Finite Group Databasebeta

← #36324540

Subgroups — #36324540

Order 21376.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#19128820C211yesmaximal, characteristic
2#24923746C211yesmaximal, characteristic, complemented
3#24923275C211yesmaximal, characteristic
4#24923273C211yesmaximal, characteristic
5C2×#19128750C211yesmaximal, characteristic, complemented
6#24923029C211yesmaximal, characteristic, complemented
7#339×C167C211yesmaximal, characteristic, complemented
8C22×Dic334C2211yescharacteristic
9C22×D668C2211yescharacteristic
10C2×#14581934C2211yescharacteristic
11#19128820—22no—
12#19128820C414yescomplemented
13C4×Dic334—22no—
14#19128797—22no—
15#19128791C2211yescharacteristic
16#19128788—22no—
17C2×(C2×D167)⋊C4—22no—
18#19128754—22no—
19#19128752—22no—
20#19128750—22no—
21C2×M16×C167C2211yescharacteristic
22#117×C167C2211yescharacteristic
23C2×C42×C167C2211yescharacteristic
24C23×D167—21no—
25C2×Dic334—21no—
26C2×Dic334—21no—
27C2×Dic334—21no—
28C2×Dic334—21no—
29C2×Dic334C2×C412yes—
30C2×Dic334D412yes—
31C2×D668—21no—
32C2×D668—21no—
33C2×D668—21no—
34C2×D668—21no—
35C2×D668—21no—
36C2×D668C2×C412yes—
37C22×Dic167D412yes—
38#14581934—21no—
39#14581934—21no—
40C2×C167⋊C8—21no—
41C4⋊Dic167D412yes—
42Dic167⋊C4—24no—
43C22⋊Dic167—21no—
44(C2×D167)⋊C4—28no—
45C4×Dic167—24no—
46C4×Dic167—21no—
47C22×C4×C167—21no—
48C22×C4×C167C2311yescharacteristic
49M16×C167—21no—
50M16×C167—21no—
51C2×C8×C167—21no—
52C4⋊C4×C167D412yes—
53C22⋊C4×C167—21no—
54C42×C167—21no—
55C42×C167—21no—
56C42×C167C2×C412yes—
57C22×D167—22no—
58C22×D167—44no—
59C22×D167—41no—
60C22×D167—41no—
61Dic334—22no—
62Dic334—22no—
63Dic334—22no—
64Dic334—22no—
65D668—22no—
66D668—22no—
67D668—41no—
68D668—41no—
69C2×Dic167—22no—
70C2×Dic167—22no—
71C2×Dic167—22no—
72C2×Dic167C22⋊C414yes—
73C167⋊C8—41no—
74C23×C167C2×D411yescharacteristic
75C2×C4×C167—22no—
76C2×C4×C167—22no—
77C2×C4×C167—42no—
78C2×C4×C167C22×C411yescharacteristic
79C2×C4×C167C2×D411yescharacteristic
80C2×C4×C167C2×D411yescharacteristic
81C2×C4×C167C2×D411yescharacteristic
82C2×C4×C167#4111yescharacteristic
83C2×C4×C167#4111yescharacteristic
84C8×C167—41no—
85C22×C167—22no—
86C22×C167C2×C22⋊C411yescharacteristic
87C22×C167#12011yescharacteristic
88C22×C167#12311yescharacteristic
89C2×D167—44no—
90C2×D167—42no—
91C2×D167—82no—
92C2×D167—82no—
93C4×C167—42no—
94C4×C167—42no—
95C4×C167C4×D411yescharacteristic
96C4×C167C4×D411yescharacteristic
97C4×C167#12111yescharacteristic
98C4×C167#12111yescharacteristic
99Dic167—24no—
100Dic167—42no—
101C2×C167—21no—
102C2×C167—21no—
103C2×C167#38611yescharacteristic
104C2×C167#45412yes—
105D167—82no—
106C167#188811yescharacteristic, complemented, Sylow
107#1888—1671nomaximal, Sylow
108C2×#124—1671no—
109#519—1671no—
110#418—1671no—
111#418—1671no—
112C2×#101—1671no—
113#339—1671no—
114#339D16711yescharacteristic, complemented
115C22×Q8—1671no—
116C22×D4—1671no—
117C2×M16—1671no—
118C2×M16C2×D16711yescharacteristic
119#124—3342no—
120#124—1674no—
121C4×Q8—3342no—
122C4×Q8—3342no—
123#117—3342no—
124#117—1671no—
125#117C2×D16711yescharacteristic
126C2×C22⋊C4—3342no—
127C2×C42C2×D16711yescharacteristic
128#102—3342no—
129#102—3342no—
130#101—3342no—
131C24—3341no—
132C2×Q8—3341no—
133C2×Q8—3341no—
134C2×Q8—3341no—
135C2×Q8—3341no—
136C2×Q8—1672no—
137C2×Q8—1672no—
138C2×D4—3341no—
139C2×D4—3341no—
140C2×D4—3341no—
141C2×D4—3341no—
142C2×D4—3341no—
143C2×D4—1672no—
144C22×C4—21no—
145C22×C4—1672no—
146C22×C4C22×D16711yescharacteristic
147M16—21no—
148M16—21no—
149M16—3341no—
150M16—3341no—
151C2×C8—21no—
152C2×C8—3341no—
153C4⋊C4—3344no—
154C4⋊C4—1672no—
155C4⋊C4D66812yes—
156C22⋊C4—21no—
157C22⋊C4—3348no—
158C22⋊C4—3341no—
159C42—21no—
160C42—21no—
161C42—3344no—
162C42—3341no—
163C42C4×D16712yes—
164C23—3342no—
165C23—6684no—
166C23—6681no—
167C23—6681no—
168C23D4×D16711yescharacteristic
169Q8—3342no—
170Q8—3342no—
171Q8—3342no—
172Q8—3342no—
173D4—3342no—
174D4—3342no—
175D4—6681no—
176D4—6681no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—3342no—
181C2×C4—3342no—
182C2×C4—3342no—
183C2×C4—1674no—
184C2×C4C2×C4×D16711yescharacteristic
185C2×C4D4×D16711yescharacteristic
186C2×C4D4×D16711yescharacteristic
187C2×C4C2×D66811yescharacteristic
188C2×C4#1458195011yescharacteristic
189C2×C4#1458195111yescharacteristic
190C8—41no—
191C8—6681no—
192C22—22no—
193C22—13362no—
194C22—13362no—
195C22—6684no—
196C22—6682no—
197C22C22⋊C4×D16711yescharacteristic
198C22#1912880211yescharacteristic
199C22#1912881711yescharacteristic
200C4—42no—
201C4—42no—
202C4—3344no—
203C4—6682no—
204C4#1912879411yescharacteristic
205C4C4×D66811yescharacteristic
206C4#1912880511yescharacteristic
207C4#1912880711yescharacteristic
208C2—21no—
209C2—21no—
210C2—13362no—
211C2#2492314711yescharacteristic
212C2#2492344312yes—
subgroupG/H
centerC22#19128802
commutatorC2×C4×C167C22×C4
FrattiniC22×C4C22×D167
Fitting#339×C167C2
socleC22×C167#120
prime radicalC22×D668C22
cosocleC22×C4×C167C23
hypercenter#339D167
hypocenterC167#1888
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #24923746C2semidirect
5. C2×#19128750C2semidirect
6. #24923029C2semidirect
7. #339×C167C2semidirect
12. #19128820C4semidirect
106. C167#1888semidirect
114. #339D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288202
#249237462
#249232752
#249232732
C2×#191287502
#249230292
#339×C1672
#1888167