Finite Group Databasebeta

← C2×#24923415 #36333340

Subgroups — C2×#24923415 #36333340

Order 21376.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#14581949C211yesmaximal, characteristic, complemented
2C22×#14581939C211yesmaximal, characteristic, complemented
3C22×#14581935C211yesmaximal, characteristic, complemented
4C2×#19128807C211yesmaximal, characteristic, complemented
5#24923415C218yesmaximal, direct product factor
6C2×#19128755C211yesmaximal, characteristic, complemented
7C2×(C2×D167)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C167C211yesmaximal, characteristic, complemented
9C2×#14581949—24no—
10C2×#14581949—22no—
11C2×#14581949—24no—
12C2×#14581949C2214yescomplemented
13C22×Dic334C2211yescharacteristic
14C22×D668—21no—
15C22×D668C2211yescharacteristic
16C22×C22⋊D167—21no—
17C23×Dic167—21no—
18C22×C4×D167C2211yescharacteristic
19C2×#14581939—24no—
20C2×#14581939C2214yescomplemented
21C2×#14581935—24no—
22C2×#14581935C2214yescomplemented
23C22×C167⋊C8C2211yescharacteristic
24#19128807—22no—
25#19128807C2214yescomplemented
26C2×(C2×D167)⋊C4—21no—
27#19128755C2214yescomplemented
28(C2×D167)⋊C8C2214yescomplemented
29C22×D4×C167C2211yescharacteristic
30C22×C8×C167C2211yescharacteristic
31C2×C4⋊C4×C167C2211yescharacteristic
32#102×C167C2214yescomplemented
33C23×D167—21no—
34C23×D167D411yescharacteristic
35#14581949—216no—
36#14581949—28no—
37#14581949—28no—
38C2×Dic334—21no—
39C2×Dic334—22no—
40C2×Dic334D414yescomplemented
41C2×Dic334C2312yes—
42C2×D668—24no—
43C2×D668—22no—
44C2×D668—22no—
45C2×D668—21no—
46C2×D668—44no—
47C2×D668C2312yes—
48C2×C22⋊D167—28no—
49C2×C22⋊D167—24no—
50C22×Dic167—28no—
51C22×Dic167—24no—
52C22×Dic167—21no—
53C22×Dic167D411yescharacteristic
54C2×C4×D167—22no—
55C2×C4×D167—22no—
56C2×C4×D167D412yes—
57C2×C4×D167C2312yes—
58#14581939—28no—
59#14581935—28no—
60C2×C167⋊C8—22no—
61C2×C167⋊C8C2312yes—
62(C2×D167)⋊C4—24no—
63C24×C167—21no—
64C2×D4×C167—21no—
65C2×D4×C167—22no—
66C2×D4×C167D414yescomplemented
67C2×D4×C167C2312yes—
68C22×C4×C167—21no—
69C22×C4×C167C2311yescharacteristic
70C2×C8×C167—22no—
71C2×C8×C167C2312yes—
72C4⋊C4×C167—21no—
73C4⋊C4×C167C2312yes—
74C22×D167—22no—
75C22×D167—22no—
76C22×D167—22no—
77C22×D167—22no—
78C22×D167—42no—
79C22×D167—42no—
80C22×D167—42no—
81C22×D167C2×D412yes—
82Dic334—24no—
83Dic334—22no—
84Dic334C2×D414yescomplemented
85D668—22no—
86D668—22no—
87D668—48no—
88D668—42no—
89C22⋊D167—216no—
90C2×Dic167—216no—
91C2×Dic167—22no—
92C2×Dic167—24no—
93C2×Dic167—22no—
94C2×Dic167C2×D412yes—
95C4×D167—24no—
96C4×D167—22no—
97C4×D167—22no—
98C167⋊C8—22no—
99C23×C167—24no—
100C23×C167—22no—
101C23×C167—42no—
102C23×C167—42no—
103C23×C167C2×D411yescharacteristic
104D4×C167—24no—
105D4×C167—22no—
106D4×C167C2×D414yescomplemented
107C2×C4×C167—22no—
108C2×C4×C167—42no—
109C2×C4×C167C2×D411yescharacteristic
110C2×C4×C167C2×D412yes—
111C2×C4×C167C2×D412yes—
112C2×C4×C167C2411yescharacteristic
113C8×C167—22no—
114C22×C167—24no—
115C22×C167—42no—
116C22×C167—42no—
117C22×C167—48no—
118C22×C167#12012yes—
119C22×C167#13512yes—
120C22×C167AGL(1,ℤ/8)12yes—
121C22×C167C22×D411yescharacteristic
122C2×D167—22no—
123C2×D167—22no—
124C2×D167—22no—
125C2×D167—44no—
126C2×D167—42no—
127C2×D167—42no—
128C2×D167—42no—
129C2×D167—84no—
130C4×C167—42no—
131C4×C167#12012yes—
132C4×C167C22×D411yescharacteristic
133C4×C167C22×D411yescharacteristic
134Dic167—22no—
135Dic167—24no—
136C2×C167—44no—
137C2×C167#44914yescomplemented
138C2×C167C2×#12011yescharacteristic
139C2×C167C2×#13511yescharacteristic
140C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
141D167—42no—
142D167—82no—
143C167C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1671nomaximal, Sylow
145C22×#41—1671no—
146C22×SD16—1671no—
147C22×D8—1671no—
148C2×#121—1671no—
149#449—1678no—
150C2×#103—1671no—
151C2×#102D16711yescharacteristic, complemented
152C2×C22⋊C8—1671no—
153C2×#41—3344no—
154C2×#41—3344no—
155C2×#41—3342no—
156C2×#41—1674no—
157C22×Q8—1671no—
158C22×D4—3341no—
159C22×D4—3341no—
160C22×D4—1671no—
161C22×D4C2×D16711yescharacteristic
162C23×C4—3341no—
163C23×C4—1671no—
164C2×SD16—3344no—
165C2×SD16—1674no—
166C2×D8—3344no—
167C2×D8—1674no—
168C22×C8—1671no—
169C22×C8C2×D16711yescharacteristic
170#121—3342no—
171#121—1674no—
172C2×C4⋊C4C2×D16711yescharacteristic
173C2×C22⋊C4—3341no—
174#103—1674no—
175#102C2×D16714yescomplemented
176C22⋊C8—1674no—
177C24—21no—
178C24—3341no—
179C24—1671no—
180#41—33416no—
181#41—3348no—
182#41—3348no—
183C2×Q8—3341no—
184C2×Q8—3342no—
185C2×Q8—1674no—
186C2×Q8—1672no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—3344no—
190C2×D4—3348no—
191C2×D4—3344no—
192C2×D4—3342no—
193C2×D4—3341no—
194C2×D4—3342no—
195C2×D4—6684no—
196C2×D4—1672no—
197C2×D4D66814yescomplemented
198C2×D4C22×D16712yes—
199C22×C4—21no—
200C22×C4—3348no—
201C22×C4—3341no—
202C22×C4—3344no—
203C22×C4—3342no—
204C22×C4—3342no—
205C22×C4—1671no—
206C22×C4—1672no—
207C22×C4—1672no—
208C22×C4C22×D16711yescharacteristic
209SD16—3348no—
210D8—3348no—
211C2×C8—22no—
212C2×C8—3342no—
213C2×C8—1672no—
214C2×C8C22×D16712yes—
215C4⋊C4—21no—
216C4⋊C4C22×D16712yes—
217C22⋊C4—3344no—
218C23—22no—
219C23—24no—
220C23—42no—
221C23—42no—
222C23—3342no—
223C23—3342no—
224C23—3342no—
225C23—3342no—
226C23—6682no—
227C23—6682no—
228C23—6682no—
229C23—1672no—
230C23D4×D16711yescharacteristic
231Q8—3342no—
232Q8—3344no—
233Q8—1674no—
234D4—22no—
235D4—24no—
236D4—33416no—
237D4—3342no—
238D4—3342no—
239D4—6688no—
240D4—6682no—
241D4C2×D66814yescomplemented
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—33416no—
245C2×C4—3342no—
246C2×C4—3342no—
247C2×C4—3344no—
248C2×C4—3342no—
249C2×C4—3342no—
250C2×C4—3344no—
251C2×C4—1672no—
252C2×C4D4×D16712yes—
253C2×C4D4×D16711yescharacteristic
254C2×C4C2×D66812yes—
255C2×C4C23×D16711yescharacteristic
256C8—22no—
257C8—3342no—
258C22—24no—
259C22—48no—
260C22—42no—
261C22—42no—
262C22—13364no—
263C22—3342no—
264C22—3342no—
265C22—3342no—
266C22—6682no—
267C22—6682no—
268C22—6682no—
269C22—6684no—
270C22#1912880212yes—
271C22#1912885312yes—
272C22#1912885612yes—
273C22C2×D4×D16711yescharacteristic
274C4—42no—
275C4—3342no—
276C4—3344no—
277C4#1912880212yes—
278C4C2×D4×D16711yescharacteristic
279C4C22×D66811yescharacteristic
280C2—44no—
281C2—13362no—
282C2—6682no—
283C2#2492341514yesdirect product factor
284C2C2×#1912880211yescharacteristic
285C2C2×#1912885311yescharacteristic
286C2C2×#1912885611yescharacteristic
subgroupG/H
centerC23D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×#102×C167C2
socleC23×C167C2×D4
hypercenterC2×#102D167
hypocenterC167C2×#449
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14581949C2semidirect
2. C22×#14581939C2semidirect
3. C22×#14581935C2semidirect
4. C2×#19128807C2semidirect
5. #24923415C2direct (N × H)
6. C2×#19128755C2semidirect
7. C2×(C2×D167)⋊C8C2semidirect
8. C2×#102×C167C2semidirect
12. C2×#14581949C22semidirect
20. C2×#14581939C22semidirect
22. C2×#14581935C22semidirect
25. #19128807C22semidirect
27. #19128755C22semidirect
28. (C2×D167)⋊C8C22semidirect
32. #102×C167C22semidirect
40. C2×Dic334D4semidirect
66. C2×D4×C167D4semidirect
84. Dic334C2×D4semidirect
106. D4×C167C2×D4semidirect
137. C2×C167#449semidirect
143. C167C2×#449semidirect
151. C2×#102D167semidirect
175. #102C2×D167semidirect
197. C2×D4D668semidirect
241. D4C2×D668semidirect
283. C2#24923415direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145819492
C22×#145819392
C22×#145819352
C2×#191288072
#249234152
C2×#191287552
C2×(C2×D167)⋊C82
C2×#102×C1672
C2×#449167