Finite Group Databasebeta

← #36322671

Subgroups — #36322671

Order 21376.

274 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 81 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#24923233C211yesmaximal, characteristic
2#24923132C211yesmaximal, characteristic
3#24923006C214yesmaximal, complemented
4C2×C22⋊Q8×C167C211yesmaximal, characteristic
5C2×#14581934—21no—
6C2×C22⋊Dic167—21no—
7C22⋊(C167⋊C8)—21no—
8C22⋊(C167⋊C8)C2212yes—
9C167⋊#95—21no—
10C167⋊#95C2212yes—
11C22×Q8×C167C411yescharacteristic, complemented
12C23×C4×C167C2211yescharacteristic
13C22⋊Q8×C167—21no—
14C22⋊Q8×C167—21no—
15C22⋊Q8×C167C412yescomplemented
16C22⋊Q8×C167C2212yes—
17C2×C4⋊C4×C167—21no—
18C2×C4⋊C4×C167C411yescharacteristic, complemented
19C2×C22⋊C4×C167—21no—
20C22×Dic167—22no—
21#14581934—42no—
22C2×C167⋊C8—22no—
23C22⋊Dic167—42no—
24C24×C167D411yescharacteristic
25C2×Q8×C167—21no—
26C2×Q8×C167—42no—
27C2×Q8×C167C2×C412yes—
28C22×C4×C167—21no—
29C22×C4×C167—21no—
30C22×C4×C167—42no—
31C22×C4×C167C2×C411yescharacteristic
32C22×C4×C167C2×C411yescharacteristic
33C22×C4×C167D411yescharacteristic
34C22×C4×C167D411yescharacteristic
35C22×C4×C167D411yescharacteristic
36C22×C4×C167C2311yescharacteristic
37C4⋊C4×C167—22no—
38C4⋊C4×C167—21no—
39C4⋊C4×C167—22no—
40C4⋊C4×C167C2×C412yes—
41C22⋊C4×C167—22no—
42C22⋊C4×C167—22no—
43C2×Dic167—42no—
44C2×Dic167—42no—
45C2×Dic167—42no—
46C167⋊C8—42no—
47C23×C167—22no—
48C23×C167—22no—
49C23×C167—41no—
50C23×C167C22⋊C411yescharacteristic
51C23×C167C22⋊C411yescharacteristic
52C23×C167C2×D411yescharacteristic
53Q8×C167—42no—
54Q8×C167—42no—
55C2×C4×C167—22no—
56C2×C4×C167—22no—
57C2×C4×C167—22no—
58C2×C4×C167—22no—
59C2×C4×C167—41no—
60C2×C4×C167—41no—
61C2×C4×C167—41no—
62C2×C4×C167—41no—
63C2×C4×C167—42no—
64C2×C4×C167—42no—
65C2×C4×C167C22⋊C411yescharacteristic
66C2×C4×C167C22⋊C411yescharacteristic
67C2×C4×C167C22×C411yescharacteristic
68C2×C4×C167C2×D411yescharacteristic
69C22×C167—21no—
70C22×C167—21no—
71C22×C167—21no—
72C22×C167—21no—
73C22×C167—21no—
74C22×C167—21no—
75C22×C167—21no—
76C22×C167—44no—
77C22×C167#9912yes—
78C22×C167C2×C22⋊C411yescharacteristic
79C22×C167#13712yes—
80C4×C167—21no—
81C4×C167—21no—
82C4×C167—21no—
83C4×C167—21no—
84C4×C167—42no—
85C4×C167—42no—
86Dic167—82no—
87C2×C167—22no—
88C2×C167—22no—
89C2×C167—41no—
90C2×C167C2×#9911yescharacteristic
91C2×C167#41911yescharacteristic
92C2×C167#42111yescharacteristic
93C167#151111yescharacteristic, complemented, Sylow
94#1511—1671nomaximal, Sylow
95C2×C22⋊Q8D16711yescharacteristic
96#407—1671no—
97#381—1671no—
98#328—1674no—
99C22×Q8Dic16711yescharacteristic, complemented
100C23×C4C2×D16711yescharacteristic
101C2×M16—3341no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8Dic16712yescomplemented
105C22⋊Q8C2×D16712yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4Dic16711yescharacteristic, complemented
108C2×C22⋊C4—21no—
109C2×C22⋊C4—3341no—
110C22⋊C8—3341no—
111C22⋊C8—1672no—
112#95—3341no—
113#95—1672no—
114C24C22⋊D16711yescharacteristic
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×Dic16712yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—3342no—
122C22×C4C2×Dic16711yescharacteristic
123C22×C4C2×Dic16711yescharacteristic
124C22×C4C22⋊D16711yescharacteristic
125C22×C4C22⋊D16711yescharacteristic
126C22×C4C22⋊D16711yescharacteristic
127C22×C4C22×D16711yescharacteristic
128M16—6682no—
129C2×C8—3342no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4C2×Dic16712yes—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—6682no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23C22⋊Dic16711yescharacteristic
141C23C22⋊Dic16711yescharacteristic
142C23C2×C22⋊D16711yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—6682no—
156C2×C4—6682no—
157C2×C4—6682no—
158C2×C4C22⋊Dic16711yescharacteristic
159C2×C4C22⋊Dic16711yescharacteristic
160C2×C4C22×Dic16711yescharacteristic
161C2×C4C2×C22⋊D16711yescharacteristic
162C8—6682no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#1912874712yes—
172C22C2×C22⋊Dic16711yescharacteristic
173C22#1912886312yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—42no—
180C4—13362no—
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#1912874711yescharacteristic
185C2#2492327811yescharacteristic
186C2#2492328611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic167
commutatorC2×C4×C167C22×C4
FrattiniC22×C4C22×D167
FittingC2×C22⋊Q8×C167C2
socleC22×C167C2×C22⋊C4
prime radicalC24×C167D4
cosocleC22×C4×C167C23
hypercenterC2×C22⋊Q8D167
hypocenterC167#1511
LCSG ▷ C2×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #24923006C2semidirect
11. C22×Q8×C167C4semidirect
15. C22⋊Q8×C167C4semidirect
18. C2×C4⋊C4×C167C4semidirect
93. C167#1511semidirect
99. C22×Q8Dic167semidirect
104. C22⋊Q8Dic167semidirect
107. C2×C4⋊C4Dic167semidirect

Maximal subgroups

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

subgroupindex
#249232332
#249231322
#249230062
C2×C22⋊Q8×C1672
#1511167