Finite Group Databasebeta

← #36337283

Subgroups — #36337283

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#24924082C211yesmaximal, characteristic, complemented
2#24924071C211yesmaximal, characteristic, complemented
3#24923711C212yesmaximal, complemented
4#24923707C211yesmaximal, characteristic, complemented
5#24923706C212yesmaximal, complemented
6#24923702C212yesmaximal, complemented
7#24923699C211yesmaximal, characteristic, complemented
8C2×#19128775C211yesmaximal, characteristic, complemented
9SD32×D167C212yesmaximal, complemented
10#24923687C211yesmaximal, characteristic, complemented
11#510×C167C211yesmaximal, characteristic, complemented
12#19128880—21no—
13#19128878—21no—
14#19128864—22no—
15#19128863—21no—
16#19128859—22no—
17#19128857—21no—
18#19128855C2212yes—
19#19128855C2212yescomplemented
20#19128854C2211yescharacteristic, complemented
21Q16×D167C2212yescomplemented
22C2×D1336C2211yescharacteristic
23D8×D167C2212yes—
24#19128838C2211yescharacteristic
25#19128778C2211yescharacteristic, complemented
26#19128777C2212yescomplemented
27#19128776C2211yescharacteristic, complemented
28#19128775C2212yescomplemented
29#19128775C2212yescomplemented
30#19128775C2211yescharacteristic, complemented
31D2672C2212yescomplemented
32#19128773C2211yescharacteristic, complemented
33#19128771C2212yescomplemented
34C2×C167⋊C16C2211yescharacteristic, complemented
35C16×D167C2212yescomplemented
36#135×C167C2211yescharacteristic
37C2×Q16×C167C2211yescharacteristic, complemented
38Q32×C167C2212yescomplemented
39SD32×C167C2212yescomplemented
40M32×C167C2211yescharacteristic, complemented
41#14581951—22no—
42#14581951—22no—
43#14581951—41no—
44#14581950—21no—
45#14581950—41no—
46#14581950—42no—
47#14581950D411yescharacteristic
48Q8×D167—22no—
49Q8×D167—41no—
50C2×D668—21no—
51C2×D668—41no—
52D4×D167—22no—
53D4×D167—42no—
54Dic668C2311yescharacteristic
55#14581940—22no—
56#14581939—21no—
57#14581937—22no—
58#14581937—22no—
59D1336—21no—
60D1336C2312yes—
61D1336C2311yescharacteristic
62#14581935—22no—
63#14581934D411yescharacteristic
64#14581933D412yes—
65C8×D167C2312yes—
66C167⋊C16C2311yescharacteristic
67C167⋊C16C2311yescharacteristic
68#41×C167—21no—
69C2×Q8×C167—21no—
70Q16×C167—21no—
71Q16×C167C2312yes—
72Q16×C167C2311yescharacteristic
73SD16×C167—21no—
74D8×C167C2311yescharacteristic
75C2×C8×C167C2311yescharacteristic
76C16×C167C2312yes—
77C22×D167—41no—
78C22×D167—42no—
79Dic334—21no—
80Dic334—42no—
81Dic334D811yescharacteristic
82D668—22no—
83D668—21no—
84D668—42no—
85D668—42no—
86D668—41no—
87D668D811yescharacteristic
88C22⋊D167—41no—
89C22⋊D167—42no—
90C22⋊D167D812yes—
91C4×D167—42no—
92C4×D167—41no—
93C4×D167—42no—
94C4×D167—42no—
95C4×D167C2×D412yes—
96C167⋊C8C2×D412yes—
97Q8×C167—21no—
98Q8×C167—22no—
99Q8×C167—41no—
100D4×C167—21no—
101D4×C167—41no—
102C2×C4×C167—41no—
103C2×C4×C167—41no—
104C2×C4×C167C2×D411yescharacteristic
105C8×C167C2×D411yescharacteristic
106C8×C167C2411yescharacteristic
107C22×C167—41no—
108C22×C167C2×D811yescharacteristic
109C2×D167—41no—
110C2×D167—42no—
111C2×D167—162no—
112C2×D167—161no—
113C2×D167C2×D812yes—
114C4×C167—41no—
115C4×C167—42no—
116C4×C167C2×D811yescharacteristic
117C4×C167C22×D411yescharacteristic
118Dic167—41no—
119Dic167C2×D812yes—
120C2×C167—21no—
121C2×C167—81no—
122C2×C167C22×D811yescharacteristic
123D167—22no—
124D167—81no—
125D167—82no—
126C167#341611yescharacteristic, complemented, Sylow
127#3416—1671nomaximal, Sylow
128#578—1671no—
129#576—1671no—
130#510—1672no—
131#510D16711yescharacteristic, complemented
132#509—1672no—
133#509—1671no—
134#508—1671no—
135#508—1672no—
136C2×SD32—1671no—
137C2×SD32—1672no—
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#135—1671no—
148#135C2×D16711yescharacteristic
149C2×Q16—1672no—
150C2×Q16C2×D16711yescharacteristic, complemented
151C2×D8—1672no—
152C2×D8—1671no—
153#131—1671no—
154Q32—1671no—
155Q32C2×D16712yescomplemented
156SD32—1672no—
157SD32—1672no—
158SD32—1671no—
159SD32—1671no—
160SD32—1672no—
161SD32C2×D16712yescomplemented
162D16—1671no—
163D16—1672no—
164M32—1672no—
165M32C2×D16711yescharacteristic, complemented
166C2×C16—1671no—
167C2×C16—1672no—
168#41—21no—
169#41—3342no—
170#41—3342no—
171#41—3341no—
172#41—6681no—
173#41—6682no—
174#41—6681no—
175#41—1671no—
176C2×Q8—21no—
177C2×Q8—3342no—
178C2×Q8—6681no—
179C2×D4—3342no—
180C2×D4—3341no—
181C2×D4—6681no—
182C2×D4—6682no—
183Q16—21no—
184Q16—3342no—
185Q16—1671no—
186Q16C22×D16712yes—
187Q16C22×D16711yescharacteristic
188SD16—21no—
189SD16—3342no—
190SD16—3342no—
191SD16—3341no—
192D8—3342no—
193D8—3341no—
194D8—1671no—
195D8—1672no—
196D8C22×D16711yescharacteristic
197M16—1671no—
198M16—1672no—
199C2×C8—1672no—
200C2×C8C22×D16711yescharacteristic
201C16—1671no—
202C16—1671no—
203C16C22×D16712yes—
204C23—6682no—
205C23—6681no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—3341no—
210Q8—6682no—
211Q8—1671no—
212D4—21no—
213D4—41no—
214D4—3342no—
215D4—3341no—
216D4—6682no—
217D4—6682no—
218D4—6681no—
219D4—6682no—
220D4—6681no—
221D4—1672no—
222D4—1671no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—6682no—
226C2×C4—6682no—
227C2×C4—6682no—
228C2×C4—6681no—
229C2×C4—1672no—
230C2×C4D4×D16711yescharacteristic
231C8—1672no—
232C8D4×D16711yescharacteristic
233C8C23×D16711yescharacteristic
234C22—41no—
235C22—26721no—
236C22—26722no—
237C22—6681no—
238C22—6682no—
239C22—1672no—
240C22D8×D16711yescharacteristic
241C4—41no—
242C4—42no—
243C4—6681no—
244C4—1672no—
245C4D8×D16711yescharacteristic
246C4C2×D4×D16711yescharacteristic
247C2—21no—
248C2—81no—
249C2—13362no—
250C2—13361no—
251C2—3342no—
252C2C2×D8×D16711yescharacteristic
subgroupG/H
centerC2C2×D8×D167
commutator, cosocleC8×C167C24
FrattiniC8C23×D167
Fitting#510×C167C2
socleC2×C167C22×D8
hypercenter#510D167
hypocenterC167#3416
LCSG ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #24924082C2semidirect
2. #24924071C2semidirect
3. #24923711C2semidirect
4. #24923707C2semidirect
5. #24923706C2semidirect
6. #24923702C2semidirect
7. #24923699C2semidirect
8. C2×#19128775C2semidirect
9. SD32×D167C2semidirect
10. #24923687C2semidirect
11. #510×C167C2semidirect
19. #19128855C22semidirect
20. #19128854C22semidirect
21. Q16×D167C22semidirect
25. #19128778C22semidirect
26. #19128777C22semidirect
27. #19128776C22semidirect
28. #19128775C22semidirect
29. #19128775C22semidirect
30. #19128775C22semidirect
31. D2672C22semidirect
32. #19128773C22semidirect
33. #19128771C22semidirect
34. C2×C167⋊C16C22semidirect
35. C16×D167C22semidirect
37. C2×Q16×C167C22semidirect
38. Q32×C167C22semidirect
39. SD32×C167C22semidirect
40. M32×C167C22semidirect
126. C167#3416semidirect
131. #510D167semidirect
150. C2×Q16C2×D167semidirect
155. Q32C2×D167semidirect
161. SD32C2×D167semidirect
165. M32C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249240822
#249240712
#249237112
#249237072
#249237062
#249237022
#249236992
C2×#191287752
SD32×D1672
#249236872
#510×C1672
#3416167