Finite Group Databasebeta

← #36337281

Subgroups — #36337281

Order 21376.

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

normal subgroups only

This group has 360 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#24924083C211yesmaximal, characteristic, complemented
2#24924071C211yesmaximal, characteristic, complemented
3#24923711C212yesmaximal, complemented
4#24923709C211yesmaximal, characteristic, complemented
5#24923704C211yesmaximal, characteristic, complemented
6#24923703C212yesmaximal, complemented
7#24923700C212yesmaximal, complemented
8C2×#19128777C211yesmaximal, characteristic, complemented
9SD32×D167C212yesmaximal, complemented
10#24923685C211yesmaximal, characteristic, complemented
11#508×C167C211yesmaximal, characteristic, complemented
12#19128879—21no—
13#19128878—21no—
14#19128862—21no—
15#19128861—22no—
16#19128859—22no—
17#19128857—21no—
18#19128855C2212yes—
19#19128851C2212yescomplemented
20C2×Dic668C2211yescharacteristic, complemented
21Q16×D167C2212yescomplemented
22C2×D1336C2211yescharacteristic
23D8×D167C2212yes—
24#19128838C2211yescharacteristic
25#19128778C2212yescomplemented
26#19128777C2212yescomplemented
27#19128777C2212yescomplemented
28#19128776C2212yescomplemented
29#19128775C2212yescomplemented
30#19128773C2212yescomplemented
31#19128772C2211yescharacteristic, complemented
32#19128771C2212yescomplemented
33C16×D167C2212yescomplemented
34#135×C167C2211yescharacteristic
35#135×C167C2211yescharacteristic, complemented
36Q32×C167C2211yescharacteristic, complemented
37SD32×C167C2211yescharacteristic, complemented
38SD32×C167C2211yescharacteristic, complemented
39D16×C167C2211yescharacteristic, complemented
40C2×C16×C167C2211yescharacteristic, complemented
41#14581951—22no—
42#14581950—41no—
43#14581950—41no—
44#14581950D411yescharacteristic
45#14581949—22no—
46#14581949—42no—
47C2×Dic334—21no—
48C2×Dic334—41no—
49Q8×D167—22no—
50C2×D668—21no—
51C2×D668—41no—
52D4×D167—22no—
53D4×D167—42no—
54Dic668—21no—
55Dic668C2312yes—
56#14581940—22no—
57#14581938—22no—
58#14581937—22no—
59D1336—21no—
60D1336C2312yes—
61#14581935—22no—
62#14581934D411yescharacteristic
63#14581933D412yes—
64C8×D167C2312yes—
65C167⋊C16C2312yes—
66#41×C167—21no—
67#41×C167—21no—
68Q16×C167C2311yescharacteristic
69Q16×C167C2311yescharacteristic
70SD16×C167—21no—
71SD16×C167—21no—
72D8×C167C2311yescharacteristic
73D8×C167C2311yescharacteristic
74C2×C8×C167C2311yescharacteristic
75C16×C167C2311yescharacteristic
76C16×C167C2311yescharacteristic
77C22×D167—41no—
78C22×D167—42no—
79Dic334—22no—
80Dic334—41no—
81Dic334—42no—
82Dic334D811yescharacteristic
83D668—22no—
84D668—42no—
85D668—41no—
86D668D811yescharacteristic
87C22⋊D167—42no—
88C22⋊D167—42no—
89C22⋊D167D812yes—
90C2×Dic167—41no—
91C2×Dic167—42no—
92C4×D167—42no—
93C4×D167—42no—
94C4×D167C2×D412yes—
95C167⋊C8C2×D412yes—
96Q8×C167—21no—
97Q8×C167—21no—
98D4×C167—21no—
99D4×C167—21no—
100D4×C167—41no—
101D4×C167—41no—
102C2×C4×C167—41no—
103C2×C4×C167—41no—
104C2×C4×C167C2×D411yescharacteristic
105C8×C167C2×D411yescharacteristic
106C8×C167C2411yescharacteristic
107C22×C167—41no—
108C22×C167—41no—
109C22×C167C2×D811yescharacteristic
110C2×D167—42no—
111C2×D167—162no—
112C2×D167—161no—
113C2×D167C2×D812yes—
114C4×C167—41no—
115C4×C167—41no—
116C4×C167C2×D811yescharacteristic
117C4×C167C22×D411yescharacteristic
118Dic167—42no—
119Dic167C2×D812yes—
120C2×C167—21no—
121C2×C167—81no—
122C2×C167—81no—
123C2×C167C22×D811yescharacteristic
124D167—22no—
125D167—82no—
126C167#341611yescharacteristic, complemented, Sylow
127#3416—1671nomaximal, Sylow
128#578—1671no—
129#576—1671no—
130#510—1671no—
131#510—1672no—
132#509—1671no—
133#509—1672no—
134#508—1672no—
135#508D16711yescharacteristic, complemented
136C2×SD32—1672no—
137C2×SD32—1671no—
138#504—1671no—
1392−1+4—3341no—
140H(5,𝔽2)—3341no—
141#137—3342no—
142#137—3341no—
143AGL(1,ℤ/8)—3342no—
144AGL(1,ℤ/8)—3341no—
145#135—1672no—
146#135—1672no—
147#135C2×D16711yescharacteristic
148#135C2×D16711yescharacteristic, complemented
149C2×Q16—1672no—
150C2×Q16—1671no—
151C2×D8—1672no—
152C2×D8—1671no—
153#131—1671no—
154Q32—1672no—
155Q32C2×D16711yescharacteristic, complemented
156SD32—1672no—
157SD32—1672no—
158SD32—1672no—
159SD32—1672no—
160SD32C2×D16711yescharacteristic, complemented
161SD32C2×D16711yescharacteristic, complemented
162D16—1672no—
163D16C2×D16711yescharacteristic, complemented
164M32—1671no—
165M32—1672no—
166C2×C16—1672no—
167C2×C16C2×D16711yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—3342no—
171#41—3342no—
172#41—6681no—
173#41—6681no—
174#41—6682no—
175#41—1671no—
176C2×Q8—3342no—
177C2×Q8—3341no—
178C2×Q8—6681no—
179C2×D4—3342no—
180C2×D4—3341no—
181C2×D4—6681no—
182C2×D4—6682no—
183Q16—3342no—
184Q16—3341no—
185Q16—1672no—
186Q16C22×D16711yescharacteristic
187Q16C22×D16711yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—3342no—
191SD16—3342no—
192D8—3342no—
193D8—3341no—
194D8—1672no—
195D8C22×D16711yescharacteristic
196D8C22×D16711yescharacteristic
197M16—1671no—
198M16—1672no—
199C2×C8—1672no—
200C2×C8C22×D16711yescharacteristic
201C16—1672no—
202C16C22×D16711yescharacteristic
203C16C22×D16711yescharacteristic
204C23—6682no—
205C23—6681no—
206Q8—21no—
207Q8—21no—
208Q8—3342no—
209Q8—6682no—
210Q8—6681no—
211Q8—1671no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3342no—
217D4—6682no—
218D4—6682no—
219D4—6682no—
220D4—6681no—
221D4—1672no—
222D4—1671no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—6682no—
226C2×C4—6682no—
227C2×C4—6681no—
228C2×C4—6682no—
229C2×C4—1672no—
230C2×C4D4×D16711yescharacteristic
231C8—1672no—
232C8D4×D16711yescharacteristic
233C8C23×D16711yescharacteristic
234C22—41no—
235C22—41no—
236C22—26721no—
237C22—26722no—
238C22—6682no—
239C22—1672no—
240C22D8×D16711yescharacteristic
241C4—41no—
242C4—41no—
243C4—6682no—
244C4—1672no—
245C4D8×D16711yescharacteristic
246C4C2×D4×D16711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—13362no—
251C2—3342no—
252C2C2×D8×D16711yescharacteristic
subgroupG/H
centerC2C2×D8×D167
commutator, cosocleC8×C167C24
FrattiniC8C23×D167
Fitting#508×C167C2
socleC2×C167C22×D8
hypercenter#508D167
hypocenterC167#3416
LCSG ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #24924083C2semidirect
2. #24924071C2semidirect
3. #24923711C2semidirect
4. #24923709C2semidirect
5. #24923704C2semidirect
6. #24923703C2semidirect
7. #24923700C2semidirect
8. C2×#19128777C2semidirect
9. SD32×D167C2semidirect
10. #24923685C2semidirect
11. #508×C167C2semidirect
19. #19128851C22semidirect
20. C2×Dic668C22semidirect
21. Q16×D167C22semidirect
25. #19128778C22semidirect
26. #19128777C22semidirect
27. #19128777C22semidirect
28. #19128776C22semidirect
29. #19128775C22semidirect
30. #19128773C22semidirect
31. #19128772C22semidirect
32. #19128771C22semidirect
33. C16×D167C22semidirect
35. #135×C167C22semidirect
36. Q32×C167C22semidirect
37. SD32×C167C22semidirect
38. SD32×C167C22semidirect
39. D16×C167C22semidirect
40. C2×C16×C167C22semidirect
126. C167#3416semidirect
135. #508D167semidirect
148. #135C2×D167semidirect
155. Q32C2×D167semidirect
160. SD32C2×D167semidirect
161. SD32C2×D167semidirect
163. D16C2×D167semidirect
167. C2×C16C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249240832
#249240712
#249237112
#249237092
#249237042
#249237032
#249237002
C2×#191287772
SD32×D1672
#249236852
#508×C1672
#3416167