Finite Group Databasebeta

← #36335760

Subgroups — #36335760

Order 21376.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19128860C212yesmaximal, complemented
2#24923940C211yesmaximal, characteristic, complemented
3D4×D668C211yesmaximal, characteristic, complemented
4#24923646C211yesmaximal, characteristic, complemented
5#24923522C212yesmaximal, complemented
6#24923470C211yesmaximal, characteristic, complemented
7#24923465C211yesmaximal, characteristic, complemented
8#24923416C212yesmaximal, complemented
9#24923405C212yesmaximal, complemented
10C4×#14581939C211yesmaximal, characteristic, complemented
11#436×C167C211yesmaximal, characteristic, complemented
12C2×#14581950C2212yescomplemented
13C22×D668C2212yes—
14C2×D4×D167—21no—
15#19128860—28no—
16C2×#14581939C2211yescharacteristic
17C2×#14581939C2212yescomplemented
18C2×D1336C2212yes—
19C2×D1336C2212yescomplemented
20C2×D1336C2212yescomplemented
21#19128829C2211yescharacteristic
22#19128829C2211yescharacteristic, complemented
23#19128828—21no—
24#19128807—21no—
25#19128806C2212yes—
26#19128806C2212yescomplemented
27#19128805—22no—
28#19128802—22no—
29C4×Dic334C2211yescharacteristic, complemented
30C4×D668C2211yescharacteristic
31#19128794—21no—
32#19128764C2211yescharacteristic, complemented
33#19128757C2211yescharacteristic, complemented
34#19128754C2211yescharacteristic
35#19128754C2212yescomplemented
36#19128754C2212yescomplemented
37C2×M16×C167C2212yescomplemented
38C4×D4×C167C2211yescharacteristic
39C4⋊C8×C167C2211yescharacteristic, complemented
40C22⋊C8×C167C2212yescomplemented
41C4×C8×C167C2211yescharacteristic, complemented
42C23×D167—22no—
43#14581950—24no—
44#14581950—42no—
45C2×Dic334C2311yescharacteristic
46C2×D668—24no—
47C2×D668—21no—
48C2×D668—22no—
49C2×D668—21no—
50C2×D668—21no—
51C2×D668—21no—
52C2×D668—42no—
53C2×D668C2312yes—
54C2×D668C2312yes—
55C2×D668C2311yescharacteristic
56C2×C22⋊D167—22no—
57C2×C22⋊D167—22no—
58C2×C22⋊D167—22no—
59D4×D167—44no—
60C2×C4×D167—21no—
61C2×C4×D167—22no—
62#14581939—24no—
63#14581939D414yescomplemented
64D1336—24no—
65D1336—24no—
66D1336—42no—
67C4⋊Dic167C2311yescharacteristic
68C4⋊Dic167C2311yescharacteristic
69Dic167⋊C4—21no—
70(C2×D167)⋊C4—22no—
71(C2×D167)⋊C4—21no—
72(C2×D167)⋊C4—22no—
73C4×Dic167—21no—
74C2×D4×C167D411yescharacteristic
75C22×C4×C167C2312yes—
76M16×C167—24no—
77C2×C8×C167C2312yes—
78C2×C8×C167C2311yescharacteristic
79C2×C8×C167C2311yescharacteristic
80C4⋊C4×C167D411yescharacteristic
81C22⋊C4×C167D412yes—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—22no—
85C22×D167—22no—
86C22×D167—44no—
87C22×D167—44no—
88C22×D167—82no—
89Dic334—21no—
90Dic334C2×D412yescomplemented
91D668—22no—
92D668—22no—
93D668—21no—
94D668—22no—
95D668—22no—
96D668—44no—
97D668—42no—
98D668—42no—
99D668—81no—
100D668—81no—
101D668C2×D412yes—
102C22⋊D167—44no—
103C22⋊D167—44no—
104C22⋊D167—82no—
105C2×Dic167—21no—
106C2×Dic167—21no—
107C2×Dic167—21no—
108C4×D167—44no—
109C4×D167—42no—
110C23×C167C2×D412yes—
111D4×C167—22no—
112C2×C4×C167—22no—
113C2×C4×C167—22no—
114C2×C4×C167C2×D411yescharacteristic
115C2×C4×C167C2×D412yes—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2411yescharacteristic
118C8×C167—22no—
119C8×C167—21no—
120C8×C167C2×D412yescomplemented
121C22×C167—22no—
122C22×C167—22no—
123C22×C167—22no—
124C22×C167C22×D411yescharacteristic
125C2×D167—22no—
126C2×D167—41no—
127C2×D167—41no—
128C2×D167—42no—
129C2×D167—42no—
130C2×D167—42no—
131C2×D167—42no—
132C2×D167—42no—
133C2×D167—42no—
134C2×D167—88no—
135C4×C167—22no—
136C4×C167—21no—
137C4×C167AGL(1,ℤ/8)12yescomplemented
138C4×C167C22×D411yescharacteristic
139C4×C167H(5,𝔽2)11yescharacteristic
140Dic167—22no—
141Dic167—41no—
142Dic167—41no—
143C2×C167—42no—
144C2×C167D4211yescharacteristic
145C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C167#57611yescharacteristic
147D167—42no—
148D167—82no—
149D167—82no—
150C167#327411yescharacteristic, complemented, Sylow
151#3274—1671nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1672no—
153#550—1671no—
154D42—1671no—
155#493—1671no—
156#469—1672no—
157#460—1671no—
158#459—1671no—
159#449—1672no—
160#447—1672no—
161C4×SD16—1671no—
162#436D16711yescharacteristic, complemented
163C2×#41—1672no—
164C22×D4—3341no—
165C22×D4—1672no—
166AGL(1,ℤ/8)—3348no—
167C2×SD16—1672no—
168C2×SD16—1671no—
169C2×D8—1672no—
170C2×D8—1672no—
171C2×D8—1672no—
172C2×M16C2×D16712yescomplemented
173#127—3341no—
174#127—1671no—
175#127—1671no—
176#121—3341no—
177#121—3342no—
178#121—1672no—
179#121—1672no—
180#120—3342no—
181C4×Q8—1671no—
182C4×D4—3341no—
183C4×D4—1671no—
184C4×D4C2×D16711yescharacteristic
185#106—1671no—
186C4⋊C8C2×D16711yescharacteristic, complemented
187#103—1671no—
188#102—1671no—
189#102—1672no—
190#102—1672no—
191C22⋊C8C2×D16712yescomplemented
192C4×C8C2×D16711yescharacteristic, complemented
193C24—3342no—
194#41—3344no—
195#41—6682no—
196C2×Q8—1671no—
197C2×D4—3344no—
198C2×D4—3342no—
199C2×D4—3342no—
200C2×D4—3342no—
201C2×D4—3341no—
202C2×D4—3341no—
203C2×D4—3341no—
204C2×D4—3342no—
205C2×D4—3341no—
206C2×D4—6682no—
207C2×D4—6684no—
208C2×D4—1672no—
209C2×D4—1672no—
210C2×D4—1671no—
211C2×D4D66811yescharacteristic
212C22×C4—3342no—
213C22×C4—3341no—
214C22×C4C22×D16712yes—
215SD16—3344no—
216SD16—1674no—
217D8—3344no—
218D8—3344no—
219D8—6682no—
220M16—24no—
221C2×C8C22×D16711yescharacteristic
222C2×C8C22×D16712yes—
223C2×C8C22×D16711yescharacteristic
224C4⋊C4—3341no—
225C4⋊C4—1671no—
226C4⋊C4—1671no—
227C4⋊C4D66811yescharacteristic
228C22⋊C4—3342no—
229C22⋊C4—3341no—
230C22⋊C4—3342no—
231C22⋊C4D66812yes—
232C42—3341no—
233C42C22×D16711yescharacteristic
234C23—13362no—
235C23—3341no—
236C23—3342no—
237C23—3342no—
238C23—6684no—
239C23—6684no—
240C23C2×D66812yes—
241Q8—3341no—
242Q8—1672no—
243D4—22no—
244D4—13361no—
245D4—13361no—
246D4—13362no—
247D4—3342no—
248D4—3342no—
249D4—3341no—
250D4—3342no—
251D4—3342no—
252D4—6684no—
253D4—6684no—
254D4—6684no—
255D4—6682no—
256D4—6682no—
257D4—1672no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3341no—
261C2×C4—3341no—
262C2×C4—3341no—
263C2×C4—6684no—
264C2×C4—6682no—
265C2×C4C2×D66811yescharacteristic
266C2×C4C2×D66812yes—
267C2×C4C2×D66811yescharacteristic
268C2×C4C23×D16711yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D16712yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—13368no—
276C22—3342no—
277C22—6682no—
278C22—6682no—
279C22—6682no—
280C22—6682no—
281C22—6682no—
282C22—6682no—
283C22—6681no—
284C22—6681no—
285C22C22×D66811yescharacteristic
286C4—22no—
287C4—21no—
288C4—3342no—
289C4—6681no—
290C4—6681no—
291C4#1912886012yescomplemented
292C4C2×D4×D16711yescharacteristic
293C4#1912887811yescharacteristic
294C2—42no—
295C2—13362no—
296C2—13362no—
297C2—6682no—
298C2D4×D66811yescharacteristic
299C2C2×#1912886011yescharacteristic
300C2#2492407411yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#436×C167C2
socleC22×C167C22×D4
hypercenter#436D167
hypocenterC167#3274
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19128860C2semidirect
2. #24923940C2semidirect
3. D4×D668C2semidirect
4. #24923646C2semidirect
5. #24923522C2semidirect
6. #24923470C2semidirect
7. #24923465C2semidirect
8. #24923416C2semidirect
9. #24923405C2semidirect
10. C4×#14581939C2semidirect
11. #436×C167C2semidirect
12. C2×#14581950C22semidirect
17. C2×#14581939C22semidirect
19. C2×D1336C22semidirect
20. C2×D1336C22semidirect
22. #19128829C22semidirect
26. #19128806C22semidirect
29. C4×Dic334C22semidirect
32. #19128764C22semidirect
33. #19128757C22semidirect
35. #19128754C22semidirect
36. #19128754C22semidirect
37. C2×M16×C167C22semidirect
39. C4⋊C8×C167C22semidirect
40. C22⋊C8×C167C22semidirect
41. C4×C8×C167C22semidirect
63. #14581939D4semidirect
90. Dic334C2×D4semidirect
120. C8×C167C2×D4semidirect
137. C4×C167AGL(1,ℤ/8)semidirect
150. C167#3274semidirect
162. #436D167semidirect
172. C2×M16C2×D167semidirect
186. C4⋊C8C2×D167semidirect
191. C22⋊C8C2×D167semidirect
192. C4×C8C2×D167semidirect
271. C8D4×D167semidirect
291. C4#19128860semidirect

Maximal subgroups

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

subgroupindex
C2×#191288602
#249239402
D4×D6682
#249236462
#249235222
#249234702
#249234652
#249234162
#249234052
C4×#145819392
#436×C1672
#3274167