Finite Group Databasebeta

← #36333380

Subgroups — #36333380

Order 21376.

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

normal subgroups only

This group has 758 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×#19128863C211yesmaximal, characteristic, complemented
2C2×#19128858C211yesmaximal, characteristic, complemented
3#24923780C211yesmaximal, characteristic, complemented
4#24923431C214yesmaximal, complemented
5#24923414C214yesmaximal, complemented
6#24923278C211yesmaximal, characteristic, complemented
7#24923274C211yesmaximal, characteristic, complemented
8#24923233C211yesmaximal, characteristic, complemented
9C22×#41×C167C211yesmaximal, characteristic, complemented
10C2×#14581950C2211yescharacteristic
11#19128863—24no—
12#19128858—24no—
13C2×#14581940C2212yescomplemented
14C2×#14581938C2212yescomplemented
15C2×#14581937C2212yescomplemented
16C2×#14581935C2212yescomplemented
17C2×#14581934C2211yescharacteristic
18C2×#14581934C2211yescharacteristic
19#19128815—21no—
20#19128811C2212yescomplemented
21#19128806C2212yescomplemented
22#19128801—21no—
23C4×C22⋊D167—22no—
24#19128791C2211yescharacteristic
25#19128756C2214yescomplemented
26#19128753C2214yescomplemented
27C22⋊(C167⋊C8)C2214yescomplemented
28C2×#41×C167—24no—
29C2×#41×C167—24no—
30C2×#41×C167—22no—
31C2×#41×C167C2214yes—
32C22×Q8×C167C2211yescharacteristic
33C22×D4×C167—21no—
34C22×D4×C167C2211yescharacteristic
35C23×C4×C167—21no—
36C23×C4×C167C2211yescharacteristic
37#14581950—22no—
38#14581950—22no—
39C2×Dic334C2311yescharacteristic
40C2×D668C2311yescharacteristic
41C2×C22⋊D167—21no—
42C2×C22⋊D167—22no—
43C2×C4×D167—21no—
44#14581940—24no—
45#14581938—24no—
46#14581937—24no—
47#14581935—24no—
48#14581934—22no—
49#14581934—22no—
50C2×C167⋊C8C2312yes—
51C2×C167⋊C8C2312yes—
52C4⋊Dic167C2312yes—
53Dic167⋊C4—22no—
54C22⋊Dic167—21no—
55C22⋊Dic167—22no—
56(C2×D167)⋊C4—22no—
57C4×Dic167—21no—
58C24×C167—21no—
59C24×C167D411yescharacteristic
60#41×C167—216no—
61#41×C167—28no—
62#41×C167—28no—
63C2×Q8×C167—22no—
64C2×Q8×C167—21no—
65C2×Q8×C167D414yescomplemented
66C2×Q8×C167C2312yes—
67C2×D4×C167—28no—
68C2×D4×C167—24no—
69C2×D4×C167—21no—
70C2×D4×C167—22no—
71C2×D4×C167D414yescomplemented
72C2×D4×C167C2312yes—
73C22×C4×C167—28no—
74C22×C4×C167—21no—
75C22×C4×C167—24no—
76C22×C4×C167—22no—
77C22×C4×C167—22no—
78C22×C4×C167D411yescharacteristic
79C22×C4×C167D412yes—
80C22×C4×C167C2312yes—
81C22×C4×C167C2311yescharacteristic
82C22×D167—21no—
83Dic334—21no—
84Dic334—21no—
85D668—21no—
86D668—21no—
87C22⋊D167—44no—
88C22⋊D167—42no—
89C2×Dic167—22no—
90C2×Dic167—21no—
91C4×D167—42no—
92C167⋊C8—22no—
93C167⋊C8—22no—
94C23×C167—22no—
95C23×C167—24no—
96C23×C167—22no—
97C23×C167—42no—
98C23×C167—44no—
99C23×C167C2×D412yes—
100C23×C167C2×D411yescharacteristic
101Q8×C167—24no—
102Q8×C167—22no—
103Q8×C167C2×D414yes—
104D4×C167—216no—
105D4×C167—24no—
106D4×C167—22no—
107D4×C167C2×D414yes—
108C2×C4×C167—216no—
109C2×C4×C167—22no—
110C2×C4×C167—24no—
111C2×C4×C167—22no—
112C2×C4×C167—24no—
113C2×C4×C167—22no—
114C2×C4×C167—22no—
115C2×C4×C167C2×D412yes—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2×D412yes—
118C2×C4×C167C2×D412yes—
119C2×C4×C167C2411yescharacteristic
120C22×C167—22no—
121C22×C167—22no—
122C22×C167—24no—
123C22×C167—22no—
124C22×C167—22no—
125C22×C167—44no—
126C22×C167—48no—
127C22×C167—44no—
128C22×C167#12012yes—
129C22×C167C22×D411yescharacteristic
130C2×D167—41no—
131C2×D167—42no—
132C4×C167—22no—
133C4×C167—24no—
134C4×C167#12012yes—
135C4×C167C22×D411yescharacteristic
136C4×C167C22×D411yescharacteristic
137Dic167—41no—
138Dic167—42no—
139C2×C167—22no—
140C2×C167—42no—
141C2×C167—44no—
142C2×C167C2×#12011yescharacteristic
143C2×C167#57512yes—
144D167—81no—
145C167#300511yescharacteristic, complemented, Sylow
146#3005—1671nomaximal, Sylow
147C22×#41D16711yescharacteristic, complemented
148C2×#137—1671no—
149C2×AGL(1,ℤ/8)—1671no—
150#525—1671no—
151#452—1674no—
152#449—1674no—
153#419—1671no—
154#418—1671no—
155#407—1671no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1671no—
160C2×#41C2×D16714yes—
161C22×Q8C2×D16711yescharacteristic
162C22×D4—21no—
163C22×D4C2×D16711yescharacteristic
164C23×C4—21no—
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—3341no—
175C22⋊Q8—1672no—
176#121—1672no—
177#120—3341no—
178C4×D4—3342no—
179#117—1671no—
180#103—1674no—
181#102—1674no—
182C22⋊C8—1674no—
183C24—21no—
184C24C22⋊D16711yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—3342no—
189#41—3342no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1671no—
193C2×Q8C22⋊D16714yescomplemented
194C2×Q8C22×D16712yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—3342no—
200C2×D4—3341no—
201C2×D4—1671no—
202C2×D4C22⋊D16714yescomplemented
203C2×D4C22×D16712yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—3341no—
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—3342no—
223C4⋊C4—1672no—
224C22⋊C4—3341no—
225C22⋊C4—3342no—
226C22⋊C4—3342no—
227C42—3341no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—42no—
232C23—44no—
233C23—3341no—
234C23C2×C22⋊D16712yes—
235C23C2×C22⋊D16711yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—3341no—
239Q8—3341no—
240Q8C2×C22⋊D16714yes—
241D4—216no—
242D4—22no—
243D4—24no—
244D4—3341no—
245D4—3341no—
246D4—6684no—
247D4—6682no—
248D4C2×C22⋊D16714yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—3342no—
257C2×C4—3341no—
258C2×C4—6682no—
259C2×C4C2×C22⋊D16712yes—
260C2×C4C2×C22⋊D16712yes—
261C2×C4C2×C22⋊D16712yes—
262C2×C4C2×C22⋊D16711yescharacteristic
263C2×C4C23×D16711yescharacteristic
264C8—3342no—
265C8—3342no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—6681no—
275C22—6682no—
276C22#1912880112yes—
277C22C22×C22⋊D16711yescharacteristic
278C4—22no—
279C4—24no—
280C4—6681no—
281C4—6682no—
282C4#1912880112yes—
283C4C22×C22⋊D16711yescharacteristic
284C4C22×C22⋊D16711yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—13361no—
289C2C2×#1912880111yescharacteristic
290C2#2492406712yes—
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC22×#41×C167C2
socleC22×C167C22×D4
hypercenterC22×#41D167
hypocenterC167#3005
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19128863C2semidirect
2. C2×#19128858C2semidirect
3. #24923780C2semidirect
4. #24923431C2semidirect
5. #24923414C2semidirect
6. #24923278C2semidirect
7. #24923274C2semidirect
8. #24923233C2semidirect
9. C22×#41×C167C2semidirect
13. C2×#14581940C22semidirect
14. C2×#14581938C22semidirect
15. C2×#14581937C22semidirect
16. C2×#14581935C22semidirect
20. #19128811C22semidirect
21. #19128806C22semidirect
25. #19128756C22semidirect
26. #19128753C22semidirect
27. C22⋊(C167⋊C8)C22semidirect
65. C2×Q8×C167D4semidirect
71. C2×D4×C167D4semidirect
145. C167#3005semidirect
147. C22×#41D167semidirect
193. C2×Q8C22⋊D167semidirect
202. C2×D4C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288632
C2×#191288582
#249237802
#249234312
#249234142
#249232782
#249232742
#249232332
C22×#41×C1672
#3005167