Finite Group Databasebeta

← #36324584

Subgroups — #36324584

Order 21376.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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#24923780C211yesmaximal, characteristic
2C2×#19128791C211yesmaximal, characteristic, complemented
3C2×#19128758C212yesmaximal, complemented
4#24923030C212yesmaximal, complemented
5#407×C167C211yesmaximal, characteristic, complemented
6C2×#14581950C2211yescharacteristic
7#19128815—21no—
8#19128811C412yescomplemented
9#19128806C412yescomplemented
10#19128801—21no—
11C4×C22⋊D167—22no—
12#19128791—24no—
13#19128791C2211yescharacteristic
14C2×C4⋊Dic167—21no—
15C2×C22⋊Dic167—21no—
16C2×C4×Dic167C2212yes—
17#19128758—24no—
18C23×C4×C167C2211yescharacteristic
19C2×M16×C167C2212yes—
20C22⋊C8×C167—22no—
21#14581950—22no—
22#14581950—22no—
23C2×Dic334C2×C411yescharacteristic
24C2×D668C2×C411yescharacteristic
25C2×C22⋊D167—22no—
26C2×C22⋊D167—21no—
27C22×Dic167—22no—
28C2×C4×D167—21no—
29C4⋊Dic167—24no—
30C4⋊Dic167C2×C412yes—
31Dic167⋊C4—22no—
32C22⋊Dic167—24no—
33C22⋊Dic167—22no—
34C22⋊Dic167—21no—
35(C2×D167)⋊C4—22no—
36C4×Dic167—22no—
37C4×Dic167—21no—
38C4×Dic167D414yescomplemented
39C24×C167D411yescharacteristic
40C22×C4×C167—22no—
41C22×C4×C167—22no—
42C22×C4×C167C2×C412yes—
43C22×C4×C167D412yes—
44C22×C4×C167D411yescharacteristic
45C22×C4×C167C2311yescharacteristic
46M16×C167—22no—
47M16×C167—22no—
48C2×C8×C167—22no—
49C22×D167—21no—
50Dic334—22no—
51D668—22no—
52C22⋊D167—44no—
53C22⋊D167—41no—
54C22⋊D167—41no—
55C2×Dic167—22no—
56C2×Dic167—21no—
57C2×Dic167—24no—
58C2×Dic167—44no—
59C4×D167—42no—
60C23×C167—21no—
61C23×C167—21no—
62C23×C167—42no—
63C23×C167C22⋊C412yes—
64C23×C167C2×D411yescharacteristic
65C2×C4×C167—21no—
66C2×C4×C167—21no—
67C2×C4×C167—24no—
68C2×C4×C167—24no—
69C2×C4×C167C22⋊C412yes—
70C2×C4×C167C22×C411yescharacteristic
71C2×C4×C167C2×D412yes—
72C2×C4×C167C2×D411yescharacteristic
73C2×C4×C167#4112yes—
74C8×C167—42no—
75C22×C167—22no—
76C22×C167—22no—
77C22×C167—22no—
78C22×C167—22no—
79C22×C167—42no—
80C22×C167—42no—
81C22×C167C2×C22⋊C411yescharacteristic
82C22×C167#12011yescharacteristic
83C22×C167#12311yescharacteristic
84C2×D167—41no—
85C2×D167—42no—
86C4×C167—22no—
87C4×C167C4×D412yes—
88C4×C167#12112yes—
89Dic167—44no—
90Dic167—41no—
91Dic167—42no—
92C2×C167—21no—
93C2×C167—21no—
94C2×C167—42no—
95C2×C167#38611yescharacteristic
96C2×C167#42112yes—
97D167—81no—
98C167#189711yescharacteristic, complemented, Sylow
99#1897—1671nomaximal, Sylow
100#525—1671no—
101C2×#117—1671no—
102C2×#104—1672no—
103#407D16711yescharacteristic, complemented
104#339—1672no—
105C2×#41—1671no—
106C23×C4C2×D16711yescharacteristic
107C2×M16C2×D16712yes—
108#123—3341no—
109C22⋊Q8—1672no—
110#121—1672no—
111#120—3341no—
112C4×D4—3342no—
113#117—3344no—
114#117—1671no—
115C2×C4⋊C4—3341no—
116C2×C22⋊C4—3341no—
117C2×C42—1672no—
118#104—3344no—
119C22⋊C8—22no—
120C24D66811yescharacteristic
121#41—3342no—
122#41—3342no—
123C2×Q8—1671no—
124C2×D4—3342no—
125C2×D4—3341no—
126C2×D4—1671no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—3342no—
130C22×C4—3341no—
131C22×C4C4×D16712yes—
132C22×C4C22⋊D16712yes—
133C22×C4D66811yescharacteristic
134C22×C4C22×D16711yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—3344no—
139C4⋊C4—3342no—
140C4⋊C4—1672no—
141C22⋊C4—3344no—
142C22⋊C4—3342no—
143C22⋊C4—3341no—
144C22⋊C4—3342no—
145C42—3342no—
146C42—3341no—
147C42—1674no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—3341no—
152C23(C2×D167)⋊C412yes—
153C23C2×D66811yescharacteristic
154Q8—3342no—
155D4—3342no—
156D4—6684no—
157D4—6681no—
158D4—6681no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—3342no—
164C2×C4—3341no—
165C2×C4—3344no—
166C2×C4—6684no—
167C2×C4—6682no—
168C2×C4(C2×D167)⋊C412yes—
169C2×C4C2×C4×D16711yescharacteristic
170C2×C4D4×D16712yes—
171C2×C4C2×C22⋊D16711yescharacteristic
172C2×C4#1458194912yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—6681no—
181C22—6682no—
182C22C2×(C2×D167)⋊C411yescharacteristic
183C22#1912880211yescharacteristic
184C22#1912881611yescharacteristic
185C4—22no—
186C4—6684no—
187C4—6681no—
188C4—6682no—
189C4(C22⋊D167)⋊C412yes—
190C4#1912880412yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—13361no—
195C2#2492314611yescharacteristic
196C2#2492328512yes—
subgroupG/H
centerC2×C4C2×C22⋊D167
commutatorC2×C4×C167C22×C4
FrattiniC22×C4C22×D167
Fitting#407×C167C2
socleC22×C167C2×C22⋊C4
prime radical#24923780C2
cosocleC22×C4×C167C23
hypercenter#407D167
hypocenterC167#1897
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#19128791C2semidirect
3. C2×#19128758C2semidirect
4. #24923030C2semidirect
5. #407×C167C2semidirect
8. #19128811C4semidirect
9. #19128806C4semidirect
38. C4×Dic167D4semidirect
98. C167#1897semidirect
103. #407D167semidirect

Maximal subgroups

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

subgroupindex
#249237802
C2×#191287912
C2×#191287582
#249230302
#407×C1672
#1897167