Finite Group Databasebeta

← #36334586

Subgroups — #36334586

Order 21376.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#19128860C211yesmaximal, characteristic, complemented
2C2×#19128858C211yesmaximal, characteristic, complemented
3C2×#19128857C211yesmaximal, characteristic, complemented
4#24923815C211yesmaximal, characteristic, complemented
5C2×#19128828C211yesmaximal, characteristic, complemented
6#24923657C212yesmaximal, complemented
7#24923655C212yesmaximal, complemented
8#24923644C212yesmaximal, complemented
9#24923640C212yesmaximal, complemented
10M16×Dic167C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C167C211yesmaximal, characteristic, complemented
12C2×#14581950C2211yescharacteristic
13C22×D668C2211yescharacteristic
14C22×C22⋊D167—22no—
15#19128860—24no—
16#19128858—24no—
17#19128857—24no—
18C2×#14581939C2212yescomplemented
19C2×#14581938C2212yescomplemented
20C2×#14581937C2212yescomplemented
21C2×D1336C2212yescomplemented
22C2×#14581935C2212yescomplemented
23C2×#14581935C2212yescomplemented
24C2×#14581934C2211yescharacteristic
25#19128830C2211yescharacteristic, complemented
26#19128828—22no—
27#19128828C2212yes—
28#19128828C2212yes—
29#19128828C2211yescharacteristic, complemented
30#19128821C2211yescharacteristic, complemented
31#19128819C2211yescharacteristic, complemented
32#19128804—22no—
33C4×C22⋊D167—22no—
34C2×C4×Dic167C2211yescharacteristic
35#19128742C2212yescomplemented
36C8×Dic167C2212yescomplemented
37C2×#41×C167C2211yescharacteristic
38C22×D4×C167C2211yescharacteristic
39AGL(1,ℤ/8)×C167—24no—
40C2×SD16×C167C2212yescomplemented
41C2×D8×C167C2212yescomplemented
42C2×M16×C167C2211yescharacteristic
43C23×D167—21no—
44#14581950—22no—
45#14581950—22no—
46C2×Dic334C2311yescharacteristic
47C2×D668—22no—
48C2×D668—22no—
49C2×D668—21no—
50C2×D668C2311yescharacteristic
51C2×D668C2312yes—
52C2×C22⋊D167—24no—
53C2×C22⋊D167—24no—
54C2×C22⋊D167—22no—
55C2×C22⋊D167—21no—
56C2×C22⋊D167—48no—
57C22×Dic167D412yes—
58C2×C4×D167—21no—
59#14581939—24no—
60#14581938—24no—
61#14581937—24no—
62D1336—24no—
63#14581935—24no—
64#14581935—24no—
65#14581934D414yescomplemented
66C2×C167⋊C8C2312yes—
67Dic167⋊C4—22no—
68C22⋊Dic167—22no—
69(C2×D167)⋊C4—22no—
70C4×Dic167D412yes—
71C4×Dic167C2311yescharacteristic
72C4×Dic167C2311yescharacteristic
73C24×C167—21no—
74#41×C167—22no—
75#41×C167—22no—
76C2×Q8×C167C2311yescharacteristic
77C2×D4×C167—21no—
78C2×D4×C167—22no—
79C2×D4×C167—22no—
80C2×D4×C167—21no—
81C2×D4×C167C2311yescharacteristic
82C2×D4×C167C2312yes—
83C22×C4×C167—21no—
84C22×C4×C167C2311yescharacteristic
85SD16×C167—24no—
86D8×C167—24no—
87M16×C167D414yescomplemented
88C2×C8×C167C2312yes—
89C22×D167—21no—
90C22×D167—22no—
91C22×D167—44no—
92C22×D167—42no—
93Dic334—21no—
94Dic334—21no—
95D668—21no—
96D668—22no—
97D668—21no—
98D668—22no—
99D668—42no—
100C22⋊D167—48no—
101C22⋊D167—42no—
102C22⋊D167—44no—
103C22⋊D167—48no—
104C2×Dic167—24no—
105C2×Dic167—21no—
106C2×Dic167C2×D412yes—
107C2×Dic167C2×D412yes—
108C4×D167—42no—
109C167⋊C8C2×D414yescomplemented
110C23×C167—21no—
111C23×C167—22no—
112C23×C167—44no—
113C23×C167—42no—
114C23×C167C2×D411yescharacteristic
115Q8×C167—21no—
116Q8×C167—21no—
117D4×C167—21no—
118D4×C167—22no—
119D4×C167—22no—
120D4×C167—21no—
121D4×C167—42no—
122D4×C167—42no—
123C2×C4×C167—21no—
124C2×C4×C167—42no—
125C2×C4×C167C2×D411yescharacteristic
126C2×C4×C167C2×D412yes—
127C2×C4×C167C2×D412yes—
128C2×C4×C167C2411yescharacteristic
129C8×C167C2×D414yescomplemented
130C22×C167—22no—
131C22×C167—41no—
132C22×C167—42no—
133C22×C167—44no—
134C22×C167—42no—
135C22×C167—84no—
136C22×C167#12712yes—
137C22×C167C22×D411yescharacteristic
138C2×D167—41no—
139C2×D167—42no—
140C2×D167—44no—
141C2×D167—42no—
142C2×D167—84no—
143C4×C167—41no—
144C4×C167#12712yes—
145C4×C167C22×D411yescharacteristic
146C4×C167C22×D411yescharacteristic
147Dic167—22no—
148Dic167—41no—
149Dic167AGL(1,ℤ/8)14yescomplemented
150C2×C167—22no—
151C2×C167—81no—
152C2×C167—82no—
153C2×C167C2×#12711yescharacteristic
154C2×C167C2×AGL(1,ℤ/8)12yes—
155D167—81no—
156D167—82no—
157C167#315111yescharacteristic, complemented, Sylow
158#3151—1671nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1671no—
160C2×AGL(1,ℤ/8)—1671no—
161C2×AGL(1,ℤ/8)—1671no—
162C2×AGL(1,ℤ/8)D16711yescharacteristic, complemented
163#532—1671no—
164C2×#127—1671no—
165#496—1672no—
166#496—1672no—
167#493—1672no—
168#492—1672no—
169C4×M16—1671no—
170C2×#41—1671no—
171C2×#41C2×D16711yescharacteristic
172C22×D4—3342no—
173C22×D4—1671no—
174C22×D4C2×D16711yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3344no—
177AGL(1,ℤ/8)—3344no—
178AGL(1,ℤ/8)—3344no—
179C2×SD16—1672no—
180C2×SD16—1672no—
181C2×SD16—1672no—
182C2×SD16C2×D16712yescomplemented
183C2×D8—1672no—
184C2×D8—1672no—
185C2×D8—1672no—
186C2×D8C2×D16712yescomplemented
187C2×M16—1671no—
188C2×M16C2×D16711yescharacteristic
189C4⋊Q8—1671no—
190#127—3342no—
191#127—1672no—
192#127—1672no—
193#127—1671no—
194#124—1671no—
195#124—1671no—
196#121—3342no—
197C4×D4—3342no—
198C2×C42—1671no—
199#97—1672no—
200C4×C8—1672no—
201C24—21no—
202C24—3341no—
203#41—22no—
204#41—22no—
205#41—3342no—
206#41—3342no—
207C2×Q8—1671no—
208C2×Q8C22×D16711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—3341no—
214C2×D4—3342no—
215C2×D4—3344no—
216C2×D4—3344no—
217C2×D4—3342no—
218C2×D4—3341no—
219C2×D4—3342no—
220C2×D4—6688no—
221C2×D4—1671no—
222C2×D4—1672no—
223C2×D4C22×D16711yescharacteristic
224C2×D4C22×D16712yes—
225C22×C4—21no—
226C22×C4—3341no—
227C22×C4—1672no—
228C22×C4C22×D16711yescharacteristic
229SD16—24no—
230SD16—3344no—
231SD16—3344no—
232SD16—3344no—
233D8—24no—
234D8—3344no—
235D8—3344no—
236D8—3344no—
237M16—1674no—
238M16C22⋊D16714yescomplemented
239C2×C8—1672no—
240C2×C8C22×D16712yes—
241C4⋊C4—3342no—
242C22⋊C4—3342no—
243C22⋊C4—3342no—
244C42—1672no—
245C42—1671no—
246C42—1671no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3341no—
252C23—3342no—
253C23—6684no—
254C23—6682no—
255C23D4×D16711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3341no—
259Q8—3341no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3341no—
267D4—3342no—
268D4—3341no—
269D4—3342no—
270D4—6688no—
271D4—6682no—
272D4—6688no—
273D4—6684no—
274D4—6682no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—3344no—
278C2×C4—3341no—
279C2×C4—6682no—
280C2×C4—1672no—
281C2×C4—1672no—
282C2×C4D4×D16712yes—
283C2×C4D4×D16711yescharacteristic
284C2×C4C2×C22⋊D16712yes—
285C2×C4C23×D16711yescharacteristic
286C8—1674no—
287C8C2×C22⋊D16714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—13364no—
295C22—6684no—
296C22—6682no—
297C22—6681no—
298C22—6682no—
299C22#1912882812yes—
300C22C2×D4×D16711yescharacteristic
301C4—41no—
302C4—3342no—
303C4—6681no—
304C4—1674no—
305C4#1912882812yes—
306C4C2×D4×D16711yescharacteristic
307C4C22×C22⋊D16711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—13362no—
312C2—13361no—
313C2C2×#1912882811yescharacteristic
314C2AGL(1,ℤ/8)×D16712yes—
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×AGL(1,ℤ/8)×C167C2
socleC22×C167C22×D4
hypercenterC2×AGL(1,ℤ/8)D167
hypocenterC167#3151
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19128860C2semidirect
2. C2×#19128858C2semidirect
3. C2×#19128857C2semidirect
4. #24923815C2semidirect
5. C2×#19128828C2semidirect
6. #24923657C2semidirect
7. #24923655C2semidirect
8. #24923644C2semidirect
9. #24923640C2semidirect
10. M16×Dic167C2semidirect
11. C2×AGL(1,ℤ/8)×C167C2semidirect
18. C2×#14581939C22semidirect
19. C2×#14581938C22semidirect
20. C2×#14581937C22semidirect
21. C2×D1336C22semidirect
22. C2×#14581935C22semidirect
23. C2×#14581935C22semidirect
25. #19128830C22semidirect
29. #19128828C22semidirect
30. #19128821C22semidirect
31. #19128819C22semidirect
35. #19128742C22semidirect
36. C8×Dic167C22semidirect
40. C2×SD16×C167C22semidirect
41. C2×D8×C167C22semidirect
65. #14581934D4semidirect
87. M16×C167D4semidirect
109. C167⋊C8C2×D4semidirect
129. C8×C167C2×D4semidirect
149. Dic167AGL(1,ℤ/8)semidirect
157. C167#3151semidirect
162. C2×AGL(1,ℤ/8)D167semidirect
182. C2×SD16C2×D167semidirect
186. C2×D8C2×D167semidirect
238. M16C22⋊D167semidirect
287. C8C2×C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288602
C2×#191288582
C2×#191288572
#249238152
C2×#191288282
#249236572
#249236552
#249236442
#249236402
M16×Dic1672
C2×AGL(1,ℤ/8)×C1672
#3151167