Finite Group Databasebeta

← #36322648

Subgroups — #36322648

Order 21376.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#24923821C211yesmaximal, characteristic
2#24923323C211yesmaximal, characteristic
3#24923114C211yesmaximal, characteristic
4#24923007C211yesmaximal, characteristic
5#24923006C211yesmaximal, characteristic
6#24923005C211yesmaximal, characteristic
7#328×C167C211yesmaximal, characteristic
8#19128831C411yescharacteristic, complemented
9#19128830C411yescharacteristic, complemented
10#19128822—21no—
11#19128811C2211yescharacteristic
12#19128809—21no—
13Q8×Dic167C411yescharacteristic, complemented
14#19128797—21no—
15#19128797C411yescharacteristic, complemented
16#19128788—21no—
17C2×C4×Dic167—21no—
18C2×C4×Dic167C2211yescharacteristic
19Dic167⋊C8—21no—
20C22⋊(C167⋊C8)C2211yescharacteristic
21#19128742—21no—
22C167⋊#95—21no—
23C167⋊#95C2211yescharacteristic
24C22⋊Q8×C167C2211yescharacteristic
25C22⋊C8×C167C2211yescharacteristic
26#95×C167C2211yescharacteristic
27C2×Dic334—21no—
28C2×Dic334C2×C411yescharacteristic
29C22×Dic167—21no—
30C22×Dic167D411yescharacteristic
31C22×Dic167D411yescharacteristic
32C2×C167⋊C8—21no—
33C4⋊Dic167—21no—
34C4⋊Dic167C2×C411yescharacteristic
35Dic167⋊C4—21no—
36Dic167⋊C4—21no—
37Dic167⋊C4—21no—
38Dic167⋊C4—21no—
39Dic167⋊C4—21no—
40C22⋊Dic167—21no—
41C4×Dic167—24no—
42C4×Dic167—21no—
43C4×Dic167—21no—
44C4×Dic167C2×C411yescharacteristic
45C4×Dic167C2×C411yescharacteristic
46C4×Dic167D411yescharacteristic
47C4×Dic167D411yescharacteristic
48C2×Q8×C167C2×C411yescharacteristic
49C22×C4×C167—21no—
50C22×C4×C167C2311yescharacteristic
51C2×C8×C167—21no—
52C4⋊C4×C167—21no—
53C4⋊C4×C167C2×C411yescharacteristic
54C22⋊C4×C167—21no—
55Dic334—42no—
56Dic334—41no—
57C2×Dic167—21no—
58C2×Dic167—21no—
59C2×Dic167—22no—
60C2×Dic167—22no—
61C2×Dic167—21no—
62C2×Dic167—21no—
63C2×Dic167—41no—
64C2×Dic167—41no—
65C2×Dic167C22⋊C411yescharacteristic
66C2×Dic167C22⋊C411yescharacteristic
67C2×Dic167C22⋊C411yescharacteristic
68C2×Dic167C22⋊C411yescharacteristic
69C167⋊C8—41no—
70C23×C167C2×D411yescharacteristic
71Q8×C167—41no—
72C2×C4×C167—22no—
73C2×C4×C167—21no—
74C2×C4×C167—21no—
75C2×C4×C167—41no—
76C2×C4×C167—41no—
77C2×C4×C167—41no—
78C2×C4×C167C22×C411yescharacteristic
79C2×C4×C167C2×D411yescharacteristic
80C8×C167—41no—
81C22×C167—21no—
82C22×C167—21no—
83C22×C167C2×C22⋊C411yescharacteristic
84C22×C167#13711yescharacteristic
85C22×C167#13711yescharacteristic
86C4×C167—21no—
87C4×C167—21no—
88C4×C167—42no—
89C4×C167—41no—
90C4×C167—41no—
91Dic167—21no—
92Dic167—21no—
93Dic167—21no—
94Dic167—42no—
95Dic167—41no—
96Dic167—41no—
97Dic167#10412yes—
98C2×C167—21no—
99C2×C167—21no—
100C2×C167#41011yescharacteristic
101C2×C167#41911yescharacteristic
102C2×C167C2×#10411yescharacteristic
103C167#150611yescharacteristic, complemented, Sylow
104#1506—1671nomaximal, Sylow
105#533—1671no—
106#432—1671no—
107#376—1671no—
108#328—1671no—
109#328—1671no—
110#328—1671no—
111#328D16711yescharacteristic
112C4⋊Q8—1671no—
113C4⋊Q8—1671no—
114#125—3341no—
115C22⋊Q8—3341no—
116C22⋊Q8—1671no—
117C22⋊Q8C2×D16711yescharacteristic
118C4×Q8—3341no—
119C4×Q8—1671no—
120C4×Q8—1671no—
121#117—3341no—
122C2×C42—3341no—
123C2×C42—1671no—
124C4⋊C8—3341no—
125C22⋊C8—1671no—
126C22⋊C8C2×D16711yescharacteristic
127#97—3341no—
128#95—3341no—
129#95—1671no—
130#95C2×D16711yescharacteristic
131C2×Q8—3341no—
132C2×Q8—1671no—
133C2×Q8C4×D16711yescharacteristic
134C22×C4—21no—
135C22×C4—3341no—
136C22×C4—1671no—
137C22×C4—1671no—
138C22×C4C22×D16711yescharacteristic
139C2×C8—21no—
140C2×C8—3341no—
141C4⋊C4—21no—
142C4⋊C4—3341no—
143C4⋊C4—3341no—
144C4⋊C4—3341no—
145C4⋊C4—3341no—
146C4⋊C4—3341no—
147C4⋊C4—3341no—
148C4⋊C4—1671no—
149C4⋊C4C4×D16711yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—3341no—
152C42—3344no—
153C42—3341no—
154C42—3341no—
155C42—1671no—
156C42—1671no—
157C42—1671no—
158C42—1671no—
159C23D4×D16711yescharacteristic
160Q8—41no—
161Q8—6681no—
162Q8—6682no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3341no—
170C2×C4—3341no—
171C2×C4—3342no—
172C2×C4—3342no—
173C2×C4—3341no—
174C2×C4—3341no—
175C2×C4—6681no—
176C2×C4—6681no—
177C2×C4—1671no—
178C2×C4—1671no—
179C2×C4—1671no—
180C2×C4—1671no—
181C2×C4C2×C4×D16711yescharacteristic
182C2×C4D4×D16711yescharacteristic
183C8—41no—
184C8—6681no—
185C22—21no—
186C22—21no—
187C22C22⋊C4×D16711yescharacteristic
188C22#1912886111yescharacteristic
189C22#1912886411yescharacteristic
190C4—21no—
191C4—21no—
192C4—42no—
193C4—41no—
194C4—41no—
195C4—3341no—
196C4—3341no—
197C4—3341no—
198C4—6682no—
199C4—6681no—
200C4—6681no—
201C4—1672no—
202C2—21no—
203C2—21no—
204C2#2492324111yescharacteristic
205C2#2492327611yescharacteristic
206C2#104×D16711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D167
commutatorC2×C4×C167C22×C4
FrattiniC22×C4C22×D167
Fitting#328×C167C2
socleC22×C167C2×C22⋊C4
prime radicalC23×C167C2×D4
cosocleC22×C4×C167C23
hypercenter#328D167
hypocenterC167#1506
LCSG ▷ C2×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
8. #19128831C4semidirect
9. #19128830C4semidirect
13. Q8×Dic167C4semidirect
15. #19128797C4semidirect
103. C167#1506semidirect

Maximal subgroups

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

subgroupindex
#249238212
#249233232
#249231142
#249230072
#249230062
#249230052
#328×C1672
#1506167