Finite Group Databasebeta

← #36335741

Subgroups — #36335741

Order 21376.

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

normal subgroups only

This group has 546 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×#19128859C212yesmaximal, complemented
2#24923922C211yesmaximal, characteristic, complemented
3D4×D668C211yesmaximal, characteristic, complemented
4#24923643C211yesmaximal, characteristic, complemented
5#24923518C212yesmaximal, complemented
6#24923465C211yesmaximal, characteristic, complemented
7#24923418C212yesmaximal, complemented
8#24923417C212yesmaximal, complemented
9C4×#14581935C211yesmaximal, characteristic, complemented
10#24923346C211yesmaximal, characteristic, complemented
11#461×C167C211yesmaximal, characteristic, complemented
12C2×#14581951C2212yescomplemented
13C22×D668—21no—
14C2×D4×D167C2212yes—
15#19128859—28no—
16C2×#14581937C2212yescomplemented
17C2×#14581937C2212yescomplemented
18C2×D1336C2212yescomplemented
19C2×#14581935C2211yescharacteristic
20C2×#14581933C2212yescomplemented
21#19128829C2211yescharacteristic
22#19128817—22no—
23#19128808C2212yescomplemented
24#19128807—21no—
25#19128807C2212yes—
26#19128806—21no—
27#19128802—22no—
28C4×D668C2211yescharacteristic
29C4×D668C2211yescharacteristic, complemented
30C4⋊C4×D167—21no—
31#19128765C2211yescharacteristic, complemented
32#19128754C2212yescomplemented
33#19128753C2211yescharacteristic
34#19128752C2211yescharacteristic, complemented
35(C2×D167)⋊C8C2212yescomplemented
36C4×C167⋊C8C2211yescharacteristic, complemented
37C2×SD16×C167C2212yes—
38C4⋊Q8×C167C2211yescharacteristic, complemented
39C4×D4×C167C2211yescharacteristic
40C4⋊C8×C167C2211yescharacteristic, complemented
41#103×C167C2212yescomplemented
42C23×D167—22no—
43#14581951—24no—
44#14581951—42no—
45C2×D668—22no—
46C2×D668—22no—
47C2×D668—21no—
48C2×D668—21no—
49C2×D668—44no—
50C2×D668D411yescharacteristic
51C2×D668C2312yes—
52C2×D668C2311yescharacteristic
53C2×C22⋊D167—22no—
54D4×D167—24no—
55D4×D167—42no—
56C2×C4×D167—22no—
57C2×C4×D167—21no—
58C2×C4×D167C2312yes—
59#14581937—24no—
60#14581937—42no—
61D1336—24no—
62#14581935D414yescomplemented
63#14581933—24no—
64C2×C167⋊C8C2311yescharacteristic
65C2×C167⋊C8C2311yescharacteristic
66C4⋊Dic167D411yescharacteristic
67C4⋊Dic167C2311yescharacteristic
68Dic167⋊C4—22no—
69(C2×D167)⋊C4—22no—
70(C2×D167)⋊C4—22no—
71(C2×D167)⋊C4—21no—
72(C2×D167)⋊C4D412yes—
73C2×Q8×C167C2312yes—
74C2×D4×C167C2311yescharacteristic
75C22×C4×C167—21no—
76SD16×C167—24no—
77C2×C8×C167C2312yes—
78C4⋊C4×C167—21no—
79C4⋊C4×C167—21no—
80C4⋊C4×C167C2311yescharacteristic
81C22⋊C4×C167—21no—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—22no—
85C22×D167—44no—
86C22×D167—44no—
87C22×D167—82no—
88C22×D167C2×D412yes—
89D668—22no—
90D668—22no—
91D668—21no—
92D668—22no—
93D668—44no—
94D668—44no—
95D668—42no—
96D668—81no—
97D668C2×D412yescomplemented
98C22⋊D167—44no—
99C2×Dic167—21no—
100C2×Dic167C2×D412yes—
101C4×D167—22no—
102C4×D167—22no—
103C4×D167—44no—
104C4×D167—42no—
105C167⋊C8—21no—
106C167⋊C8C2×D412yescomplemented
107C23×C167—21no—
108Q8×C167—22no—
109Q8×C167—22no—
110D4×C167—21no—
111D4×C167C2×D412yes—
112C2×C4×C167—21no—
113C2×C4×C167—22no—
114C2×C4×C167—42no—
115C2×C4×C167C2×D411yescharacteristic
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2411yescharacteristic
118C8×C167—22no—
119C22×C167—22no—
120C22×C167—41no—
121C22×C167—41no—
122C22×C167C22×D411yescharacteristic
123C2×D167—22no—
124C2×D167—22no—
125C2×D167—22no—
126C2×D167—22no—
127C2×D167—41no—
128C2×D167—41no—
129C2×D167—42no—
130C2×D167—42no—
131C2×D167—42no—
132C2×D167—88no—
133C4×C167—21no—
134C4×C167—42no—
135C4×C167—41no—
136C4×C167AGL(1,ℤ/8)12yescomplemented
137C4×C167C22×D411yescharacteristic
138C4×C167H(5,𝔽2)11yescharacteristic
139Dic167—22no—
140Dic167—41no—
141C2×C167—42no—
142C2×C167D4211yescharacteristic
143C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C167#57711yescharacteristic
145D167—42no—
146D167—42no—
147D167—82no—
148C167#327211yescharacteristic, complemented, Sylow
149#3272—1671nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1672no—
151#547—1671no—
152D42—1671no—
153#492—1671no—
154#468—1672no—
155#461D16711yescharacteristic, complemented
156#459—1671no—
157#450—1672no—
158#449—1672no—
159C4×D8—1671no—
160#436—1671no—
161C2×#41—1672no—
162C22×D4—3341no—
163C22×D4—1672no—
164AGL(1,ℤ/8)—3348no—
165C2×SD16—1672no—
166C2×SD16—1672no—
167C2×SD16C2×D16712yes—
168C2×D8—1671no—
169C2×D8—1672no—
170C2×M16—1672no—
171C4⋊Q8C2×D16711yescharacteristic, complemented
172#127—1671no—
173#123—3342no—
174C22⋊Q8—1672no—
175#121—3341no—
176#121—3341no—
177#121—1672no—
178#120—3342no—
179C4×D4—1671no—
180C4×D4—1671no—
181C4×D4C2×D16711yescharacteristic
182C2×C4⋊C4—3341no—
183#107—1671no—
184C4⋊C8C2×D16711yescharacteristic, complemented
185#103C2×D16712yescomplemented
186#102—1671no—
187#102—1672no—
188#102—1671no—
189C22⋊C8—1672no—
190C4×C8—1671no—
191C24—3342no—
192#41—3344no—
193#41—6682no—
194C2×Q8C22×D16712yes—
195C2×D4—3342no—
196C2×D4—3342no—
197C2×D4—3344no—
198C2×D4—3342no—
199C2×D4—3341no—
200C2×D4—3341no—
201C2×D4—6684no—
202C2×D4—6682no—
203C2×D4—1671no—
204C2×D4—1672no—
205C2×D4—1671no—
206C2×D4C22×D16711yescharacteristic
207C22×C4—21no—
208C22×C4—3342no—
209C22×C4—3341no—
210C22×C4—1672no—
211SD16—24no—
212SD16—3344no—
213SD16—6682no—
214D8—3344no—
215D8—1674no—
216M16—3344no—
217C2×C8—1671no—
218C2×C8—1671no—
219C2×C8C22×D16712yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3342no—
223C4⋊C4—1671no—
224C4⋊C4—1671no—
225C4⋊C4C22×D16711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3342no—
228C22⋊C4—3342no—
229C22⋊C4—3341no—
230C22⋊C4—1672no—
231C42C22×D16711yescharacteristic
232C23—21no—
233C23—13362no—
234C23—3341no—
235C23—3342no—
236C23—6684no—
237C23—6684no—
238C23—1672no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—13361no—
243D4—3342no—
244D4—3342no—
245D4—3341no—
246D4—3342no—
247D4—6684no—
248D4—6684no—
249D4—6684no—
250D4—6682no—
251D4—1672no—
252D4D4×D16712yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—3342no—
257C2×C4—3342no—
258C2×C4—3341no—
259C2×C4—6684no—
260C2×C4—6682no—
261C2×C4—1672no—
262C2×C4D4×D16711yescharacteristic
263C2×C4D4×D16711yescharacteristic
264C2×C4C23×D16711yescharacteristic
265C8—22no—
266C8—3341no—
267C8—1672no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—13368no—
272C22—3342no—
273C22—3342no—
274C22—3342no—
275C22—3342no—
276C22—6682no—
277C22—6682no—
278C22—6682no—
279C22—6681no—
280C22—6681no—
281C22C2×D4×D16711yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—3342no—
286C4—6681no—
287C4#1912885912yescomplemented
288C4C2×D4×D16711yescharacteristic
289C4#1912887811yescharacteristic
290C2—42no—
291C2—13362no—
292C2—6682no—
293C2—6682no—
294C2#2492390311yescharacteristic
295C2C2×#1912885911yescharacteristic
296C2#2492407811yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#461×C167C2
socleC22×C167C22×D4
hypercenter#461D167
hypocenterC167#3272
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#19128859C2semidirect
2. #24923922C2semidirect
3. D4×D668C2semidirect
4. #24923643C2semidirect
5. #24923518C2semidirect
6. #24923465C2semidirect
7. #24923418C2semidirect
8. #24923417C2semidirect
9. C4×#14581935C2semidirect
10. #24923346C2semidirect
11. #461×C167C2semidirect
12. C2×#14581951C22semidirect
16. C2×#14581937C22semidirect
17. C2×#14581937C22semidirect
18. C2×D1336C22semidirect
20. C2×#14581933C22semidirect
23. #19128808C22semidirect
29. C4×D668C22semidirect
31. #19128765C22semidirect
32. #19128754C22semidirect
34. #19128752C22semidirect
35. (C2×D167)⋊C8C22semidirect
36. C4×C167⋊C8C22semidirect
38. C4⋊Q8×C167C22semidirect
40. C4⋊C8×C167C22semidirect
41. #103×C167C22semidirect
62. #14581935D4semidirect
97. D668C2×D4semidirect
106. C167⋊C8C2×D4semidirect
136. C4×C167AGL(1,ℤ/8)semidirect
148. C167#3272semidirect
155. #461D167semidirect
171. C4⋊Q8C2×D167semidirect
184. C4⋊C8C2×D167semidirect
185. #103C2×D167semidirect
287. C4#19128859semidirect

Maximal subgroups

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

subgroupindex
C2×#191288592
#249239222
D4×D6682
#249236432
#249235182
#249234652
#249234182
#249234172
C4×#145819352
#249233462
#461×C1672
#3272167