Finite Group Databasebeta

← #36337278

Subgroups — #36337278

Order 21376.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#24924071C211yesmaximal, characteristic, complemented
2#24924070C211yesmaximal, characteristic, complemented
3#24923707C211yesmaximal, characteristic, complemented
4#24923706C212yesmaximal, complemented
5#24923703C212yesmaximal, complemented
6#24923700C212yesmaximal, complemented
7#24923699C211yesmaximal, characteristic, complemented
8C2×#19128773C211yesmaximal, characteristic, complemented
9D16×D167C212yesmaximal, complemented
10#24923687C211yesmaximal, characteristic, complemented
11#509×C167C211yesmaximal, characteristic, complemented
12#19128878—21no—
13#19128877—21no—
14#19128859—22no—
15#19128858—21no—
16#19128857—21no—
17#19128856—22no—
18#19128855C2212yescomplemented
19#19128854C2211yescharacteristic, complemented
20#19128851C2212yescomplemented
21C2×D1336C2211yescharacteristic, complemented
22D8×D167C2212yescomplemented
23D8×D167C2212yescomplemented
24#19128838C2211yescharacteristic
25#19128778C2211yescharacteristic, complemented
26#19128777C2212yescomplemented
27#19128776C2211yescharacteristic, complemented
28#19128775C2211yescharacteristic, complemented
29D2672C2212yescomplemented
30#19128773C2211yescharacteristic, complemented
31#19128773C2212yescomplemented
32#19128773C2212yescomplemented
33#19128771C2212yescomplemented
34C2×C167⋊C16C2211yescharacteristic, complemented
35C16×D167C2212yescomplemented
36#135×C167C2211yescharacteristic, complemented
37C2×D8×C167C2211yescharacteristic, complemented
38SD32×C167C2212yescomplemented
39D16×C167C2212yescomplemented
40M32×C167C2211yescharacteristic, complemented
41#14581951—22no—
42#14581950—21no—
43#14581950—41no—
44#14581950D411yescharacteristic
45#14581949—22no—
46#14581949—41no—
47C2×D668—21no—
48C2×D668—41no—
49C2×C22⋊D167—42no—
50D4×D167—22no—
51D4×D167—22no—
52D4×D167—41no—
53D4×D167—42no—
54Dic668C2311yescharacteristic
55#14581939—21no—
56#14581938—22no—
57#14581937—22no—
58D1336—21no—
59D1336C2311yescharacteristic
60D1336C2312yes—
61#14581935—22no—
62#14581935—22no—
63#14581934D411yescharacteristic
64#14581933D412yes—
65C8×D167C2312yes—
66C167⋊C16C2311yescharacteristic
67C167⋊C16C2311yescharacteristic
68#41×C167—21no—
69C2×D4×C167—21no—
70Q16×C167C2311yescharacteristic
71SD16×C167—21no—
72D8×C167—21no—
73D8×C167C2311yescharacteristic
74D8×C167C2312yes—
75C2×C8×C167C2311yescharacteristic
76C16×C167C2312yes—
77C22×D167—41no—
78C22×D167—42no—
79C22×D167—42no—
80Dic334—21no—
81Dic334D811yescharacteristic
82D668—21no—
83D668—22no—
84D668—42no—
85D668—41no—
86D668D811yescharacteristic
87C22⋊D167—41no—
88C22⋊D167—42no—
89C22⋊D167—42no—
90C22⋊D167—42no—
91C22⋊D167D812yes—
92C2×Dic167—42no—
93C4×D167—41no—
94C4×D167—42no—
95C4×D167C2×D412yes—
96C167⋊C8C2×D412yes—
97C23×C167—41no—
98Q8×C167—21no—
99D4×C167—22no—
100D4×C167—21no—
101D4×C167—41no—
102D4×C167—41no—
103C2×C4×C167—41no—
104C2×C4×C167C2×D411yescharacteristic
105C8×C167C2×D411yescharacteristic
106C8×C167C2411yescharacteristic
107C22×C167—41no—
108C22×C167—42no—
109C22×C167—161no—
110C22×C167C2×D811yescharacteristic
111C2×D167—41no—
112C2×D167—42no—
113C2×D167—162no—
114C2×D167—162no—
115C2×D167—161no—
116C2×D167C2×D812yes—
117C4×C167—41no—
118C4×C167C2×D811yescharacteristic
119C4×C167C22×D411yescharacteristic
120Dic167—41no—
121Dic167C2×D812yes—
122C2×C167—21no—
123C2×C167—81no—
124C2×C167—82no—
125C2×C167C22×D811yescharacteristic
126D167—22no—
127D167—81no—
128D167—82no—
129C167#341511yescharacteristic, complemented, Sylow
130#3415—1671nomaximal, Sylow
131#576—1671no—
132#576—1671no—
133#509—1672no—
134#509—1672no—
135#509—1671no—
136#509D16711yescharacteristic, complemented
137#508—1671no—
138#508—1672no—
139C2×D16—1671no—
140C2×D16—1672no—
141#504—1671no—
142H(5,𝔽2)—3341no—
143H(5,𝔽2)—3341no—
144AGL(1,ℤ/8)—3342no—
145AGL(1,ℤ/8)—3342no—
146AGL(1,ℤ/8)—3341no—
147AGL(1,ℤ/8)—3341no—
148#135—1672no—
149#135—1672no—
150#135—1671no—
151#135C2×D16711yescharacteristic, complemented
152C2×D8—1672no—
153C2×D8—1672no—
154C2×D8—1671no—
155C2×D8C2×D16711yescharacteristic, complemented
156#131—1671no—
157Q32—1671no—
158SD32—1671no—
159SD32—1671no—
160SD32—1672no—
161SD32C2×D16712yescomplemented
162D16—1672no—
163D16—1671no—
164D16—1672no—
165D16—1672no—
166D16C2×D16712yescomplemented
167M32—1672no—
168M32C2×D16711yescharacteristic, complemented
169C2×C16—1671no—
170C2×C16—1672no—
171#41—21no—
172#41—3342no—
173#41—3342no—
174#41—3341no—
175#41—6681no—
176#41—6681no—
177#41—1671no—
178C2×D4—21no—
179C2×D4—3342no—
180C2×D4—3342no—
181C2×D4—3341no—
182C2×D4—6681no—
183C2×D4—6682no—
184C2×D4—6681no—
185C2×D4—6682no—
186Q16—1671no—
187Q16C22×D16711yescharacteristic
188SD16—21no—
189SD16—3342no—
190SD16—3342no—
191SD16—3341no—
192D8—21no—
193D8—3342no—
194D8—3342no—
195D8—3341no—
196D8—1672no—
197D8—1671no—
198D8C22×D16712yes—
199D8C22×D16711yescharacteristic
200M16—1671no—
201M16—1672no—
202C2×C8—1672no—
203C2×C8C22×D16711yescharacteristic
204C16—1671no—
205C16—1671no—
206C16C22×D16712yes—
207C23—41no—
208C23—6682no—
209C23—6682no—
210C23—6681no—
211Q8—21no—
212Q8—3341no—
213Q8—1671no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—3341no—
219D4—3342no—
220D4—6682no—
221D4—6682no—
222D4—6681no—
223D4—6682no—
224D4—6682no—
225D4—6681no—
226D4—1672no—
227D4—1671no—
228C2×C4—41no—
229C2×C4—6682no—
230C2×C4—6682no—
231C2×C4—6681no—
232C2×C4—1672no—
233C2×C4D4×D16711yescharacteristic
234C8—1672no—
235C8D4×D16711yescharacteristic
236C8C23×D16711yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—26721no—
241C22—26722no—
242C22—26722no—
243C22—6682no—
244C22—6681no—
245C22—1672no—
246C22D8×D16711yescharacteristic
247C4—41no—
248C4—6681no—
249C4—1672no—
250C4D8×D16711yescharacteristic
251C4C2×D4×D16711yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—13362no—
256C2—13361no—
257C2—3342no—
258C2C2×D8×D16711yescharacteristic
subgroupG/H
centerC2C2×D8×D167
commutator, cosocleC8×C167C24
FrattiniC8C23×D167
Fitting#509×C167C2
socleC2×C167C22×D8
hypercenter#509D167
hypocenterC167#3415
LCSG ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #24924071C2semidirect
2. #24924070C2semidirect
3. #24923707C2semidirect
4. #24923706C2semidirect
5. #24923703C2semidirect
6. #24923700C2semidirect
7. #24923699C2semidirect
8. C2×#19128773C2semidirect
9. D16×D167C2semidirect
10. #24923687C2semidirect
11. #509×C167C2semidirect
18. #19128855C22semidirect
19. #19128854C22semidirect
20. #19128851C22semidirect
21. C2×D1336C22semidirect
22. D8×D167C22semidirect
23. D8×D167C22semidirect
25. #19128778C22semidirect
26. #19128777C22semidirect
27. #19128776C22semidirect
28. #19128775C22semidirect
29. D2672C22semidirect
30. #19128773C22semidirect
31. #19128773C22semidirect
32. #19128773C22semidirect
33. #19128771C22semidirect
34. C2×C167⋊C16C22semidirect
35. C16×D167C22semidirect
36. #135×C167C22semidirect
37. C2×D8×C167C22semidirect
38. SD32×C167C22semidirect
39. D16×C167C22semidirect
40. M32×C167C22semidirect
129. C167#3415semidirect
136. #509D167semidirect
151. #135C2×D167semidirect
155. C2×D8C2×D167semidirect
161. SD32C2×D167semidirect
166. D16C2×D167semidirect
168. M32C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249240712
#249240702
#249237072
#249237062
#249237032
#249237002
#249236992
C2×#191287732
D16×D1672
#249236872
#509×C1672
#3415167