Finite Group Databasebeta

← #36335953

Subgroups — #36335953

Order 21376.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#24923924C211yesmaximal, characteristic, complemented
2D4×D668C211yesmaximal, characteristic, complemented
3#24923616C211yesmaximal, characteristic, complemented
4#24923584C212yesmaximal, complemented
5#24923549C211yesmaximal, characteristic, complemented
6#24923504C212yesmaximal, complemented
7#24923464C211yesmaximal, characteristic, complemented
8#24923401C212yesmaximal, complemented
9C8×D668C211yesmaximal, characteristic, complemented
10#102×D167C212yesmaximal, complemented
11C4×D8×C167C211yesmaximal, characteristic, complemented
12C2×#14581950—21no—
13C22×D668—21no—
14C2×D4×D167C2212yes—
15C2×#14581935C2212yescomplemented
16C2×C8×D167C2212yescomplemented
17#19128832C2211yescharacteristic, complemented
18#19128829C2211yescharacteristic
19#19128814—22no—
20#19128811—21no—
21#19128807C2212yes—
22#19128806—21no—
23#19128803C2212yescomplemented
24#19128802—22no—
25C4×D668C2211yescharacteristic
26C4⋊C4×D167C2212yescomplemented
27#19128765C2212yescomplemented
28C4⋊(C167⋊C8)C2211yescharacteristic, complemented
29#19128754C2212yescomplemented
30#19128754C2212yescomplemented
31#19128753C2212yes—
32#19128752C2212yescomplemented
33(C2×D167)⋊C8C2212yescomplemented
34C2×D8×C167C2211yescharacteristic
35C4×D4×C167C2211yescharacteristic
36C4×D4×C167C2211yescharacteristic, complemented
37#107×C167C2211yescharacteristic, complemented
38#102×C167C2211yescharacteristic
39#102×C167C2211yescharacteristic, complemented
40C4×C8×C167C2211yescharacteristic, complemented
41C23×D167—22no—
42#14581950—44no—
43C2×Dic334—21no—
44C2×D668—22no—
45C2×D668—21no—
46C2×D668—21no—
47C2×D668—44no—
48C2×D668D411yescharacteristic
49C2×D668C2312yes—
50C2×C22⋊D167—22no—
51C2×C22⋊D167—22no—
52D4×D167—24no—
53D4×D167—42no—
54C2×C4×D167—22no—
55C2×C4×D167C2312yes—
56#14581935—42no—
57C2×C167⋊C8C2312yes—
58C8×D167—42no—
59C4⋊Dic167—21no—
60C4⋊Dic167D411yescharacteristic
61C4⋊Dic167C2312yes—
62Dic167⋊C4—22no—
63C22⋊Dic167—22no—
64(C2×D167)⋊C4—22no—
65(C2×D167)⋊C4D412yes—
66C2×D4×C167C2311yescharacteristic
67C2×D4×C167C2311yescharacteristic
68C22×C4×C167—21no—
69C22×C4×C167—21no—
70D8×C167—22no—
71C2×C8×C167C2311yescharacteristic
72C2×C8×C167C2311yescharacteristic
73C4⋊C4×C167C2311yescharacteristic
74C4⋊C4×C167C2311yescharacteristic
75C22⋊C4×C167—21no—
76C22⋊C4×C167—21no—
77C42×C167C2311yescharacteristic
78C22×D167—22no—
79C22×D167—44no—
80C22×D167—44no—
81C22×D167—82no—
82C22×D167C2×D412yes—
83Dic334—42no—
84D668—22no—
85D668—22no—
86D668—44no—
87D668—42no—
88D668—81no—
89D668D814yescomplemented
90C22⋊D167—44no—
91C22⋊D167—44no—
92C2×Dic167—22no—
93C2×Dic167C2×D412yes—
94C4×D167—22no—
95C4×D167—22no—
96C4×D167—44no—
97C167⋊C8—22no—
98C23×C167—21no—
99C23×C167—21no—
100D4×C167—21no—
101D4×C167—21no—
102D4×C167—21no—
103D4×C167#4112yescomplemented
104C2×C4×C167—21no—
105C2×C4×C167—21no—
106C2×C4×C167—42no—
107C2×C4×C167—42no—
108C2×C4×C167C2×D411yescharacteristic
109C2×C4×C167C2×D411yescharacteristic
110C2×C4×C167C2411yescharacteristic
111C8×C167—21no—
112C8×C167—21no—
113C22×C167—22no—
114C22×C167—41no—
115C22×C167—41no—
116C22×C167—41no—
117C22×C167—41no—
118C22×C167—41no—
119C22×C167C22×D411yescharacteristic
120C2×D167—22no—
121C2×D167—22no—
122C2×D167—42no—
123C2×D167—42no—
124C2×D167—42no—
125C2×D167—88no—
126C2×D167C2×D814yes—
127C4×C167—21no—
128C4×C167—41no—
129C4×C167—41no—
130C4×C167C2×D812yes—
131C4×C167C2×#4111yescharacteristic
132C4×C167H(5,𝔽2)11yescharacteristic
133Dic167—22no—
134Dic167—42no—
135C2×C167—42no—
136C2×C167—81no—
137C2×C167#54611yescharacteristic
138C2×C167C22×D811yescharacteristic
139C2×C167#57711yescharacteristic
140D167—24no—
141D167—82no—
142C167#329411yescharacteristic, complemented, Sylow
143#3294—1671nomaximal, Sylow
144#547—1671no—
145D42—1671no—
146#486—1671no—
147#480—1672no—
148#474—1671no—
149#466—1672no—
150#459—1671no—
151#447—1672no—
152C4×D8D16711yescharacteristic, complemented
153C8×D4—1671no—
154C2×#102—1672no—
155C2×#41—3341no—
156C22×D4—3341no—
157C22×D4—1672no—
158C2×D8—1672no—
159C2×D8C2×D16711yescharacteristic
160C22×C8—1672no—
161C4⋊Q8—1671no—
162#127—1671no—
163#123—3342no—
164C22⋊Q8—3341no—
165#121—3341no—
166#121—1672no—
167#121—1672no—
168#120—3342no—
169C4×D4—1671no—
170C4×D4C2×D16711yescharacteristic
171C4×D4C2×D16711yescharacteristic, complemented
172C2×C4⋊C4—1672no—
173#107—1672no—
174#107C2×D16711yescharacteristic, complemented
175C4⋊C8—1671no—
176#102—1672no—
177#102—1672no—
178#102—1672no—
179#102—1672no—
180#102C2×D16711yescharacteristic
181#102C2×D16711yescharacteristic, complemented
182C22⋊C8—1672no—
183C4×C8C2×D16711yescharacteristic, complemented
184C24—3342no—
185#41—6684no—
186C2×Q8—3341no—
187C2×D4—3342no—
188C2×D4—3344no—
189C2×D4—3342no—
190C2×D4—3342no—
191C2×D4—3341no—
192C2×D4—3341no—
193C2×D4—6684no—
194C2×D4—6682no—
195C2×D4—1671no—
196C2×D4—1672no—
197C2×D4C22×D16711yescharacteristic
198C2×D4C22×D16711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—3342no—
202C22×C4—1672no—
203D8—22no—
204D8—6682no—
205C2×C8—6682no—
206C2×C8—1672no—
207C2×C8C22×D16711yescharacteristic
208C2×C8C22×D16711yescharacteristic
209C4⋊C4—3342no—
210C4⋊C4—3341no—
211C4⋊C4—1672no—
212C4⋊C4—1671no—
213C4⋊C4C22×D16711yescharacteristic
214C4⋊C4C22×D16711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—3342no—
218C22⋊C4—3342no—
219C22⋊C4—1672no—
220C42C22×D16711yescharacteristic
221C23—21no—
222C23—21no—
223C23—13362no—
224C23—3342no—
225C23—6684no—
226C23—6684no—
227C23—1672no—
228Q8—6682no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—13361no—
233D4—3342no—
234D4—3342no—
235D4—6684no—
236D4—6684no—
237D4—6684no—
238D4—6682no—
239D4—1674no—
240D4#1458195012yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—3342no—
246C2×C4—3342no—
247C2×C4—3342no—
248C2×C4—6684no—
249C2×C4—1672no—
250C2×C4D4×D16711yescharacteristic
251C2×C4D4×D16711yescharacteristic
252C2×C4C23×D16711yescharacteristic
253C8—21no—
254C8—21no—
255C8—3342no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—13368no—
263C22—3342no—
264C22—3342no—
265C22—6682no—
266C22—6682no—
267C22—6682no—
268C22—1674no—
269C22C2×D4×D16711yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—3342no—
274C4—6682no—
275C4D8×D16712yes—
276C4C2×#1458195011yescharacteristic
277C4#1912887811yescharacteristic
278C2—42no—
279C2—81no—
280C2—13362no—
281C2—3344no—
282C2#2492391511yescharacteristic
283C2C2×D8×D16711yescharacteristic
284C2#2492407611yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC4×D8×C167C2
socleC22×C167C22×D4
hypercenterC4×D8D167
hypocenterC167#3294
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #24923924C2semidirect
2. D4×D668C2semidirect
3. #24923616C2semidirect
4. #24923584C2semidirect
5. #24923549C2semidirect
6. #24923504C2semidirect
7. #24923464C2semidirect
8. #24923401C2semidirect
9. C8×D668C2semidirect
10. #102×D167C2semidirect
11. C4×D8×C167C2semidirect
15. C2×#14581935C22semidirect
16. C2×C8×D167C22semidirect
17. #19128832C22semidirect
23. #19128803C22semidirect
26. C4⋊C4×D167C22semidirect
27. #19128765C22semidirect
28. C4⋊(C167⋊C8)C22semidirect
29. #19128754C22semidirect
30. #19128754C22semidirect
32. #19128752C22semidirect
33. (C2×D167)⋊C8C22semidirect
36. C4×D4×C167C22semidirect
37. #107×C167C22semidirect
39. #102×C167C22semidirect
40. C4×C8×C167C22semidirect
89. D668D8semidirect
103. D4×C167#41semidirect
142. C167#3294semidirect
152. C4×D8D167semidirect
171. C4×D4C2×D167semidirect
174. #107C2×D167semidirect
181. #102C2×D167semidirect
183. C4×C8C2×D167semidirect
240. D4#14581950semidirect

Maximal subgroups

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

subgroupindex
#249239242
D4×D6682
#249236162
#249235842
#249235492
#249235042
#249234642
#249234012
C8×D6682
#102×D1672
C4×D8×C1672
#3294167