Finite Group Databasebeta

← #36333406

Subgroups — #36333406

Order 21376.

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

normal subgroups only

This group has 784 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
1C2×#19128857C211yesmaximal, characteristic, complemented
2C22×#14581935C211yesmaximal, characteristic, complemented
3#24923833C211yesmaximal, characteristic, complemented
4#24923414C214yesmaximal, complemented
5#24923403C214yesmaximal, complemented
6#24923274C211yesmaximal, characteristic, complemented
7#24923265C211yesmaximal, characteristic, complemented
8#24923236C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C167C211yesmaximal, characteristic, complemented
10C22×D668C2211yescharacteristic, complemented
11#19128857—24no—
12C2×#14581937C2212yescomplemented
13C2×#14581935—24no—
14C2×#14581935C2212yescomplemented
15C2×#14581935C2214yescomplemented
16C2×#14581934C2211yescharacteristic, complemented
17C22×C167⋊C8C2211yescharacteristic, complemented
18#19128828—21no—
19#19128821—21no—
20#19128806C2214yescomplemented
21#19128800—22no—
22#19128791C2211yescharacteristic, complemented
23#19128756C2212yescomplemented
24#19128753C2212yescomplemented
25#19128753C2214yescomplemented
26C22⋊(C167⋊C8)C2214yescomplemented
27H(5,𝔽2)×C167—24no—
28H(5,𝔽2)×C167—24no—
29C2×#41×C167—21no—
30C2×#41×C167C2211yescharacteristic
31C2×#41×C167C2211yescharacteristic
32C2×#41×C167C2212yescomplemented
33C22×D4×C167—21no—
34C22×D4×C167—22no—
35C22×D4×C167C2211yescharacteristic
36C22×D4×C167C2212yescomplemented
37C23×D167—21no—
38C2×D668—22no—
39C2×D668—22no—
40C2×D668—21no—
41C2×D668C2312yes—
42C2×C22⋊D167—24no—
43#14581937—24no—
44#14581935—28no—
45#14581935—24no—
46#14581934—22no—
47C2×C167⋊C8—22no—
48C2×C167⋊C8C2312yes—
49C2×C167⋊C8C2312yes—
50C4⋊Dic167C2312yes—
51C22⋊Dic167—21no—
52(C2×D167)⋊C4—24no—
53C4×Dic167—21no—
54C24×C167—21no—
55C24×C167—22no—
56#41×C167—28no—
57#41×C167—24no—
58#41×C167—24no—
59#41×C167—22no—
60#41×C167—22no—
61#41×C167—22no—
62#41×C167D414yescomplemented
63C2×Q8×C167D411yescharacteristic
64C2×Q8×C167C2311yescharacteristic
65C2×D4×C167—216no—
66C2×D4×C167—28no—
67C2×D4×C167—21no—
68C2×D4×C167—22no—
69C2×D4×C167—24no—
70C2×D4×C167—24no—
71C2×D4×C167—21no—
72C2×D4×C167—22no—
73C2×D4×C167—22no—
74C2×D4×C167D412yes—
75C2×D4×C167D411yescharacteristic
76C2×D4×C167D414yescomplemented
77C2×D4×C167C2312yes—
78C2×D4×C167C2311yescharacteristic
79C22×C4×C167—21no—
80C22×C4×C167—22no—
81C22×C4×C167C2312yes—
82C22×C4×C167C2311yescharacteristic
83C22×D167—22no—
84C22×D167—42no—
85C22×D167—82no—
86D668—22no—
87D668—22no—
88D668—42no—
89C22⋊D167—84no—
90C2×Dic167—22no—
91C167⋊C8—22no—
92C167⋊C8—22no—
93C23×C167—28no—
94C23×C167—21no—
95C23×C167—22no—
96C23×C167—24no—
97C23×C167—82no—
98C23×C167—84no—
99C23×C167C2×D412yes—
100C23×C167C2×D411yescharacteristic
101Q8×C167—22no—
102Q8×C167—21no—
103Q8×C167C2×D412yes—
104D4×C167—28no—
105D4×C167—28no—
106D4×C167—24no—
107D4×C167—24no—
108D4×C167—22no—
109D4×C167—24no—
110D4×C167—22no—
111D4×C167—21no—
112D4×C167C2×D412yes—
113D4×C167C2×D414yes—
114C2×C4×C167—28no—
115C2×C4×C167—21no—
116C2×C4×C167—24no—
117C2×C4×C167—22no—
118C2×C4×C167—22no—
119C2×C4×C167C2×D412yes—
120C2×C4×C167C2×D411yescharacteristic
121C2×C4×C167C2×D412yes—
122C2×C4×C167C2×D412yes—
123C2×C4×C167C2411yescharacteristic
124C22×C167—22no—
125C22×C167—22no—
126C22×C167—24no—
127C22×C167—22no—
128C22×C167—24no—
129C22×C167—44no—
130C22×C167—42no—
131C22×C167—84no—
132C22×C167—88no—
133C22×C167#12012yes—
134C22×C167C22×D411yescharacteristic
135C2×D167—42no—
136C2×D167—44no—
137C2×D167—84no—
138C4×C167—22no—
139C4×C167—22no—
140C4×C167#12012yes—
141C4×C167C22×D411yescharacteristic
142C4×C167C22×D411yescharacteristic
143Dic167—42no—
144C2×C167—22no—
145C2×C167—42no—
146C2×C167—44no—
147C2×C167—42no—
148C2×C167C2×#12011yescharacteristic
149C2×C167#57612yes—
150D167—82no—
151C167#300811yescharacteristic, complemented, Sylow
152#3008—1671nomaximal, Sylow
153C2×H(5,𝔽2)D16711yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—1671no—
155C22×D8—1671no—
156#535—1671no—
157#449—1674no—
158#447—1674no—
159#418—1671no—
160#416—1671no—
161#408—1671no—
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D16711yescharacteristic
166C2×#41C2×D16711yescharacteristic
167C2×#41C2×D16712yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—1671no—
171C22×D4C2×D16711yescharacteristic
172C22×D4C2×D16712yescomplemented
173AGL(1,ℤ/8)—3344no—
174C2×SD16—1672no—
175C2×D8—3344no—
176C2×D8—1674no—
177C2×D8—1672no—
178C2×M16—1671no—
179C22×C8—1671no—
180#127—3341no—
181#124—3341no—
182#121—1674no—
183#120—3342no—
184#117—1671no—
185#103—1672no—
186#102—1672no—
187#102—1674no—
188C22⋊C8—1674no—
189C24—21no—
190C24—22no—
191C24—3341no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D16714yescomplemented
199C2×Q8C22⋊D16711yescharacteristic
200C2×Q8C22×D16711yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—3344no—
211C2×D4—3342no—
212C2×D4—3342no—
213C2×D4—3341no—
214C2×D4—1672no—
215C2×D4C22⋊D16712yes—
216C2×D4C22⋊D16711yescharacteristic
217C2×D4C22⋊D16714yescomplemented
218C2×D4C22×D16712yes—
219C2×D4C22×D16711yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D16712yes—
223C22×C4C22×D16711yescharacteristic
224SD16—3344no—
225D8—3348no—
226D8—3344no—
227M16—3342no—
228C2×C8—3342no—
229C2×C8—1672no—
230C2×C8—1672no—
231C4⋊C4—1672no—
232C22⋊C4—3341no—
233C22⋊C4—3344no—
234C42—3341no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—13362no—
242C23—3342no—
243C23—6682no—
244C23C2×C22⋊D16712yes—
245C23C2×C22⋊D16711yescharacteristic
246Q8—21no—
247Q8—22no—
248Q8C2×C22⋊D16712yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—24no—
257D4—13364no—
258D4—3342no—
259D4—3342no—
260D4—6682no—
261D4C2×C22⋊D16712yes—
262D4C2×C22⋊D16714yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—3342no—
269C2×C4C2×C22⋊D16712yes—
270C2×C4C2×C22⋊D16712yes—
271C2×C4C2×C22⋊D16712yes—
272C2×C4C2×C22⋊D16711yescharacteristic
273C2×C4C23×D16711yescharacteristic
274C8—3342no—
275C8—3342no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—13364no—
286C22—6682no—
287C22—6684no—
288C22#1912880112yes—
289C22C22×C22⋊D16711yescharacteristic
290C4—22no—
291C4—22no—
292C4—6682no—
293C4#1912880112yes—
294C4C22×C22⋊D16711yescharacteristic
295C4C22×C22⋊D16711yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—13362no—
301C2C2×#1912880111yescharacteristic
302C2#2492407212yes—
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×H(5,𝔽2)×C167C2
socleC22×C167C22×D4
hypercenterC2×H(5,𝔽2)D167
hypocenterC167#3008
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#19128857C2semidirect
2. C22×#14581935C2semidirect
3. #24923833C2semidirect
4. #24923414C2semidirect
5. #24923403C2semidirect
6. #24923274C2semidirect
7. #24923265C2semidirect
8. #24923236C2semidirect
9. C2×H(5,𝔽2)×C167C2semidirect
10. C22×D668C22semidirect
12. C2×#14581937C22semidirect
14. C2×#14581935C22semidirect
15. C2×#14581935C22semidirect
16. C2×#14581934C22semidirect
17. C22×C167⋊C8C22semidirect
20. #19128806C22semidirect
22. #19128791C22semidirect
23. #19128756C22semidirect
24. #19128753C22semidirect
25. #19128753C22semidirect
26. C22⋊(C167⋊C8)C22semidirect
32. C2×#41×C167C22semidirect
36. C22×D4×C167C22semidirect
62. #41×C167D4semidirect
76. C2×D4×C167D4semidirect
151. C167#3008semidirect
153. C2×H(5,𝔽2)D167semidirect
167. C2×#41C2×D167semidirect
172. C22×D4C2×D167semidirect
198. #41C22⋊D167semidirect
217. C2×D4C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288572
C22×#145819352
#249238332
#249234142
#249234032
#249232742
#249232652
#249232362
C2×H(5,𝔽2)×C1672
#3008167