Finite Group Databasebeta

← #36323113

Subgroups — #36323113

Order 21376.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#24923898C211yesmaximal, characteristic
2#24923427C211yesmaximal, characteristic
3#24923424C211yesmaximal, characteristic
4#24923190C211yesmaximal, characteristic
5#24923005C211yesmaximal, characteristic
6#24922994C211yesmaximal, characteristic
7#328×C167C211yesmaximal, characteristic
8C22×Dic334C2211yescharacteristic
9C2×Dic668—21no—
10C2×#14581940—21no—
11#19128831—21no—
12#19128830—21no—
13#19128822—21no—
14#19128822—21no—
15#19128811C2211yescharacteristic
16#19128809—21no—
17#19128788—21no—
18C2×C4⋊Dic167—21no—
19C2×Dic167⋊C4C2211yescharacteristic
20#19128757—21no—
21#19128755—21no—
22C22⋊(C167⋊C8)C2211yescharacteristic
23C167⋊#95—21no—
24C22⋊Q8×C167C2211yescharacteristic
25C22⋊C8×C167C2211yescharacteristic
26#95×C167C2211yescharacteristic
27C2×Dic334—21no—
28C2×Dic334—21no—
29C2×Dic334—21no—
30C2×Dic334—42no—
31C2×Dic334D411yescharacteristic
32C22×Dic167—21no—
33C22×Dic167D411yescharacteristic
34C22×Dic167D411yescharacteristic
35Dic668—42no—
36#14581940—42no—
37C2×C167⋊C8—21no—
38C4⋊Dic167—21no—
39C4⋊Dic167—42no—
40C4⋊Dic167D411yescharacteristic
41Dic167⋊C4—21no—
42Dic167⋊C4—21no—
43Dic167⋊C4—21no—
44Dic167⋊C4—21no—
45Dic167⋊C4—21no—
46Dic167⋊C4—21no—
47Dic167⋊C4—21no—
48C22⋊Dic167—21no—
49C4×Dic167—21no—
50C4×Dic167—21no—
51C2×Q8×C167D411yescharacteristic
52C22×C4×C167—21no—
53C22×C4×C167C2311yescharacteristic
54C2×C8×C167—21no—
55C4⋊C4×C167—21no—
56C4⋊C4×C167D411yescharacteristic
57C22⋊C4×C167—21no—
58Dic334—22no—
59Dic334—22no—
60Dic334—41no—
61Dic334—41no—
62Dic334—41no—
63Dic334—81no—
64C2×Dic167—21no—
65C2×Dic167—21no—
66C2×Dic167—22no—
67C2×Dic167—21no—
68C2×Dic167—21no—
69C2×Dic167—21no—
70C2×Dic167—41no—
71C2×Dic167—41no—
72C2×Dic167—41no—
73C2×Dic167Q1612yescomplemented
74C167⋊C8—41no—
75C23×C167C2×D411yescharacteristic
76Q8×C167—41no—
77C2×C4×C167—22no—
78C2×C4×C167—21no—
79C2×C4×C167—21no—
80C2×C4×C167—41no—
81C2×C4×C167—41no—
82C2×C4×C167—41no—
83C2×C4×C167C2×D411yescharacteristic
84C2×C4×C167C2×D411yescharacteristic
85C8×C167—41no—
86C22×C167—21no—
87C22×C167—21no—
88C22×C167#12011yescharacteristic
89C22×C167C2×Q1611yescharacteristic
90C22×C167#13711yescharacteristic
91C4×C167—21no—
92C4×C167—21no—
93C4×C167—42no—
94C4×C167—41no—
95C4×C167—41no—
96Dic167—22no—
97Dic167—41no—
98Dic167—42no—
99Dic167—41no—
100Dic167—41no—
101C2×C167—21no—
102C2×C167—21no—
103C2×C167#45111yescharacteristic
104C2×C167#45611yescharacteristic
105C2×C167B(4,𝔽2)11yescharacteristic
106C167#160511yescharacteristic, complemented, Sylow
107#1605—1671nomaximal, Sylow
108#544—1671no—
109#451—1671no—
110#451—1671no—
111#395—1671no—
112#328—1671no—
113#328D16711yescharacteristic
114#324—1671no—
115C22×Q8—1671no—
116C2×Q16—3341no—
117C2×Q16—3341no—
118C4⋊Q8—3341no—
119C4⋊Q8—3341no—
120#125—3341no—
121#125—3341no—
122C22⋊Q8—3341no—
123C22⋊Q8—1671no—
124C22⋊Q8C2×D16711yescharacteristic
125#117—3341no—
126C2×C4⋊C4—3341no—
127C2×C4⋊C4—1671no—
128#103—3341no—
129#103—3341no—
130C22⋊C8—1671no—
131C22⋊C8C2×D16711yescharacteristic
132#95—3341no—
133#95C2×D16711yescharacteristic
134C2×Q8—3341no—
135C2×Q8—3341no—
136C2×Q8—3341no—
137C2×Q8—6682no—
138C2×Q8—1671no—
139C2×Q8D66811yescharacteristic
140C22×C4—21no—
141C22×C4—3341no—
142C22×C4—1671no—
143C22×C4—1671no—
144C22×C4C22×D16711yescharacteristic
145Q16—6682no—
146Q16—6682no—
147C2×C8—21no—
148C2×C8—3341no—
149C4⋊C4—21no—
150C4⋊C4—3341no—
151C4⋊C4—3341no—
152C4⋊C4—3341no—
153C4⋊C4—3341no—
154C4⋊C4—3341no—
155C4⋊C4—3341no—
156C4⋊C4—3341no—
157C4⋊C4—3341no—
158C4⋊C4—6682no—
159C4⋊C4—1671no—
160C4⋊C4D66811yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3341no—
163C42—3341no—
164C42—3341no—
165C23D4×D16711yescharacteristic
166Q8—41no—
167Q8—13361no—
168Q8—3342no—
169Q8—3342no—
170Q8—6681no—
171Q8—6681no—
172Q8—6681no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—3341no—
180C2×C4—3341no—
181C2×C4—3342no—
182C2×C4—3341no—
183C2×C4—3341no—
184C2×C4—3341no—
185C2×C4—6681no—
186C2×C4—6681no—
187C2×C4—6681no—
188C2×C4—1672no—
189C2×C4D4×D16711yescharacteristic
190C2×C4C2×D66811yescharacteristic
191C8—41no—
192C8—6681no—
193C22—21no—
194C22—21no—
195C22#1912880211yescharacteristic
196C22Q16×D16711yescharacteristic
197C22#1912886111yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—3342no—
204C4—6681no—
205C4—6682no—
206C4—6681no—
207C4—6681no—
208C2—21no—
209C2—21no—
210C2#2492342611yescharacteristic
211C2#2492345311yescharacteristic
212C2#2492345711yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#328×C167C2
socleC22×C167#120
prime radicalC23×C167C2×D4
hypercenter#328D167
hypocenterC167#1605
LCSG ▷ C22×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
73. C2×Dic167Q16semidirect
106. C167#1605semidirect

Maximal subgroups

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

subgroupindex
#249238982
#249234272
#249234242
#249231902
#249230052
#249229942
#328×C1672
#1605167