Finite Group Databasebeta

← #36327403

Subgroups — #36327403

Order 21376.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×(C22⋊D167)⋊C4C211yesmaximal, characteristic
2C2×#19128790C211yesmaximal, characteristic, complemented
3#24923207C211yesmaximal, characteristic, complemented
4#24923201C211yesmaximal, characteristic, complemented
5#24923195C212yesmaximal, complemented
6#24923194C211yesmaximal, characteristic, complemented
7#24923185C212yesmaximal
8C4⋊C4×Dic167C212yesmaximal, complemented
9C22⋊C4×Dic167C212yesmaximal, complemented
10#24923116C211yesmaximal, characteristic, complemented
11#399×C167C211yesmaximal, characteristic, complemented
12C22×C22⋊D167C2211yescharacteristic
13C23×Dic167C2211yescharacteristic
14C22×C4×D167C2211yescharacteristic
15(C22⋊D167)⋊C4—24no—
16#19128790—24no—
17C2×C4⋊Dic167C2212yes—
18C2×C4⋊Dic167C2211yescharacteristic
19C2×Dic167⋊C4C2211yescharacteristic
20C2×C22⋊Dic167C2212yes—
21C2×(C2×D167)⋊C4C2212yes—
22C2×(C2×D167)⋊C4C2212yes—
23C2×(C2×D167)⋊C4C2211yescharacteristic
24C2×C4×Dic167C2212yes—
25C2×C4×Dic167C2211yescharacteristic
26C2×C4×Dic167C2212yes—
27C167⋊#95C2212yes—
28C167⋊#95C2212yes—
29C167⋊#95C2211yescharacteristic
30C167⋊#95C2212yes—
31C167⋊#95C2212yes—
32C2×C4⋊C4×C167C2212yes—
33C2×C4⋊C4×C167C2211yescharacteristic
34C2×C22⋊C4×C167C2212yes—
35C2×C22⋊C4×C167C2211yescharacteristic
36#95×C167C2211yescharacteristic
37C23×D167C2311yescharacteristic
38C2×C22⋊D167—22no—
39C2×C22⋊D167—22no—
40C2×C22⋊D167—22no—
41C22×Dic167—22no—
42C22×Dic167—24no—
43C22×Dic167C2311yescharacteristic
44C22×Dic167C2312yes—
45C22×Dic167C2312yes—
46C22×Dic167C2311yescharacteristic
47C22×Dic167C2311yescharacteristic
48C2×C4×D167—24no—
49C2×C4×D167D414yes—
50C4⋊Dic167—24no—
51C4⋊Dic167—22no—
52Dic167⋊C4—22no—
53C22⋊Dic167—24no—
54(C2×D167)⋊C4—24no—
55(C2×D167)⋊C4—22no—
56(C2×D167)⋊C4—24no—
57C4×Dic167—24no—
58C4×Dic167—22no—
59C4×Dic167—24no—
60C24×C167C2311yescharacteristic
61C22×C4×C167C2311yescharacteristic
62C22×C4×C167C2312yes—
63C22×C4×C167C2311yescharacteristic
64C22×C4×C167C2312yes—
65C4⋊C4×C167—22no—
66C4⋊C4×C167—24no—
67C22⋊C4×C167—22no—
68C22⋊C4×C167—24no—
69C22×D167—22no—
70C22×D167—22no—
71C22×D167—21no—
72C22×D167—21no—
73C22×D167C2×D412yes—
74C22⋊D167—28no—
75C2×Dic167—21no—
76C2×Dic167—21no—
77C2×Dic167—22no—
78C2×Dic167—21no—
79C2×Dic167—22no—
80C2×Dic167—22no—
81C2×Dic167—22no—
82C2×Dic167—22no—
83C2×Dic167—22no—
84C2×Dic167—22no—
85C2×Dic167—44no—
86C2×Dic167C2×D412yes—
87C2×Dic167#4112yes—
88C2×Dic167#4114yes—
89C4×D167—28no—
90C23×C167—21no—
91C23×C167—22no—
92C23×C167—22no—
93C23×C167—21no—
94C23×C167#4112yes—
95C23×C167C2411yescharacteristic
96C2×C4×C167—22no—
97C2×C4×C167—22no—
98C2×C4×C167—22no—
99C2×C4×C167—21no—
100C2×C4×C167—22no—
101C2×C4×C167—22no—
102C2×C4×C167—22no—
103C2×C4×C167—22no—
104C2×C4×C167C2×D412yes—
105C2×C4×C167#4114yes—
106C22×C167—22no—
107C22×C167—24no—
108C22×C167—41no—
109C22×C167—42no—
110C22×C167—41no—
111C22×C167C22×D411yescharacteristic
112C22×C167C2×#4112yes—
113C22×C167C2×#4111yescharacteristic
114C22×C167C2×#4112yes—
115C22×C167C2×#4111yescharacteristic
116C2×D167—22no—
117C2×D167—22no—
118C2×D167—22no—
119C2×D167—42no—
120C2×D167—42no—
121C4×C167—22no—
122C4×C167—24no—
123C4×C167—41no—
124C4×C167—42no—
125Dic167—22no—
126Dic167—41no—
127Dic167—42no—
128Dic167—42no—
129Dic167#12314yes—
130C2×C167—42no—
131C2×C167C2×#12311yescharacteristic
132C2×C167#52511yescharacteristic
133C2×C167#52912yes—
134C2×C167#52912yes—
135C2×C167#53211yescharacteristic
136D167—42no—
137C167#239511yescharacteristic, complemented, Sylow
138#2395—1671nomaximal, Sylow
139C2×C4×D4—1671no—
140C2×#117—1671no—
141#400—1671no—
142#399D16711yescharacteristic, complemented
143#398—1671no—
144#397—1672no—
145#397—1671no—
146#394—1672no—
147C4×C4⋊C4—1672no—
148C4×C22⋊C4—1672no—
149C4×C22⋊C4—1671no—
150C22×D4—1671no—
151C23×C4—1671no—
152C23×C4—1671no—
153C4×D4—3344no—
154#117—3344no—
155C2×C4⋊C4—1671no—
156C2×C4⋊C4—1671no—
157C2×C4⋊C4—1672no—
158C2×C4⋊C4C2×D16711yescharacteristic
159C2×C4⋊C4C2×D16712yes—
160C2×C22⋊C4—1672no—
161C2×C22⋊C4—1672no—
162C2×C22⋊C4—1671no—
163C2×C22⋊C4—1672no—
164C2×C22⋊C4C2×D16711yescharacteristic
165C2×C22⋊C4C2×D16712yes—
166C2×C42—1672no—
167C2×C42—1671no—
168C2×C42—1672no—
169#95—1672no—
170#95—1671no—
171#95—1672no—
172#95—1672no—
173#95—1672no—
174#95C2×D16711yescharacteristic
175C24—1671no—
176C24C22×D16711yescharacteristic
177C2×D4—3342no—
178C2×D4—3342no—
179C2×D4—3342no—
180C22×C4—3344no—
181C22×C4—3342no—
182C22×C4—3344no—
183C22×C4—1672no—
184C22×C4—1671no—
185C22×C4—1672no—
186C22×C4—1671no—
187C22×C4—1674no—
188C22×C4—1671no—
189C22×C4C22×D16711yescharacteristic
190C22×C4C22×D16712yes—
191C22×C4C22×D16711yescharacteristic
192C22×C4C22×D16712yes—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—3344no—
196C4⋊C4—3342no—
197C4⋊C4—3342no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—3344no—
201C22⋊C4—3342no—
202C22⋊C4—3344no—
203C22⋊C4—3344no—
204C42—3344no—
205C42—3342no—
206C42—3344no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—3342no—
212C23—3341no—
213C23—3342no—
214C23—3341no—
215C23—1672no—
216C23#1458195112yes—
217C23C23×D16711yescharacteristic
218D4—3348no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—3348no—
228C2×C4—3341no—
229C2×C4—3341no—
230C2×C4—3342no—
231C2×C4—3341no—
232C2×C4—3342no—
233C2×C4—3342no—
234C2×C4—3342no—
235C2×C4—3342no—
236C2×C4—3342no—
237C2×C4—3342no—
238C2×C4—6684no—
239C2×C4—1672no—
240C2×C4—1672no—
241C2×C4—1674no—
242C2×C4D4×D16712yes—
243C2×C4#1458194914yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—3342no—
250C22—3342no—
251C22—3342no—
252C22—6682no—
253C22—6682no—
254C22C2×D4×D16711yescharacteristic
255C22#41×D16712yes—
256C22#41×D16711yescharacteristic
257C22C2×#1458194912yes—
258C22C2×#1458195111yescharacteristic
259C4—22no—
260C4—24no—
261C4—41no—
262C4—42no—
263C4—3342no—
264C4—6681no—
265C4—6682no—
266C4—6682no—
267C4—1674no—
268C2—42no—
269C2—6682no—
270C2#123×D16711yescharacteristic
271C2#2492378211yescharacteristic
272C2#2492379612yes—
273C2#2492380512yes—
274C2#2492381711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D167
commutator, socle, cosocleC23×C167C24
Fitting#399×C167C2
prime radicalC22×C22⋊D167C22
hypercenter#399D167
hypocenterC167#2395
LCSG ▷ C23×C167 ▷ C167
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#19128790C2semidirect
3. #24923207C2semidirect
4. #24923201C2semidirect
5. #24923195C2semidirect
6. #24923194C2semidirect
8. C4⋊C4×Dic167C2semidirect
9. C22⋊C4×Dic167C2semidirect
10. #24923116C2semidirect
11. #399×C167C2semidirect
137. C167#2395semidirect
142. #399D167semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D167)⋊C42
C2×#191287902
#249232072
#249232012
#249231952
#249231942
#249231852
C4⋊C4×Dic1672
C22⋊C4×Dic1672
#249231162
#399×C1672
#2395167