Finite Group Databasebeta

← #36334061

Subgroups — #36334061

Order 21376.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
2#24923604C212yesmaximal, complemented
3#24923599C212yesmaximal, complemented
4#24923587C212yesmaximal, complemented
5#24923582C212yesmaximal, complemented
6#24923309C212yesmaximal, complemented
7#24923268C212yesmaximal, complemented
8#24923233C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C167C211yesmaximal, characteristic, complemented
10C2×#14581950C2211yescharacteristic
11C2×#14581934C2212yes—
12#19128815—21no—
13#19128811C2211yescharacteristic
14#19128811C2211yescharacteristic
15#19128806C2211yescharacteristic
16#19128806C2211yescharacteristic
17#19128801—21no—
18C4×C22⋊D167—21no—
19C4×C22⋊D167—21no—
20#19128791C2211yescharacteristic
21#19128765C2214yescomplemented
22#19128763C2214yescomplemented
23#19128755C2214yescomplemented
24#19128752C2214yescomplemented
25C22⋊(C167⋊C8)C2212yes—
26C22⋊(C167⋊C8)C2212yes—
27C23×C4×C167—21no—
28C23×C4×C167C2211yescharacteristic
29C2×C4⋊C4×C167—21no—
30C2×C4⋊C4×C167—21no—
31C2×C4⋊C4×C167—21no—
32C2×C4⋊C4×C167C2212yes—
33C2×C4⋊C4×C167C2212yes—
34C2×C4⋊C4×C167C2212yes—
35#14581950—24no—
36C2×Dic334C2311yescharacteristic
37C2×D668C2311yescharacteristic
38C2×C22⋊D167—21no—
39C2×C22⋊D167—21no—
40C2×C22⋊D167—21no—
41C2×C4×D167—21no—
42#14581934—24no—
43C2×C167⋊C8C2314yes—
44C4⋊Dic167C2311yescharacteristic
45C4⋊Dic167C2311yescharacteristic
46Dic167⋊C4—21no—
47Dic167⋊C4—21no—
48C22⋊Dic167—21no—
49C22⋊Dic167—21no—
50C22⋊Dic167—21no—
51(C2×D167)⋊C4—21no—
52(C2×D167)⋊C4—21no—
53C4×Dic167—21no—
54C24×C167D411yescharacteristic
55C22×C4×C167—24no—
56C22×C4×C167—22no—
57C22×C4×C167—22no—
58C22×C4×C167—22no—
59C22×C4×C167—44no—
60C22×C4×C167D411yescharacteristic
61C22×C4×C167D411yescharacteristic
62C22×C4×C167D411yescharacteristic
63C22×C4×C167C2311yescharacteristic
64C22×C4×C167C2311yescharacteristic
65C22×C4×C167C2311yescharacteristic
66C4⋊C4×C167—22no—
67C4⋊C4×C167—22no—
68C4⋊C4×C167—22no—
69C4⋊C4×C167C2314yes—
70C22×D167—21no—
71Dic334—22no—
72D668—22no—
73C22⋊D167—42no—
74C22⋊D167—42no—
75C22⋊D167—42no—
76C2×Dic167—21no—
77C2×Dic167—21no—
78C2×Dic167—21no—
79C4×D167—42no—
80C167⋊C8—24no—
81C23×C167—22no—
82C23×C167—22no—
83C23×C167—22no—
84C23×C167C2×D411yescharacteristic
85C23×C167C2×D411yescharacteristic
86C23×C167C2×D411yescharacteristic
87C2×C4×C167—22no—
88C2×C4×C167—22no—
89C2×C4×C167—22no—
90C2×C4×C167—22no—
91C2×C4×C167—22no—
92C2×C4×C167—24no—
93C2×C4×C167—44no—
94C2×C4×C167—44no—
95C2×C4×C167—44no—
96C2×C4×C167C2×D411yescharacteristic
97C2×C4×C167C2×D411yescharacteristic
98C2×C4×C167C2×D411yescharacteristic
99C2×C4×C167#4114yes—
100C2×C4×C167C2411yescharacteristic
101C22×C167—28no—
102C22×C167—21no—
103C22×C167—21no—
104C22×C167—21no—
105C22×C167—21no—
106C22×C167—21no—
107C22×C167—21no—
108C22×C167#12312yes—
109C22×C167AGL(1,ℤ/8)12yes—
110C22×C167#13712yes—
111C22×C167C22×D411yescharacteristic
112C2×D167—41no—
113C2×D167—41no—
114C2×D167—41no—
115C4×C167—21no—
116C4×C167—21no—
117C4×C167—44no—
118C4×C167#12312yes—
119C4×C167C2×#4112yes—
120Dic167—41no—
121Dic167—41no—
122Dic167—41no—
123C2×C167—22no—
124C2×C167—22no—
125C2×C167—22no—
126C2×C167C2×#12311yescharacteristic
127C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C167C2×#13711yescharacteristic
129D167—81no—
130C167#309211yescharacteristic, complemented, Sylow
131#3092—1671nomaximal, Sylow
132#525—1671no—
133C22×C4⋊C4D16711yescharacteristic, complemented
134#484—1672no—
135#483—1672no—
136#481—1672no—
137#480—1672no—
138#428—1672no—
139#417—1672no—
140#407—1671no—
141C2×#41—1671no—
142C23×C4—21no—
143C23×C4C2×D16711yescharacteristic
144C2×M16—1672no—
145#123—3341no—
146C22⋊Q8—1671no—
147C22⋊Q8—1671no—
148#121—1671no—
149#121—1671no—
150#120—3341no—
151C4×D4—3341no—
152C4×D4—3341no—
153#117—1671no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D16712yes—
158C2×C4⋊C4C2×D16712yes—
159C2×C4⋊C4C2×D16712yes—
160#107—1674no—
161#106—1674no—
162#103—1674no—
163#102—1674no—
164C22⋊C8—1672no—
165C22⋊C8—1672no—
166C24C22⋊D16711yescharacteristic
167#41—3344no—
168C2×Q8—1671no—
169C2×D4—3341no—
170C2×D4—3341no—
171C2×D4—3341no—
172C2×D4—1671no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—3341no—
179C22×C4C22⋊D16711yescharacteristic
180C22×C4C22⋊D16711yescharacteristic
181C22×C4C22⋊D16711yescharacteristic
182C22×C4C22×D16711yescharacteristic
183C22×C4C22×D16711yescharacteristic
184C22×C4C22×D16711yescharacteristic
185M16—3344no—
186C2×C8—1674no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—3341no—
191C4⋊C4—3341no—
192C4⋊C4—1671no—
193C4⋊C4—1671no—
194C4⋊C4C22×D16714yes—
195C22⋊C4—3341no—
196C22⋊C4—3341no—
197C22⋊C4—3341no—
198C22⋊C4—3341no—
199C22⋊C4—3341no—
200C42—3341no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—3341no—
205C23C2×C22⋊D16711yescharacteristic
206C23C2×C22⋊D16711yescharacteristic
207C23C2×C22⋊D16711yescharacteristic
208Q8—3342no—
209D4—3342no—
210D4—6682no—
211D4—6682no—
212D4—6682no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—3341no—
223C2×C4—3341no—
224C2×C4—3341no—
225C2×C4—6682no—
226C2×C4C2×C22⋊D16711yescharacteristic
227C2×C4C2×C22⋊D16711yescharacteristic
228C2×C4C2×C22⋊D16711yescharacteristic
229C2×C4#1458195014yes—
230C2×C4C23×D16711yescharacteristic
231C8—3344no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—6681no—
240C22—6681no—
241C22—6681no—
242C22#1912881512yes—
243C22#1912885812yes—
244C22#1912886312yes—
245C22C22×C22⋊D16711yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—6681no—
250C4—6681no—
251C4—6681no—
252C4#1912881512yes—
253C4C2×#1458195012yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—13361no—
258C2C2×#1912881511yescharacteristic
259C2C2×#1912885811yescharacteristic
260C2C2×#1912886311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC22×C4⋊C4×C167C2
socleC22×C167C22×D4
prime radical#24923780C2
hypercenterC22×C4⋊C4D167
hypocenterC167#3092
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #24923604C2semidirect
3. #24923599C2semidirect
4. #24923587C2semidirect
5. #24923582C2semidirect
6. #24923309C2semidirect
7. #24923268C2semidirect
8. #24923233C2semidirect
9. C22×C4⋊C4×C167C2semidirect
21. #19128765C22semidirect
22. #19128763C22semidirect
23. #19128755C22semidirect
24. #19128752C22semidirect
130. C167#3092semidirect
133. C22×C4⋊C4D167semidirect

Maximal subgroups

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

subgroupindex
#249237802
#249236042
#249235992
#249235872
#249235822
#249233092
#249232682
#249232332
C22×C4⋊C4×C1672
#3092167