Finite Group Databasebeta

← #36335817

Subgroups — #36335817

Order 21376.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×D8×D167C211yesmaximal, characteristic, complemented
2#24923907C212yesmaximal, complemented
3#24923654C211yesmaximal, characteristic, complemented
4#24923507C211yesmaximal, characteristic, complemented
5#24923505C211yesmaximal, characteristic, complemented
6#24923486C212yesmaximal, complemented
7#24923418C212yesmaximal, complemented
8#24923414C212yesmaximal, complemented
9D8×Dic167C211yesmaximal, characteristic, complemented
10C8×C22⋊D167C211yesmaximal, characteristic, complemented
11C2×#135×C167C211yesmaximal, characteristic, complemented
12C2×D4×D167C2212yes—
13C2×#14581937C2212yescomplemented
14C2×D1336C2211yescharacteristic
15C2×#14581935C2212yes—
16D8×D167—24no—
17C2×C8×D167C2211yescharacteristic
18#19128821C2212yescomplemented
19#19128815—22no—
20#19128808C2212yes—
21#19128806C2212yescomplemented
22#19128804—22no—
23#19128800—22no—
24D4×Dic167C2212yescomplemented
25C4×C22⋊D167C2211yescharacteristic
26C2×(C2×D167)⋊C4—22no—
27#19128766C2211yescharacteristic, complemented
28Dic167⋊C8C2211yescharacteristic, complemented
29#19128756C2212yescomplemented
30#19128754C2212yescomplemented
31#19128753C2212yescomplemented
32C22⋊(C167⋊C8)C2211yescharacteristic, complemented
33(C2×D167)⋊C8C2211yescharacteristic
34C8×Dic167C2211yescharacteristic, complemented
35C2×#41×C167C2212yescomplemented
36#135×C167—24no—
37C2×Q16×C167C2211yescharacteristic, complemented
38C2×SD16×C167C2212yescomplemented
39C2×D8×C167C2211yescharacteristic
40C22×C8×C167C2211yescharacteristic, complemented
41C23×D167—22no—
42C2×D668C2312yes—
43C2×C22⋊D167—22no—
44C2×C22⋊D167—22no—
45C2×C22⋊D167D411yescharacteristic
46D4×D167—24no—
47D4×D167—42no—
48C22×Dic167—22no—
49C2×C4×D167C2311yescharacteristic
50#14581937—42no—
51D1336—22no—
52#14581935—24no—
53C2×C167⋊C8C2311yescharacteristic
54C2×C167⋊C8C2311yescharacteristic
55C8×D167—22no—
56C4⋊Dic167C2312yes—
57Dic167⋊C4—22no—
58Dic167⋊C4D411yescharacteristic
59C22⋊Dic167—22no—
60C22⋊Dic167D411yescharacteristic
61(C2×D167)⋊C4—22no—
62(C2×D167)⋊C4—22no—
63(C2×D167)⋊C4—22no—
64(C2×D167)⋊C4—22no—
65(C2×D167)⋊C4D411yescharacteristic
66C4×Dic167C2311yescharacteristic
67#41×C167—24no—
68#41×C167—42no—
69C2×Q8×C167C2312yes—
70C2×D4×C167—22no—
71C2×D4×C167C2312yes—
72C22×C4×C167—22no—
73C22×C4×C167C2311yescharacteristic
74Q16×C167—22no—
75SD16×C167—24no—
76D8×C167D414yescomplemented
77C2×C8×C167—22no—
78C2×C8×C167C2311yescharacteristic
79C2×C8×C167C2311yescharacteristic
80C22×D167—22no—
81C22×D167—44no—
82C22×D167—44no—
83C22×D167—82no—
84C22×D167C2×D411yescharacteristic
85D668—22no—
86D668—22no—
87C22⋊D167—22no—
88C22⋊D167—44no—
89C22⋊D167—82no—
90C2×Dic167—22no—
91C2×Dic167—82no—
92C2×Dic167C2×D411yescharacteristic
93C2×Dic167C2×D411yescharacteristic
94C2×Dic167C2×D411yescharacteristic
95C4×D167—21no—
96C4×D167—21no—
97C167⋊C8—21no—
98C167⋊C8—21no—
99C23×C167—22no—
100C23×C167C2×D411yescharacteristic
101Q8×C167—22no—
102Q8×C167—22no—
103D4×C167—22no—
104D4×C167—44no—
105D4×C167C2×D414yes—
106C2×C4×C167—21no—
107C2×C4×C167—21no—
108C2×C4×C167—22no—
109C2×C4×C167—44no—
110C2×C4×C167C2×D411yescharacteristic
111C2×C4×C167C2411yescharacteristic
112C8×C167—21no—
113C8×C167C2×D412yes—
114C22×C167—21no—
115C22×C167—24no—
116C22×C167—21no—
117C22×C167—21no—
118C22×C167—42no—
119C22×C167—42no—
120C22×C167C22×D411yescharacteristic
121C2×D167—21no—
122C2×D167—21no—
123C2×D167—42no—
124C2×D167—42no—
125C2×D167—42no—
126C2×D167—88no—
127C2×D167#13512yescomplemented
128C4×C167—21no—
129C4×C167—42no—
130C4×C167C22×D411yescharacteristic
131C4×C167H(5,𝔽2)11yescharacteristic
132Dic167—21no—
133Dic167—21no—
134Dic167—21no—
135Dic167—42no—
136C2×C167—44no—
137C2×C167—41no—
138C2×C167D4211yescharacteristic
139C2×C167C2×#13511yescharacteristic
140C2×C167#57611yescharacteristic
141D167—22no—
142D167—82no—
143C167#328011yescharacteristic, complemented, Sylow
144#3280—1671nomaximal, Sylow
145C2×#135D16711yescharacteristic, complemented
146C22×D8—1671no—
147#546—1672no—
148#495—1671no—
149#467—1671no—
150#466—1671no—
151#463—1672no—
152#450—1672no—
153#449—1672no—
154C4×D8—1671no—
155C8×D4—1671no—
156C2×#41C2×D16712yescomplemented
157C22×D4—1672no—
158#135—24no—
159C2×Q16C2×D16711yescharacteristic, complemented
160C2×SD16—1672no—
161C2×SD16C2×D16712yescomplemented
162C2×D8—3344no—
163C2×D8—1672no—
164C2×D8—1671no—
165C2×D8C2×D16711yescharacteristic
166C22×C8—1671no—
167C22×C8C2×D16711yescharacteristic, complemented
168#124—1672no—
169#123—3342no—
170C22⋊Q8—1672no—
171#121—3342no—
172#121—1672no—
173#120—3342no—
174C4×D4—1672no—
175C4×D4—1671no—
176C2×C22⋊C4—3342no—
177#107—1671no—
178C4⋊C8—1671no—
179#103—1672no—
180#102—1672no—
181#102—1672no—
182C22⋊C8—1671no—
183C22⋊C8—1671no—
184C4×C8—1671no—
185C24—3342no—
186#41—24no—
187#41—42no—
188C2×Q8C22×D16712yes—
189C2×D4—22no—
190C2×D4—3342no—
191C2×D4—3344no—
192C2×D4—3342no—
193C2×D4—6682no—
194C2×D4—1671no—
195C2×D4—1672no—
196C2×D4C22×D16712yes—
197C22×C4—22no—
198C22×C4—3342no—
199C22×C4—1671no—
200C22×C4C22×D16711yescharacteristic
201Q16—22no—
202SD16—24no—
203SD16—6682no—
204D8—3344no—
205D8—3342no—
206D8C22⋊D16714yescomplemented
207C2×C8—22no—
208C2×C8—3342no—
209C2×C8—1671no—
210C2×C8—1671no—
211C2×C8C22×D16711yescharacteristic
212C2×C8C22×D16711yescharacteristic
213C4⋊C4—3342no—
214C4⋊C4—1672no—
215C4⋊C4—1671no—
216C22⋊C4—3342no—
217C22⋊C4—3342no—
218C22⋊C4—3342no—
219C22⋊C4—3342no—
220C22⋊C4—3342no—
221C22⋊C4—1671no—
222C22⋊C4—1671no—
223C42—1671no—
224C23—22no—
225C23—13362no—
226C23—3342no—
227C23—6684no—
228C23—6684no—
229C23—1671no—
230C23D4×D16711yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—44no—
235D4—13362no—
236D4—3342no—
237D4—3342no—
238D4—3342no—
239D4—6684no—
240D4C2×C22⋊D16714yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—13362no—
246C2×C4—3341no—
247C2×C4—3341no—
248C2×C4—3342no—
249C2×C4—1671no—
250C2×C4—1671no—
251C2×C4—1671no—
252C2×C4D4×D16711yescharacteristic
253C2×C4C23×D16711yescharacteristic
254C8—21no—
255C8—3341no—
256C8—3341no—
257C8C2×C22⋊D16712yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22—13368no—
265C22—3341no—
266C22—3341no—
267C22—6682no—
268C22—6682no—
269C22—6682no—
270C22—1672no—
271C22C2×D4×D16711yescharacteristic
272C4—21no—
273C4—42no—
274C4—3341no—
275C4—3341no—
276C4—3341no—
277C4—6682no—
278C4C22×C22⋊D16711yescharacteristic
279C4#1912887711yescharacteristic
280C2—41no—
281C2—44no—
282C2—13362no—
283C2—3342no—
284C2D4×C22⋊D16711yescharacteristic
285C2#135×D16711yescharacteristic
286C2#2492407111yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×#135×C167C2
socleC22×C167C22×D4
hypercenterC2×#135D167
hypocenterC167#3280
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×D8×D167C2semidirect
2. #24923907C2semidirect
3. #24923654C2semidirect
4. #24923507C2semidirect
5. #24923505C2semidirect
6. #24923486C2semidirect
7. #24923418C2semidirect
8. #24923414C2semidirect
9. D8×Dic167C2semidirect
10. C8×C22⋊D167C2semidirect
11. C2×#135×C167C2semidirect
13. C2×#14581937C22semidirect
18. #19128821C22semidirect
21. #19128806C22semidirect
24. D4×Dic167C22semidirect
27. #19128766C22semidirect
28. Dic167⋊C8C22semidirect
29. #19128756C22semidirect
30. #19128754C22semidirect
31. #19128753C22semidirect
32. C22⋊(C167⋊C8)C22semidirect
34. C8×Dic167C22semidirect
35. C2×#41×C167C22semidirect
37. C2×Q16×C167C22semidirect
38. C2×SD16×C167C22semidirect
40. C22×C8×C167C22semidirect
76. D8×C167D4semidirect
127. C2×D167#135semidirect
143. C167#3280semidirect
145. C2×#135D167semidirect
156. C2×#41C2×D167semidirect
159. C2×Q16C2×D167semidirect
161. C2×SD16C2×D167semidirect
167. C22×C8C2×D167semidirect
206. D8C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1672
#249239072
#249236542
#249235072
#249235052
#249234862
#249234182
#249234142
D8×Dic1672
C8×C22⋊D1672
C2×#135×C1672
#3280167