Finite Group Databasebeta

← #36335734

Subgroups — #36335734

Order 21376.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19128858C212yesmaximal, complemented
2#24923924C211yesmaximal, characteristic, complemented
3#24923641C211yesmaximal, characteristic, complemented
4#24923513C212yesmaximal, complemented
5#24923474C211yesmaximal, characteristic, complemented
6#24923461C211yesmaximal, characteristic, complemented
7#24923420C212yesmaximal, complemented
8#24923412C212yesmaximal, complemented
9C4×#14581935C211yesmaximal, characteristic, complemented
10#24923347C211yesmaximal, characteristic, complemented
11D42×C167C211yesmaximal, characteristic, complemented
12C2×#14581950C2212yescomplemented
13#19128858—28no—
14C2×#14581938C2212yes—
15C2×#14581938C2212yescomplemented
16C2×#14581938C2212yescomplemented
17C2×#14581935C2211yescharacteristic
18C2×#14581935C2212yescomplemented
19C2×#14581934C2212yescomplemented
20#19128832C2211yescharacteristic, complemented
21#19128814—22no—
22#19128811C2212yescomplemented
23#19128803—21no—
24C4×D668C2211yescharacteristic, complemented
25C4⋊C4×D167—21no—
26#19128765C2211yescharacteristic, complemented
27C4⋊(C167⋊C8)C2211yescharacteristic, complemented
28#19128755C2212yescomplemented
29#19128753C2211yescharacteristic
30#19128753C2212yescomplemented
31#19128752C2211yescharacteristic, complemented
32C22⋊(C167⋊C8)C2212yescomplemented
33C4×C167⋊C8C2211yescharacteristic, complemented
34C22×D4×C167—21no—
35C22×D4×C167C2212yes—
36#127×C167C2211yescharacteristic
37#121×C167—21no—
38#121×C167C2212yes—
39#120×C167—22no—
40C4×D4×C167C2211yescharacteristic
41C4×D4×C167C2211yescharacteristic
42#14581950—24no—
43#14581950—42no—
44C2×Dic334C2312yes—
45C2×D668C2311yescharacteristic
46C2×C22⋊D167—22no—
47C2×C4×D167—22no—
48C2×C4×D167—21no—
49#14581938—24no—
50#14581938—24no—
51#14581938—42no—
52#14581935—24no—
53#14581935D414yescomplemented
54#14581934—24no—
55C2×C167⋊C8C2312yes—
56C2×C167⋊C8C2311yescharacteristic
57C2×C167⋊C8C2311yescharacteristic
58C4⋊Dic167—21no—
59C4⋊Dic167—21no—
60C4⋊Dic167C2311yescharacteristic
61Dic167⋊C4—22no—
62C22⋊Dic167—22no—
63(C2×D167)⋊C4—21no—
64C24×C167—22no—
65C2×D4×C167—24no—
66C2×D4×C167—22no—
67C2×D4×C167—22no—
68C2×D4×C167—21no—
69C2×D4×C167—21no—
70C2×D4×C167—42no—
71C2×D4×C167—44no—
72C2×D4×C167D411yescharacteristic
73C2×D4×C167C2312yes—
74C2×D4×C167C2311yescharacteristic
75C22×C4×C167—21no—
76C22×C4×C167C2312yes—
77C4⋊C4×C167D411yescharacteristic
78C4⋊C4×C167C2311yescharacteristic
79C22⋊C4×C167—21no—
80C22⋊C4×C167—22no—
81C22⋊C4×C167D412yes—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84Dic334—22no—
85Dic334—22no—
86D668—21no—
87D668C2×D412yescomplemented
88C22⋊D167—44no—
89C2×Dic167—21no—
90C2×Dic167—22no—
91C4×D167—44no—
92C4×D167—42no—
93C167⋊C8—22no—
94C167⋊C8—21no—
95C167⋊C8C2×D412yescomplemented
96C23×C167—21no—
97C23×C167—22no—
98C23×C167—44no—
99C23×C167—44no—
100C23×C167—82no—
101C23×C167C2×D412yes—
102D4×C167—22no—
103D4×C167—22no—
104D4×C167—21no—
105D4×C167—22no—
106D4×C167—44no—
107D4×C167—44no—
108D4×C167—42no—
109D4×C167—81no—
110D4×C167C2×D412yes—
111C2×C4×C167—22no—
112C2×C4×C167—22no—
113C2×C4×C167—21no—
114C2×C4×C167—42no—
115C2×C4×C167C2×D411yescharacteristic
116C2×C4×C167C2×D412yes—
117C2×C4×C167C2×D411yescharacteristic
118C2×C4×C167C2411yescharacteristic
119C22×C167—22no—
120C22×C167—22no—
121C22×C167—22no—
122C22×C167—22no—
123C22×C167—41no—
124C22×C167—41no—
125C22×C167—42no—
126C22×C167—42no—
127C22×C167—42no—
128C22×C167—88no—
129C22×C167C22×D411yescharacteristic
130C2×D167—22no—
131C2×D167—41no—
132C2×D167—41no—
133C4×C167—22no—
134C4×C167—21no—
135C4×C167—41no—
136C4×C167AGL(1,ℤ/8)12yescomplemented
137C4×C167C22×D411yescharacteristic
138C4×C167H(5,𝔽2)11yescharacteristic
139Dic167—42no—
140Dic167—41no—
141C2×C167—42no—
142C2×C167—42no—
143C2×C167—82no—
144C2×C167D4211yescharacteristic
145C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C167#57711yescharacteristic
147D167—42no—
148C167#327211yescharacteristic, complemented, Sylow
149#3272—1671nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1672no—
151#547—1671no—
152D42D16711yescharacteristic, complemented
153#492—1671no—
154#468—1672no—
155#461—1671no—
156#459—1671no—
157#450—1672no—
158#449—1672no—
159C4×D8—1671no—
160#436—1671no—
161C2×#41—1672no—
162C22×D4—21no—
163C22×D4C2×D16712yes—
164AGL(1,ℤ/8)—3348no—
165C2×SD16—1672no—
166C2×SD16—1672no—
167C2×SD16—1672no—
168C2×D8—1672no—
169C2×D8—1671no—
170C2×M16—1672no—
171C4⋊Q8—1671no—
172#127C2×D16711yescharacteristic
173#123—3342no—
174C22⋊Q8—1672no—
175#121—21no—
176#121—3341no—
177#121C2×D16712yes—
178#120—22no—
179C4×D4—1671no—
180C4×D4C2×D16711yescharacteristic
181C4×D4C2×D16711yescharacteristic
182C2×C4⋊C4—3341no—
183#107—1671no—
184C4⋊C8—1671no—
185#103—1672no—
186#102—1671no—
187#102—1672no—
188#102—1671no—
189C22⋊C8—1672no—
190C4×C8—1671no—
191C24—22no—
192#41—3344no—
193#41—6682no—
194C2×Q8—1672no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—3342no—
203C2×D4—1671no—
204C2×D4C22⋊D16711yescharacteristic
205C2×D4C22×D16712yes—
206C2×D4C22×D16711yescharacteristic
207C22×C4—21no—
208C22×C4—3341no—
209C22×C4—3342no—
210C22×C4C22×D16712yes—
211SD16—3344no—
212SD16—3344no—
213SD16—6682no—
214D8—3344no—
215D8—1674no—
216M16—3344no—
217C2×C8—1671no—
218C2×C8—1672no—
219C2×C8—1671no—
220C4⋊C4—3341no—
221C4⋊C4—3342no—
222C4⋊C4—3341no—
223C4⋊C4—1671no—
224C4⋊C4C22⋊D16711yescharacteristic
225C4⋊C4C22×D16711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—3342no—
229C22⋊C4—3341no—
230C22⋊C4C22⋊D16712yes—
231C42C22×D16711yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—3341no—
238C23C2×C22⋊D16712yes—
239Q8—3342no—
240Q8—3342no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—3341no—
250D4—6684no—
251D4—1672no—
252D4D4×D16712yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3341no—
258C2×C4—3342no—
259C2×C4—6684no—
260C2×C4—6682no—
261C2×C4C2×C22⋊D16711yescharacteristic
262C2×C4C2×C22⋊D16712yes—
263C2×C4C2×C22⋊D16711yescharacteristic
264C2×C4C23×D16711yescharacteristic
265C8—3342no—
266C8—3341no—
267C8—1672no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—3342no—
279C22—6681no—
280C22—6681no—
281C22C22×C22⋊D16711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—6681no—
286C4—6682no—
287C4#1912885812yescomplemented
288C4C2×D4×D16711yescharacteristic
289C4#1912887711yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—6682no—
294C2D4×C22⋊D16711yescharacteristic
295C2C2×#1912885811yescharacteristic
296C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingD42×C167C2
socleC22×C167C22×D4
hypercenterD42D167
hypocenterC167#3272
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19128858C2semidirect
2. #24923924C2semidirect
3. #24923641C2semidirect
4. #24923513C2semidirect
5. #24923474C2semidirect
6. #24923461C2semidirect
7. #24923420C2semidirect
8. #24923412C2semidirect
9. C4×#14581935C2semidirect
10. #24923347C2semidirect
11. D42×C167C2semidirect
12. C2×#14581950C22semidirect
15. C2×#14581938C22semidirect
16. C2×#14581938C22semidirect
18. C2×#14581935C22semidirect
19. C2×#14581934C22semidirect
20. #19128832C22semidirect
22. #19128811C22semidirect
24. C4×D668C22semidirect
26. #19128765C22semidirect
27. C4⋊(C167⋊C8)C22semidirect
28. #19128755C22semidirect
30. #19128753C22semidirect
31. #19128752C22semidirect
32. C22⋊(C167⋊C8)C22semidirect
33. C4×C167⋊C8C22semidirect
53. #14581935D4semidirect
87. D668C2×D4semidirect
95. C167⋊C8C2×D4semidirect
136. C4×C167AGL(1,ℤ/8)semidirect
148. C167#3272semidirect
152. D42D167semidirect
287. C4#19128858semidirect

Maximal subgroups

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

subgroupindex
C2×#191288582
#249239242
#249236412
#249235132
#249234742
#249234612
#249234202
#249234122
C4×#145819352
#249233472
D42×C1672
#3272167