Finite Group Databasebeta

← #36334587

Subgroups — #36334587

Order 21376.

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

normal subgroups only

This group has 656 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
1C2×#19128859C212yesmaximal, complemented
2C2×#19128856C212yesmaximal, complemented
3#24923818C211yesmaximal, characteristic, complemented
4#127×D167C211yesmaximal, characteristic, complemented
5#24923659C211yesmaximal, characteristic, complemented
6#24923658C211yesmaximal, characteristic, complemented
7#24923656C211yesmaximal, characteristic, complemented
8#24923644C212yesmaximal, complemented
9#24923640C212yesmaximal, complemented
10#24923221C211yesmaximal, characteristic, complemented
11#496×C167C211yesmaximal, characteristic, complemented
12C2×#14581951C2211yescharacteristic
13C2×#14581949C2211yescharacteristic
14C2×D4×D167—22no—
15C2×D4×D167C2212yes—
16#19128859—28no—
17#19128856—28no—
18C2×#14581939C2212yescomplemented
19C2×#14581938C2212yescomplemented
20C2×#14581937C2212yescomplemented
21C2×D1336C2212yescomplemented
22C2×#14581935C2212yescomplemented
23C2×#14581935C2212yescomplemented
24C2×#14581933C2212yes—
25#19128830C2211yescharacteristic, complemented
26#19128829C2211yescharacteristic
27#19128828—22no—
28#19128828C2212yes—
29#19128828C2211yescharacteristic, complemented
30#19128820C2211yescharacteristic, complemented
31#19128805—22no—
32(C22⋊D167)⋊C4—22no—
33C42×D167C2211yescharacteristic
34#19128743C2211yescharacteristic, complemented
35C8×Dic167C2212yescomplemented
36C2×SD16×C167C2212yescomplemented
37C2×D8×C167C2212yescomplemented
38#127×C167C2211yescharacteristic
39#124×C167C2211yescharacteristic, complemented
40#97×C167C2211yescharacteristic, complemented
41C23×D167—22no—
42#14581951—24no—
43#14581949—24no—
44C2×Dic334C2311yescharacteristic
45C2×D668—21no—
46C2×D668—22no—
47C2×D668C2311yescharacteristic
48C2×D668C2312yes—
49C2×C22⋊D167—21no—
50C2×C22⋊D167—24no—
51C2×C22⋊D167—22no—
52C2×C22⋊D167—22no—
53D4×D167—24no—
54D4×D167—24no—
55D4×D167—48no—
56C22×Dic167—21no—
57C2×C4×D167—21no—
58C2×C4×D167D412yes—
59C2×C4×D167C2311yescharacteristic
60#14581939—24no—
61#14581938—24no—
62#14581937—24no—
63D1336—24no—
64#14581935—24no—
65#14581935—24no—
66#14581933D418yescomplemented
67C2×C167⋊C8C2312yes—
68Dic167⋊C4—22no—
69(C2×D167)⋊C4—22no—
70C4×Dic167D412yes—
71C4×Dic167C2311yescharacteristic
72C2×Q8×C167C2311yescharacteristic
73C2×D4×C167—22no—
74C2×D4×C167C2311yescharacteristic
75C2×D4×C167C2312yes—
76SD16×C167—24no—
77D8×C167—24no—
78C2×C8×C167C2312yes—
79C22⋊C4×C167—22no—
80C42×C167C2311yescharacteristic
81C22×D167—21no—
82C22×D167—22no—
83C22×D167—48no—
84C22×D167—44no—
85C22×D167C2×D411yescharacteristic
86Dic334—22no—
87D668—22no—
88D668—22no—
89D668—22no—
90D668—44no—
91D668—42no—
92C22⋊D167—42no—
93C22⋊D167—44no—
94C22⋊D167—48no—
95C22⋊D167—44no—
96C2×Dic167—21no—
97C2×Dic167—42no—
98C2×Dic167C2×D411yescharacteristic
99C2×Dic167C2×D412yes—
100C4×D167—24no—
101C4×D167—42no—
102C4×D167C2×D414yes—
103C167⋊C8C2×D414yescomplemented
104C23×C167—21no—
105C23×C167—22no—
106Q8×C167—22no—
107D4×C167—22no—
108D4×C167—22no—
109D4×C167—22no—
110D4×C167—44no—
111C2×C4×C167—21no—
112C2×C4×C167C2×D412yes—
113C2×C4×C167C2411yescharacteristic
114C8×C167C2×D414yescomplemented
115C22×C167—41no—
116C22×C167—42no—
117C22×C167—44no—
118C22×C167—42no—
119C22×C167C22×D411yescharacteristic
120C2×D167—22no—
121C2×D167—41no—
122C2×D167—42no—
123C2×D167—44no—
124C2×D167—42no—
125C2×D167—88no—
126C2×D167#12712yes—
127C4×C167—22no—
128C4×C167—41no—
129C4×C167C22×D412yes—
130Dic167—41no—
131Dic167#12712yes—
132Dic167AGL(1,ℤ/8)14yescomplemented
133C2×C167—82no—
134C2×C167—81no—
135C2×C167C2×#12711yescharacteristic
136C2×C167C2×AGL(1,ℤ/8)12yes—
137D167—22no—
138D167—81no—
139D167—82no—
140C167#315111yescharacteristic, complemented, Sylow
141#3151—1671nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1672no—
143C2×AGL(1,ℤ/8)—1672no—
144#532—1671no—
145C2×#127—1671no—
146#496—1671no—
147#496—1671no—
148#496—1671no—
149#496D16711yescharacteristic, complemented
150#493—1672no—
151#492—1672no—
152C4×M16—1671no—
153C2×#41—1671no—
154C2×#41—1671no—
155C22×D4—3342no—
156C22×D4—1672no—
157AGL(1,ℤ/8)—3348no—
158AGL(1,ℤ/8)—3348no—
159C2×SD16—1672no—
160C2×SD16—1672no—
161C2×SD16—1672no—
162C2×SD16C2×D16712yescomplemented
163C2×D8—1672no—
164C2×D8—1672no—
165C2×D8—1672no—
166C2×D8C2×D16712yescomplemented
167C2×M16—1672no—
168C4⋊Q8—1671no—
169#127—3342no—
170#127—1672no—
171#127—1671no—
172#127—1671no—
173#127C2×D16711yescharacteristic
174#124—1671no—
175#124C2×D16711yescharacteristic, complemented
176#121—3342no—
177C4×D4—3342no—
178C2×C42—1671no—
179#97—1671no—
180#97C2×D16711yescharacteristic, complemented
181C4×C8—1672no—
182C24—3342no—
183#41—3344no—
184#41—3344no—
185C2×Q8—1671no—
186C2×Q8C22×D16711yescharacteristic
187C2×D4—22no—
188C2×D4—3341no—
189C2×D4—3341no—
190C2×D4—3344no—
191C2×D4—3344no—
192C2×D4—3342no—
193C2×D4—3342no—
194C2×D4—3344no—
195C2×D4—3342no—
196C2×D4—6688no—
197C2×D4—1671no—
198C2×D4—1672no—
199C2×D4C22×D16711yescharacteristic
200C2×D4C22×D16712yes—
201C22×C4—3341no—
202C22×C4—3341no—
203C22×C4—1672no—
204C22×C4—1671no—
205SD16—24no—
206SD16—3344no—
207SD16—3344no—
208SD16—3344no—
209D8—24no—
210D8—3344no—
211D8—3344no—
212D8—3344no—
213M16—1678no—
214C2×C8—1672no—
215C2×C8C22×D16712yes—
216C4⋊C4—3342no—
217C22⋊C4—22no—
218C22⋊C4—3342no—
219C42—1672no—
220C42—1671no—
221C42C22×D16711yescharacteristic
222C23—21no—
223C23—22no—
224C23—3341no—
225C23—3342no—
226C23—6688no—
227C23—6684no—
228C23—1671no—
229Q8—22no—
230Q8—3342no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3342no—
236D4—3342no—
237D4—3342no—
238D4—6684no—
239D4—6682no—
240D4—6682no—
241D4—6684no—
242D4—6688no—
243D4—6684no—
244C2×C4—21no—
245C2×C4—3344no—
246C2×C4—3341no—
247C2×C4—6682no—
248C2×C4—6682no—
249C2×C4—1672no—
250C2×C4—1671no—
251C2×C4—1674no—
252C2×C4D4×D16712yes—
253C2×C4C23×D16711yescharacteristic
254C8—1674no—
255C8D4×D16714yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—13368no—
261C22—3342no—
262C22—6684no—
263C22—6682no—
264C22—6681no—
265C22—6682no—
266C22—1672no—
267C22C2×D4×D16711yescharacteristic
268C4—22no—
269C4—41no—
270C4—6681no—
271C4—1674no—
272C4—1672no—
273C4C2×D4×D16712yes—
274C2—82no—
275C2—81no—
276C2—13362no—
277C2—13361no—
278C2—3342no—
279C2#127×D16711yescharacteristic
280C2AGL(1,ℤ/8)×D16712yes—
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#496×C167C2
socleC22×C167C22×D4
hypercenter#496D167
hypocenterC167#3151
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19128859C2semidirect
2. C2×#19128856C2semidirect
3. #24923818C2semidirect
4. #127×D167C2semidirect
5. #24923659C2semidirect
6. #24923658C2semidirect
7. #24923656C2semidirect
8. #24923644C2semidirect
9. #24923640C2semidirect
10. #24923221C2semidirect
11. #496×C167C2semidirect
18. C2×#14581939C22semidirect
19. C2×#14581938C22semidirect
20. C2×#14581937C22semidirect
21. C2×D1336C22semidirect
22. C2×#14581935C22semidirect
23. C2×#14581935C22semidirect
25. #19128830C22semidirect
29. #19128828C22semidirect
30. #19128820C22semidirect
34. #19128743C22semidirect
35. C8×Dic167C22semidirect
36. C2×SD16×C167C22semidirect
37. C2×D8×C167C22semidirect
39. #124×C167C22semidirect
40. #97×C167C22semidirect
66. #14581933D4semidirect
103. C167⋊C8C2×D4semidirect
114. C8×C167C2×D4semidirect
132. Dic167AGL(1,ℤ/8)semidirect
140. C167#3151semidirect
149. #496D167semidirect
162. C2×SD16C2×D167semidirect
166. C2×D8C2×D167semidirect
175. #124C2×D167semidirect
180. #97C2×D167semidirect
255. C8D4×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288592
C2×#191288562
#249238182
#127×D1672
#249236592
#249236582
#249236562
#249236442
#249236402
#249232212
#496×C1672
#3151167