no—
65C22×C167—41no—
66C22×C167—41no—
67C22×C167—81no—
68C22×C167#12011yescharacteristic
69C22×C167C2×SD1611yescharacteristic
70C22×C167#13711yescharacteristic
71C4×C167—22no—
72C4×C167—41no—
73C4×C167—42no—
74C4×C167—42no—
75Dic167—81no—
76Dic167—81no—
77C2×C167—21no—
78C2×C167—21no—
79C2×C167—42no—
80C2×C167#44811yescharacteristic
81C2×C167#45511yescharacteristic
82C2×C167#45811yescharacteristic
83C167#160811yescharacteristic, complemented, Sylow
84#1608—1671nomaximal, Sylow
85#543D16711yescharacteristic
86#483—1672no—
87#397—1671no—
88#328—1672no—
89#323—1671no—
90C22⋊Q8—22no—
91C22⋊Q8—21no—
92C22⋊Q8—22no—
93C22⋊Q8C2×D16712yes—
94C2×C4⋊C4—1671no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—3341no—
97C2×C22⋊C4C2×D16711yescharacteristic
98#106—3342no—
99#103—3342no—
100C22⋊C8—1672no—
101#95—3341no—
102#95—1671no—
103C24C22⋊D16711yescharacteristic
104C2×Q8—21no—
105C2×Q8C22⋊D16712yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—3341no—
109C22×C4—3341no—
110C22×C4C22⋊D16711yescharacteristic
111C22×C4C22×D16711yescharacteristic
112C2×C8—3342no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—3342no—
117C4⋊C4C22⋊D16712yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—6682no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1458193712yes—
127C23C2×C22⋊D16711yescharacteristic
128Q8—42no—
129Q8—81no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—6681no—
138C2×C4—6681no—
139C2×C4—6681no—
140C2×C4—6681no—
141C2×C4—6681no—
142C2×C4—6681no—
143C2×C4C2×C22⋊D16712yes—
144C8—6682no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#1912880111yescharacteristic
153C22C2×#1458193711yescharacteristic
154C22#1912886311yescharacteristic
155C4—22no—
156C4—41no—
157C4—42no—
158C4—42no—
159C4—13361no—
160C4—13361no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#2492340811yescharacteristic
165C2#2492344711yescharacteristic
166C2#2492345911yescharacteristic
subgroupG/H
centerC22#19128801
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#543×C167C2
socleC22×C167#120
prime radicalC24×C167D4
hypercenter#543D167
hypocenterC167#1608
LCSG ▷ C22×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #24923597C2semidirect
3. #24923006C2semidirect
83. C167#1608semidirect

Maximal subgroups

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

subgroupindex
#249235972
#249231942
#249230062
#249229932
#543×C1672
#1608167