Finite Group Databasebeta

← #36333378

Subgroups — #36333378

Order 21376.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581950C211yesmaximal, characteristic, complemented
2C2×#19128862C211yesmaximal, characteristic, complemented
3C2×#19128857C211yesmaximal, characteristic, complemented
4#24923429C214yesmaximal, complemented
5#24923412C214yesmaximal, complemented
6#24923277C211yesmaximal, characteristic, complemented
7#24923273C211yesmaximal, characteristic, complemented
8#24923233C211yesmaximal, characteristic, complemented
9#525×C167C211yesmaximal, characteristic, complemented
10C2×#14581950—24no—
11C2×#14581950—24no—
12C2×#14581950—22no—
13C2×#14581950C2214yes—
14C22×Dic334C2211yescharacteristic
15C22×D668C2211yescharacteristic
16C22×C22⋊D167—21no—
17C22×C4×D167—21no—
18#19128862—24no—
19#19128857—24no—
20C2×#14581940C2212yescomplemented
21C2×#14581938C2212yescomplemented
22C2×#14581937C2212yescomplemented
23C2×#14581935C2212yescomplemented
24C2×#14581934C2211yescharacteristic
25C2×#14581934C2211yescharacteristic
26#19128755C2214yescomplemented
27#19128752C2214yescomplemented
28C22⋊(C167⋊C8)C2214yescomplemented
29C2×#41×C167C2211yescharacteristic
30C23×C4×C167C2211yescharacteristic
31#123×C167—21no—
32C22⋊Q8×C167C2212yescomplemented
33#121×C167C2212yescomplemented
34#120×C167—21no—
35C4×D4×C167—22no—
36#117×C167C2211yescharacteristic
37C23×D167—21no—
38#14581950—216no—
39#14581950—28no—
40#14581950—28no—
41C2×Dic334—22no—
42C2×Dic334—21no—
43C2×Dic334D414yescomplemented
44C2×Dic334C2312yes—
45C2×D668—22no—
46C2×D668—21no—
47C2×D668D414yescomplemented
48C2×D668C2312yes—
49C2×C22⋊D167—28no—
50C2×C22⋊D167—24no—
51C22×Dic167—21no—
52C2×C4×D167—28no—
53C2×C4×D167—24no—
54#14581940—24no—
55#14581938—24no—
56#14581937—24no—
57#14581935—24no—
58#14581934—22no—
59#14581934—22no—
60C2×C167⋊C8C2312yes—
61C2×C167⋊C8C2312yes—
62C24×C167D411yescharacteristic
63#41×C167—22no—
64#41×C167—22no—
65C2×Q8×C167C2311yescharacteristic
66C2×D4×C167—22no—
67C2×D4×C167—21no—
68C2×D4×C167C2311yescharacteristic
69C22×C4×C167—22no—
70C22×C4×C167—22no—
71C22×C4×C167—21no—
72C22×C4×C167D411yescharacteristic
73C22×C4×C167D412yes—
74C22×C4×C167C2312yes—
75C22×C4×C167C2311yescharacteristic
76C4⋊C4×C167—22no—
77C4⋊C4×C167C2312yes—
78C22⋊C4×C167—21no—
79C22⋊C4×C167—22no—
80C22⋊C4×C167—22no—
81C42×C167—21no—
82C22×D167—24no—
83C22×D167—22no—
84C22×D167—44no—
85Dic334—24no—
86Dic334—22no—
87Dic334C2×D414yes—
88D668—24no—
89D668—22no—
90D668C2×D414yes—
91C22⋊D167—216no—
92C2×Dic167—24no—
93C2×Dic167—22no—
94C4×D167—216no—
95C167⋊C8—22no—
96C167⋊C8—22no—
97C23×C167—22no—
98C23×C167—21no—
99C23×C167—42no—
100C23×C167C2×D412yes—
101C23×C167C2×D411yescharacteristic
102Q8×C167—21no—
103Q8×C167—21no—
104D4×C167—21no—
105D4×C167—21no—
106D4×C167—44no—
107D4×C167—42no—
108C2×C4×C167—22no—
109C2×C4×C167—24no—
110C2×C4×C167—22no—
111C2×C4×C167—22no—
112C2×C4×C167—22no—
113C2×C4×C167—21no—
114C2×C4×C167—42no—
115C2×C4×C167C2×D412yes—
116C2×C4×C167C2×D412yes—
117C2×C4×C167C2×D412yes—
118C2×C4×C167C2×D411yescharacteristic
119C2×C4×C167C2411yescharacteristic
120C22×C167—22no—
121C22×C167—22no—
122C22×C167—22no—
123C22×C167—22no—
124C22×C167—44no—
125C22×C167—41no—
126C22×C167—42no—
127C22×C167#12012yes—
128C22×C167C22×D411yescharacteristic
129C2×D167—24no—
130C2×D167—48no—
131C2×D167—44no—
132C4×C167—22no—
133C4×C167—41no—
134C4×C167—42no—
135C4×C167#12012yes—
136C4×C167C22×D411yescharacteristic
137C4×C167C22×D411yescharacteristic
138Dic167—24no—
139C2×C167—22no—
140C2×C167—42no—
141C2×C167—81no—
142C2×C167C2×#12011yescharacteristic
143C2×C167#57512yes—
144D167—44no—
145C167#300511yescharacteristic, complemented, Sylow
146#3005—1671nomaximal, Sylow
147C22×#41—1671no—
148C2×#137—1671no—
149C2×AGL(1,ℤ/8)—1671no—
150#525D16711yescharacteristic, complemented
151#452—1674no—
152#449—1674no—
153#419—1671no—
154#418—1671no—
155#407—1671no—
156C2×#41—3344no—
157C2×#41—3344no—
158C2×#41—3342no—
159C2×#41—1674no—
160C2×#41C2×D16711yescharacteristic
161C22×Q8—1671no—
162C22×D4—3341no—
163C22×D4—1671no—
164C23×C4—3341no—
165C23×C4C2×D16711yescharacteristic
166#137—3344no—
167AGL(1,ℤ/8)—3344no—
168C2×Q16—1672no—
169C2×SD16—1672no—
170C2×SD16—1672no—
171C2×D8—1672no—
172C2×M16—1671no—
173C2×M16—1671no—
174#123—21no—
175C22⋊Q8C2×D16712yescomplemented
176#121C2×D16712yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D16711yescharacteristic
180#103—1674no—
181#102—1674no—
182C22⋊C8—1674no—
183C24—3341no—
184C24C22⋊D16711yescharacteristic
185#41—22no—
186#41—22no—
187#41—33416no—
188#41—3348no—
189#41—3348no—
190C2×Q8—3342no—
191C2×Q8—3341no—
192C2×Q8—1674no—
193C2×Q8—1672no—
194C2×Q8C22×D16711yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—3348no—
198C2×D4—3344no—
199C2×D4—3342no—
200C2×D4—3341no—
201C2×D4—1674no—
202C2×D4—1672no—
203C2×D4C22×D16711yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—3348no—
208C22×C4—3341no—
209C22×C4—3344no—
210C22×C4C22⋊D16712yes—
211C22×C4C22⋊D16711yescharacteristic
212C22×C4C22×D16712yes—
213C22×C4C22×D16711yescharacteristic
214Q16—3344no—
215SD16—3344no—
216SD16—3344no—
217D8—3344no—
218M16—3342no—
219M16—3342no—
220C2×C8—1672no—
221C2×C8—1672no—
222C4⋊C4—22no—
223C4⋊C4C22×D16712yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3342no—
232C23—3344no—
233C23—6684no—
234C23C2×C22⋊D16712yes—
235C23C2×C22⋊D16711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—3342no—
239Q8—3344no—
240Q8—1674no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—33416no—
246D4—3342no—
247D4—3344no—
248D4—1674no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—33416no—
257C2×C4—3342no—
258C2×C4—3344no—
259C2×C4D4×D16712yes—
260C2×C4D4×D16712yes—
261C2×C4C2×C22⋊D16712yes—
262C2×C4C2×C22⋊D16711yescharacteristic
263C2×C4C23×D16711yescharacteristic
264C8—3342no—
265C8—3342no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—3344no—
274C22—6688no—
275C22—6684no—
276C22#1912880012yes—
277C22C22×C22⋊D16711yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—3344no—
282C4#1912880012yes—
283C4C2×D4×D16711yescharacteristic
284C4C2×D4×D16711yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—6684no—
289C2C2×#1912880011yescharacteristic
290C2#2492406712yes—
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#525×C167C2
socleC22×C167C22×D4
hypercenter#525D167
hypocenterC167#3005
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#14581950C2semidirect
2. C2×#19128862C2semidirect
3. C2×#19128857C2semidirect
4. #24923429C2semidirect
5. #24923412C2semidirect
6. #24923277C2semidirect
7. #24923273C2semidirect
8. #24923233C2semidirect
9. #525×C167C2semidirect
20. C2×#14581940C22semidirect
21. C2×#14581938C22semidirect
22. C2×#14581937C22semidirect
23. C2×#14581935C22semidirect
26. #19128755C22semidirect
27. #19128752C22semidirect
28. C22⋊(C167⋊C8)C22semidirect
32. C22⋊Q8×C167C22semidirect
33. #121×C167C22semidirect
43. C2×Dic334D4semidirect
47. C2×D668D4semidirect
145. C167#3005semidirect
150. #525D167semidirect
175. C22⋊Q8C2×D167semidirect
176. #121C2×D167semidirect

Maximal subgroups

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

subgroupindex
C22×#145819502
C2×#191288622
C2×#191288572
#249234292
#249234122
#249232772
#249232732
#249232332
#525×C1672
#3005167