Finite Group Databasebeta

← #36335733

Subgroups — #36335733

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×#19128856C212yesmaximal, complemented
2#24923923C211yesmaximal, characteristic, complemented
3#24923904C211yesmaximal, characteristic, complemented
4#24923641C211yesmaximal, characteristic, complemented
5#24923514C212yesmaximal, complemented
6#24923475C211yesmaximal, characteristic, complemented
7#24923421C212yesmaximal, complemented
8#24923411C212yesmaximal, complemented
9C4×#14581935C211yesmaximal, characteristic, complemented
10#24923346C211yesmaximal, characteristic, complemented
11#459×C167C211yesmaximal, characteristic, complemented
12C2×#14581949C2212yescomplemented
13C2×D4×D167—21no—
14C2×D4×D167C2212yes—
15#19128856—28no—
16C2×#14581939C2212yes—
17C2×#14581938C2212yescomplemented
18C2×#14581938C2212yescomplemented
19C2×#14581935C2211yescharacteristic
20C2×#14581933C2212yescomplemented
21#19128832C2211yescharacteristic, complemented
22#19128816—22no—
23#19128812C2212yescomplemented
24#19128806—21no—
25#19128803—21no—
26#19128803C2212yes—
27#19128800—22no—
28C4×D668C2211yescharacteristic
29C4×D668C2211yescharacteristic
30C2×C4⋊Dic167—21no—
31#19128765C2211yescharacteristic, complemented
32#19128757C2212yescomplemented
33#19128753C2211yescharacteristic, complemented
34#19128752C2211yescharacteristic
35(C2×D167)⋊C8C2212yescomplemented
36C4×C167⋊C8C2211yescharacteristic, complemented
37C2×D8×C167C2212yescomplemented
38#127×C167C2211yescharacteristic
39C4×D4×C167C2211yescharacteristic, complemented
40C4⋊C8×C167C2211yescharacteristic, complemented
41#102×C167C2212yescomplemented
42C23×D167—22no—
43#14581949—24no—
44#14581949—42no—
45C2×Dic334C2312yes—
46C2×D668D411yescharacteristic
47C2×D668C2311yescharacteristic
48C2×C22⋊D167—22no—
49C2×C22⋊D167—22no—
50C2×C22⋊D167—22no—
51D4×D167—24no—
52D4×D167—42no—
53D4×D167—44no—
54C22×Dic167—22no—
55C2×C4×D167—21no—
56C2×C4×D167C2312yes—
57#14581939—24no—
58#14581938—24no—
59#14581938—42no—
60#14581935D414yescomplemented
61#14581933—24no—
62C2×C167⋊C8C2311yescharacteristic
63C2×C167⋊C8C2311yescharacteristic
64C4⋊Dic167—22no—
65C4⋊Dic167—21no—
66C4⋊Dic167—21no—
67C4⋊Dic167D411yescharacteristic
68C4⋊Dic167C2311yescharacteristic
69C22⋊Dic167—22no—
70(C2×D167)⋊C4—22no—
71(C2×D167)⋊C4—21no—
72(C2×D167)⋊C4D412yes—
73C2×D4×C167—21no—
74C2×D4×C167—21no—
75C2×D4×C167C2312yes—
76C2×D4×C167C2311yescharacteristic
77C22×C4×C167—21no—
78D8×C167—24no—
79C2×C8×C167C2312yes—
80C4⋊C4×C167C2311yescharacteristic
81C22⋊C4×C167—21no—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—44no—
85C22×D167—44no—
86C22×D167—82no—
87C22×D167C2×D412yes—
88Dic334—22no—
89Dic334—22no—
90D668—22no—
91D668—21no—
92D668C2×D412yes—
93C22⋊D167—44no—
94C22⋊D167—44no—
95C22⋊D167—44no—
96C2×Dic167—21no—
97C2×Dic167—22no—
98C2×Dic167—44no—
99C2×Dic167C2×D412yes—
100C4×D167—22no—
101C4×D167—22no—
102C4×D167—42no—
103C167⋊C8—21no—
104C167⋊C8C2×D412yescomplemented
105C23×C167—21no—
106C23×C167—22no—
107D4×C167—22no—
108D4×C167—21no—
109D4×C167—22no—
110D4×C167—42no—
111D4×C167—81no—
112D4×C167C2×D412yescomplemented
113C2×C4×C167—21no—
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×C167—42no—
123C22×C167—42no—
124C22×C167—42no—
125C22×C167C22×D411yescharacteristic
126C2×D167—22no—
127C2×D167—22no—
128C2×D167—22no—
129C2×D167—22no—
130C2×D167—41no—
131C2×D167—41no—
132C2×D167—88no—
133C4×C167—21no—
134C4×C167—41no—
135C4×C167AGL(1,ℤ/8)12yescomplemented
136C4×C167C22×D411yescharacteristic
137C4×C167H(5,𝔽2)11yescharacteristic
138Dic167—22no—
139Dic167—42no—
140Dic167—41no—
141C2×C167—42no—
142C2×C167—82no—
143C2×C167D4211yescharacteristic
144C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C167#57711yescharacteristic
146D167—42no—
147D167—42no—
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#461—1671no—
156#459D16711yescharacteristic, complemented
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×SD16—1672no—
168C2×D8—1671no—
169C2×D8C2×D16712yescomplemented
170C2×M16—1672no—
171C4⋊Q8—1671no—
172#127C2×D16711yescharacteristic
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, complemented
182C2×C4⋊C4—3341no—
183#107—1671no—
184C4⋊C8C2×D16711yescharacteristic, complemented
185#103—1672no—
186#102—1671no—
187#102—1671no—
188#102C2×D16712yescomplemented
189C22⋊C8—1672no—
190C4×C8—1671no—
191C24—3342no—
192#41—3344no—
193#41—6682no—
194C2×Q8—1672no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—3344no—
198C2×D4—3342no—
199C2×D4—3342no—
200C2×D4—3342no—
201C2×D4—6682no—
202C2×D4—6684no—
203C2×D4—1671no—
204C2×D4—1671no—
205C2×D4C22×D16712yes—
206C2×D4C22×D16711yescharacteristic
207C22×C4—21no—
208C22×C4—3342no—
209C22×C4—3341no—
210C22×C4—1672no—
211SD16—3344no—
212SD16—3344no—
213SD16—6682no—
214D8—24no—
215D8—1674no—
216M16—3344no—
217C2×C8—1671no—
218C2×C8—1671no—
219C2×C8C22×D16712yes—
220C4⋊C4—3342no—
221C4⋊C4—3341no—
222C4⋊C4—3341no—
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—22no—
234C23—13362no—
235C23—3341no—
236C23—6684no—
237C23—6684no—
238C23—1672no—
239Q8—3342no—
240Q8—3342no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—81no—
246D4—3342no—
247D4—3341no—
248D4—6684no—
249D4—6684no—
250D4—6684no—
251D4—1672no—
252D4D4×D16712yescomplemented
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—3342no—
256C2×C4—3342no—
257C2×C4—3341no—
258C2×C4—3342no—
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—42no—
272C22—42no—
273C22—42no—
274C22—13368no—
275C22—3342no—
276C22—3342no—
277C22—3342no—
278C22—3342no—
279C22—6681no—
280C22—6681no—
281C22C2×D4×D16711yescharacteristic
282C4—21no—
283C4—41no—
284C4—3342no—
285C4—6681no—
286C4—6682no—
287C4#1912885612yescomplemented
288C4C2×D4×D16711yescharacteristic
289C4#1912887811yescharacteristic
290C2—42no—
291C2—82no—
292C2—6682no—
293C2—6682no—
294C2#2492390311yescharacteristic
295C2C2×#1912885611yescharacteristic
296C2#2492407911yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#459×C167C2
socleC22×C167C22×D4
hypercenter#459D167
hypocenterC167#3272
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19128856C2semidirect
2. #24923923C2semidirect
3. #24923904C2semidirect
4. #24923641C2semidirect
5. #24923514C2semidirect
6. #24923475C2semidirect
7. #24923421C2semidirect
8. #24923411C2semidirect
9. C4×#14581935C2semidirect
10. #24923346C2semidirect
11. #459×C167C2semidirect
12. C2×#14581949C22semidirect
17. C2×#14581938C22semidirect
18. C2×#14581938C22semidirect
20. C2×#14581933C22semidirect
21. #19128832C22semidirect
23. #19128812C22semidirect
31. #19128765C22semidirect
32. #19128757C22semidirect
33. #19128753C22semidirect
35. (C2×D167)⋊C8C22semidirect
36. C4×C167⋊C8C22semidirect
37. C2×D8×C167C22semidirect
39. C4×D4×C167C22semidirect
40. C4⋊C8×C167C22semidirect
41. #102×C167C22semidirect
60. #14581935D4semidirect
104. C167⋊C8C2×D4semidirect
112. D4×C167C2×D4semidirect
135. C4×C167AGL(1,ℤ/8)semidirect
148. C167#3272semidirect
156. #459D167semidirect
169. C2×D8C2×D167semidirect
181. C4×D4C2×D167semidirect
184. C4⋊C8C2×D167semidirect
188. #102C2×D167semidirect
252. D4D4×D167semidirect
287. C4#19128856semidirect

Maximal subgroups

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

subgroupindex
C2×#191288562
#249239232
#249239042
#249236412
#249235142
#249234752
#249234212
#249234112
C4×#145819352
#249233462
#459×C1672
#3272167