Finite Group Databasebeta

← #36335820

Subgroups — #36335820

Order 21376.

526 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 125 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×#19128855C211yesmaximal, characteristic, complemented
2C22×D1336C211yesmaximal, characteristic, complemented
3#24923910C212yesmaximal, complemented
4#24923654C211yesmaximal, characteristic, complemented
5#24923506C211yesmaximal, characteristic, complemented
6#24923487C212yesmaximal, complemented
7#24923419C212yesmaximal, complemented
8#24923417C212yesmaximal, complemented
9#24923351C211yesmaximal, characteristic, complemented
10C8×C22⋊D167C211yesmaximal, characteristic, complemented
11#467×C167C211yesmaximal, characteristic, complemented
12C2×#14581951C2212yescomplemented
13C22×D668C2212yes—
14#19128855—24no—
15C2×#14581937C2212yescomplemented
16C2×#14581937C2212yescomplemented
17C2×D1336—24no—
18C2×D1336C2211yescharacteristic
19C2×D1336C2212yes—
20C2×D1336C2211yescharacteristic
21C2×C8×D167C2211yescharacteristic, complemented
22#19128821C2212yescomplemented
23#19128817—22no—
24#19128807—22no—
25#19128807C2212yescomplemented
26#19128802—22no—
27#19128794C2212yescomplemented
28C4×C22⋊D167C2211yescharacteristic
29C22⋊C4×D167—22no—
30Dic167⋊C8C2211yescharacteristic, complemented
31#19128752C2212yescomplemented
32#19128752C2212yescomplemented
33C22⋊(C167⋊C8)C2211yescharacteristic
34(C2×D167)⋊C8C2211yescharacteristic, complemented
35C8×Dic167C2211yescharacteristic, complemented
36C2×Q16×C167C2211yescharacteristic, complemented
37C22×C8×C167C2211yescharacteristic
38C22⋊Q8×C167C2212yes—
39#107×C167C2211yescharacteristic, complemented
40#103×C167C2212yescomplemented
41C23×D167—22no—
42#14581951—24no—
43#14581951—42no—
44C2×D668—24no—
45C2×D668—22no—
46C2×D668—22no—
47C2×D668—22no—
48C2×D668—42no—
49C2×D668C2312yes—
50C2×D668C2312yes—
51C2×C22⋊D167D411yescharacteristic
52C2×C4×D167—22no—
53C2×C4×D167—22no—
54C2×C4×D167C2311yescharacteristic
55#14581937—24no—
56#14581937—42no—
57D1336—24no—
58D1336—22no—
59D1336D414yescomplemented
60C2×C167⋊C8C2311yescharacteristic
61C2×C167⋊C8C2311yescharacteristic
62C8×D167—22no—
63Dic167⋊C4D411yescharacteristic
64C22⋊Dic167D411yescharacteristic
65(C2×D167)⋊C4—22no—
66(C2×D167)⋊C4—22no—
67(C2×D167)⋊C4—22no—
68(C2×D167)⋊C4—22no—
69(C2×D167)⋊C4D411yescharacteristic
70C4×Dic167C2311yescharacteristic
71C2×Q8×C167C2312yes—
72C22×C4×C167C2311yescharacteristic
73Q16×C167—22no—
74C2×C8×C167—22no—
75C2×C8×C167C2311yescharacteristic
76C2×C8×C167C2311yescharacteristic
77C4⋊C4×C167—22no—
78C4⋊C4×C167C2312yes—
79C22⋊C4×C167—22no—
80C22×D167—22no—
81C22×D167—22no—
82C22×D167—44no—
83C22×D167—44no—
84C22×D167—82no—
85C22×D167C2×D411yescharacteristic
86D668—22no—
87D668—22no—
88D668—22no—
89D668—44no—
90D668—44no—
91D668—82no—
92D668C2×D414yes—
93C22⋊D167—22no—
94C2×Dic167C2×D411yescharacteristic
95C2×Dic167C2×D411yescharacteristic
96C2×Dic167C2×D411yescharacteristic
97C4×D167—21no—
98C4×D167—21no—
99C4×D167—44no—
100C4×D167—82no—
101C167⋊C8—21no—
102C167⋊C8—21no—
103C23×C167C2×D411yescharacteristic
104Q8×C167—22no—
105Q8×C167—22no—
106C2×C4×C167—21no—
107C2×C4×C167—21no—
108C2×C4×C167—22no—
109C2×C4×C167—22no—
110C2×C4×C167C2×D411yescharacteristic
111C2×C4×C167C2411yescharacteristic
112C8×C167—21no—
113C8×C167C2×D412yes—
114C22×C167—21no—
115C22×C167—21no—
116C22×C167#13512yescomplemented
117C22×C167C22×D411yescharacteristic
118C2×D167—21no—
119C2×D167—24no—
120C2×D167—21no—
121C2×D167—21no—
122C2×D167—42no—
123C2×D167—42no—
124C2×D167—42no—
125C2×D167—42no—
126C2×D167—42no—
127C2×D167—88no—
128C4×C167—21no—
129C4×C167—42no—
130C4×C167—42no—
131C4×C167C22×D411yescharacteristic
132C4×C167H(5,𝔽2)11yescharacteristic
133Dic167—21no—
134Dic167—21no—
135Dic167—21no—
136C2×C167—22no—
137C2×C167D4211yescharacteristic
138C2×C167C2×#13511yescharacteristic
139C2×C167#57611yescharacteristic
140D167—41no—
141D167—44no—
142D167—82no—
143C167#328011yescharacteristic, complemented, Sylow
144#3280—1671nomaximal, Sylow
145C2×#135—1671no—
146C22×D8—1671no—
147#546—1672no—
148#495—1671no—
149#467D16711yescharacteristic, complemented
150#466—1671no—
151#463—1672no—
152#450—1672no—
153#449—1672no—
154C4×D8—1671no—
155C8×D4—1671no—
156C2×#41—1672no—
157C22×D4—1672no—
158#135—3344no—
159C2×Q16C2×D16711yescharacteristic, complemented
160C2×SD16—1672no—
161C2×SD16—1672no—
162C2×D8—3344no—
163C2×D8—1672no—
164C2×D8—1671no—
165C2×D8—1671no—
166C22×C8—1671no—
167C22×C8C2×D16711yescharacteristic
168#124—1672no—
169#123—3342no—
170C22⋊Q8C2×D16712yes—
171#121—3342no—
172#121—1672no—
173#120—3342no—
174C4×D4—1672no—
175C4×D4—1671no—
176C2×C22⋊C4—3342no—
177#107C2×D16711yescharacteristic, complemented
178C4⋊C8—1671no—
179#103C2×D16712yescomplemented
180#102—1672no—
181#102—1672no—
182C22⋊C8—1671no—
183C22⋊C8—1671no—
184C4×C8—1671no—
185C24—3342no—
186#41—3344no—
187#41—6682no—
188C2×Q8C22×D16712yes—
189C2×D4—3342no—
190C2×D4—3344no—
191C2×D4—3342no—
192C2×D4—3342no—
193C2×D4—6682no—
194C2×D4—1671no—
195C2×D4—1672no—
196C2×D4—1672no—
197C22×C4—3342no—
198C22×C4—3342no—
199C22×C4—1671no—
200C22×C4C22×D16711yescharacteristic
201Q16—22no—
202SD16—3344no—
203SD16—6682no—
204D8—3344no—
205D8—3342no—
206D8—1674no—
207C2×C8—22no—
208C2×C8—3342no—
209C2×C8—1671no—
210C2×C8—1671no—
211C2×C8C22×D16711yescharacteristic
212C2×C8C22×D16711yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—1671no—
215C4⋊C4C22×D16712yes—
216C22⋊C4—22no—
217C22⋊C4—3342no—
218C22⋊C4—3342no—
219C22⋊C4—3342no—
220C22⋊C4—3342no—
221C22⋊C4—1671no—
222C22⋊C4—1671no—
223C42—1671no—
224C23—13362no—
225C23—3342no—
226C23—3342no—
227C23—6684no—
228C23—6684no—
229C23—1671no—
230C23D4×D16711yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—13362no—
234D4—3342no—
235D4—3342no—
236D4—3342no—
237D4—3342no—
238D4—6684no—
239D4—6684no—
240D4—1674no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—13362no—
246C2×C4—3341no—
247C2×C4—3341no—
248C2×C4—6684no—
249C2×C4—1671no—
250C2×C4—1671no—
251C2×C4—1671no—
252C2×C4D4×D16711yescharacteristic
253C2×C4C23×D16711yescharacteristic
254C8—21no—
255C8—3341no—
256C8—3341no—
257C8D4×D16712yes—
258C22—21no—
259C22—21no—
260C22—13368no—
261C22—3341no—
262C22—3341no—
263C22—3341no—
264C22—3344no—
265C22—6682no—
266C22—6682no—
267C22—6682no—
268C22—6682no—
269C22—6682no—
270C22#1912885512yescomplemented
271C22C2×D4×D16711yescharacteristic
272C4—21no—
273C4—42no—
274C4—42no—
275C4—3341no—
276C4—3341no—
277C4—3341no—
278C4C2×D4×D16711yescharacteristic
279C4#1912887811yescharacteristic
280C2—22no—
281C2—13362no—
282C2—6684no—
283C2—6681no—
284C2#2492390311yescharacteristic
285C2C2×#1912885511yescharacteristic
286C2#2492407111yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#467×C167C2
socleC22×C167C22×D4
hypercenter#467D167
hypocenterC167#3280
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#19128855C2semidirect
2. C22×D1336C2semidirect
3. #24923910C2semidirect
4. #24923654C2semidirect
5. #24923506C2semidirect
6. #24923487C2semidirect
7. #24923419C2semidirect
8. #24923417C2semidirect
9. #24923351C2semidirect
10. C8×C22⋊D167C2semidirect
11. #467×C167C2semidirect
12. C2×#14581951C22semidirect
15. C2×#14581937C22semidirect
16. C2×#14581937C22semidirect
21. C2×C8×D167C22semidirect
22. #19128821C22semidirect
25. #19128807C22semidirect
27. #19128794C22semidirect
30. Dic167⋊C8C22semidirect
31. #19128752C22semidirect
32. #19128752C22semidirect
34. (C2×D167)⋊C8C22semidirect
35. C8×Dic167C22semidirect
36. C2×Q16×C167C22semidirect
39. #107×C167C22semidirect
40. #103×C167C22semidirect
59. D1336D4semidirect
116. C22×C167#135semidirect
143. C167#3280semidirect
149. #467D167semidirect
159. C2×Q16C2×D167semidirect
177. #107C2×D167semidirect
179. #103C2×D167semidirect
270. C22#19128855semidirect

Maximal subgroups

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

subgroupindex
C2×#191288552
C22×D13362
#249239102
#249236542
#249235062
#249234872
#249234192
#249234172
#249233512
C8×C22⋊D1672
#467×C1672
#3280167