Finite Group Databasebeta

← #36334618

Subgroups — #36334618

Order 21376.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19128856C212yesmaximal, complemented
2C2×#19128851C211yesmaximal, characteristic, complemented
3C2×D8×D167C211yesmaximal, characteristic, complemented
4#24923832C212yesmaximal, complemented
5#24923657C212yesmaximal, complemented
6#24923656C212yesmaximal, complemented
7#24923653C211yesmaximal, characteristic, complemented
8#24923652C211yesmaximal, characteristic, complemented
9#24923644C211yesmaximal, characteristic, complemented
10#24923223C211yesmaximal, characteristic, complemented
11#493×C167C211yesmaximal, characteristic, complemented
12C2×#14581949C2212yes—
13C2×D4×D167C2212yes—
14#19128856—28no—
15#19128851—24no—
16C2×Dic668C2211yescharacteristic, complemented
17C2×#14581939C2212yescomplemented
18C2×#14581938C2212yescomplemented
19C2×#14581938C2212yescomplemented
20C2×D1336C2211yescharacteristic
21C2×#14581935C2212yes—
22C2×#14581935C2212yescomplemented
23D8×D167—24no—
24C2×#14581933C2211yescharacteristic
25C2×C8×D167C2211yescharacteristic
26#19128828C2212yescomplemented
27#19128820C2212yescomplemented
28#19128819C2212yescomplemented
29#19128804—24no—
30#19128800—24no—
31#19128789C2211yescharacteristic
32#19128743C2211yescharacteristic, complemented
33#19128742C2211yescharacteristic, complemented
34C8×Dic167C2211yescharacteristic, complemented
35C2×D8×C167C2211yescharacteristic
36C2×D8×C167C2212yescomplemented
37C2×D8×C167C2211yescharacteristic, complemented
38#127×C167C2212yescomplemented
39C4×C8×C167C2211yescharacteristic, complemented
40C23×D167—22no—
41#14581949—24no—
42#14581949—24no—
43C2×Dic334C2312yes—
44C2×D668C2312yes—
45C2×C22⋊D167—22no—
46C2×C22⋊D167—22no—
47C2×C22⋊D167—24no—
48D4×D167—24no—
49D4×D167—24no—
50C22×Dic167—22no—
51C2×C4×D167C2311yescharacteristic
52Dic668—22no—
53#14581939—24no—
54#14581938—24no—
55#14581938—24no—
56D1336—22no—
57#14581935—24no—
58#14581935—24no—
59#14581933D414yescomplemented
60C2×C167⋊C8C2311yescharacteristic
61C2×C167⋊C8C2311yescharacteristic
62C8×D167D414yescomplemented
63Dic167⋊C4D412yes—
64C22⋊Dic167—24no—
65(C2×D167)⋊C4—24no—
66(C2×D167)⋊C4D412yes—
67C4×Dic167C2311yescharacteristic
68C2×D4×C167—24no—
69C2×D4×C167C2312yes—
70C2×D4×C167C2312yes—
71D8×C167—22no—
72D8×C167—22no—
73D8×C167—24no—
74C2×C8×C167C2311yescharacteristic
75C2×C8×C167C2311yescharacteristic
76C42×C167C2311yescharacteristic
77C22×D167—22no—
78C22×D167—44no—
79C22×D167—84no—
80C22×D167C2×D411yescharacteristic
81Dic334—22no—
82Dic334—22no—
83D668—22no—
84D668—22no—
85C22⋊D167—44no—
86C22⋊D167—44no—
87C22⋊D167—84no—
88C2×Dic167—22no—
89C2×Dic167—44no—
90C2×Dic167C2×D412yes—
91C2×Dic167C2×D411yescharacteristic
92C4×D167C2×D412yes—
93C4×D167C2×D412yes—
94C167⋊C8C2×D412yes—
95C167⋊C8C2×D412yes—
96C23×C167—22no—
97C23×C167—22no—
98D4×C167—22no—
99D4×C167—22no—
100D4×C167—22no—
101D4×C167—22no—
102D4×C167—84no—
103C2×C4×C167C2×D412yes—
104C2×C4×C167C2411yescharacteristic
105C8×C167C2×D412yes—
106C8×C167C2×D412yes—
107C22×C167—42no—
108C22×C167—42no—
109C22×C167—44no—
110C22×C167—44no—
111C22×C167C22×D411yescharacteristic
112C2×D167—22no—
113C2×D167—42no—
114C2×D167—44no—
115C2×D167—88no—
116C2×D167#12712yes—
117C4×C167—22no—
118C4×C167C22×D411yescharacteristic
119C4×C167C22×D411yescharacteristic
120Dic167—22no—
121Dic167—42no—
122Dic167#12712yes—
123C2×C167—82no—
124C2×C167—82no—
125C2×C167C2×#12711yescharacteristic
126C2×C167#57612yes—
127D167—22no—
128D167—82no—
129C167#315511yescharacteristic, complemented, Sylow
130#3155—1671nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1672no—
132C2×#135—1671no—
133C22×D8—1671no—
134#535—1672no—
135#496—1672no—
136#496—1672no—
137#495—1671no—
138#495—1671no—
139#493—1671no—
140#493D16711yescharacteristic, complemented
141#405—1671no—
142C2×#41—1672no—
143C22×D4—1672no—
144AGL(1,ℤ/8)—3348no—
145#135—3344no—
146C2×Q16—1671no—
147C2×SD16—1672no—
148C2×SD16—1672no—
149C2×SD16—1672no—
150C2×D8—3344no—
151C2×D8—1672no—
152C2×D8—1672no—
153C2×D8—1671no—
154C2×D8C2×D16711yescharacteristic
155C2×D8C2×D16711yescharacteristic, complemented
156C2×D8C2×D16712yescomplemented
157C2×M16—1671no—
158C22×C8—1671no—
159#127—1672no—
160#127C2×D16712yescomplemented
161#124—1672no—
162#124—1672no—
163#121—3344no—
164#120—3344no—
165#117—1671no—
166#97—1671no—
167#97—1671no—
168C4×C8—1671no—
169C4×C8C2×D16711yescharacteristic, complemented
170C24—3342no—
171#41—3344no—
172#41—3344no—
173C2×Q8—1672no—
174C2×D4—24no—
175C2×D4—3342no—
176C2×D4—3344no—
177C2×D4—3344no—
178C2×D4—3344no—
179C2×D4—3342no—
180C2×D4—1672no—
181C2×D4C22×D16712yes—
182C2×D4C22×D16712yes—
183C22×C4—3342no—
184C22×C4—1671no—
185Q16—3342no—
186SD16—3344no—
187SD16—3344no—
188SD16—3344no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—3344no—
193D8—3344no—
194D8—3342no—
195M16—1674no—
196C2×C8—1671no—
197C2×C8—1671no—
198C2×C8—1674no—
199C2×C8C22×D16711yescharacteristic
200C2×C8C22×D16711yescharacteristic
201C4⋊C4—1672no—
202C22⋊C4—3344no—
203C22⋊C4—3344no—
204C22⋊C4—1672no—
205C42—1671no—
206C42C22×D16711yescharacteristic
207C23—22no—
208C23—22no—
209C23—13364no—
210C23—3342no—
211C23—6684no—
212C23—1671no—
213Q8—3342no—
214Q8—3342no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—13364no—
221D4—3342no—
222D4—3342no—
223D4—6684no—
224D4—6684no—
225C2×C4—3342no—
226C2×C4—6684no—
227C2×C4—1672no—
228C2×C4—1671no—
229C2×C4—1672no—
230C2×C4—1672no—
231C2×C4D4×D16712yes—
232C2×C4C23×D16711yescharacteristic
233C8—1672no—
234C8—1672no—
235C8D4×D16712yes—
236C8D4×D16712yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—13368no—
242C22—3342no—
243C22—6682no—
244C22—6684no—
245C22—1672no—
246C22C2×D4×D16711yescharacteristic
247C4—22no—
248C4—3342no—
249C4—6682no—
250C4—1672no—
251C4C2×D4×D16711yescharacteristic
252C4C2×D4×D16711yescharacteristic
253C2—82no—
254C2—82no—
255C2—13362no—
256C2—3342no—
257C2#127×D16711yescharacteristic
258C2#2492407012yes—
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#493×C167C2
socleC22×C167C22×D4
hypercenter#493D167
hypocenterC167#3155
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#19128856C2semidirect
2. C2×#19128851C2semidirect
3. C2×D8×D167C2semidirect
4. #24923832C2semidirect
5. #24923657C2semidirect
6. #24923656C2semidirect
7. #24923653C2semidirect
8. #24923652C2semidirect
9. #24923644C2semidirect
10. #24923223C2semidirect
11. #493×C167C2semidirect
16. C2×Dic668C22semidirect
17. C2×#14581939C22semidirect
18. C2×#14581938C22semidirect
19. C2×#14581938C22semidirect
22. C2×#14581935C22semidirect
26. #19128828C22semidirect
27. #19128820C22semidirect
28. #19128819C22semidirect
32. #19128743C22semidirect
33. #19128742C22semidirect
34. C8×Dic167C22semidirect
36. C2×D8×C167C22semidirect
37. C2×D8×C167C22semidirect
38. #127×C167C22semidirect
39. C4×C8×C167C22semidirect
59. #14581933D4semidirect
62. C8×D167D4semidirect
129. C167#3155semidirect
140. #493D167semidirect
155. C2×D8C2×D167semidirect
156. C2×D8C2×D167semidirect
160. #127C2×D167semidirect
169. C4×C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288562
C2×#191288512
C2×D8×D1672
#249238322
#249236572
#249236562
#249236532
#249236522
#249236442
#249232232
#493×C1672
#3155167