Finite Group Databasebeta

← #497×D181 #37544157

Subgroups — #497×D181 #37544157

Order 23168.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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×Q16×D181C212yesmaximal, complemented
2C2×SD16×D181C212yesmaximal
3C4⋊Q8×D181C211yesmaximal, characteristic, complemented
4#124×D181C211yesmaximal, characteristic
5#26041674C211yesmaximal, characteristic, complemented
6#26041673C211yesmaximal, characteristic, complemented
7#26041672C212yesmaximal, complemented
8#26041671C211yesmaximal, characteristic, complemented
9#26041670C212yesmaximal, complemented
10#97×D181C211yesmaximal, characteristic, complemented
11#497×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181C2211yescharacteristic
13C2×Q8×D181C2212yes—
14C2×D4×D181C2211yescharacteristic
15C2×Dic724C2212yescomplemented
16C2×#15668439C2212yescomplemented
17C2×#15668439C2212yescomplemented
18Q16×D181—28no—
19C2×#15668438C2212yes—
20C2×#15668437C2212yes—
21C2×#15668436C2212yes—
22SD16×D181—28no—
23C2×C8×D181C2212yes—
24#19153816C2211yescharacteristic, complemented
25#19153815—22no—
26#19153814C2212yescomplemented
27#19153805C2211yescharacteristic
28#19153804C2211yescharacteristic
29#19153803C2211yescharacteristic
30#19153802—22no—
31C4⋊C4×D181—22no—
32C22⋊C4×D181—22no—
33C42×D181C2211yescharacteristic
34#19153727C2211yescharacteristic, complemented
35#19153726C2212yescomplemented
36C2×Q16×C181C2212yescomplemented
37C2×SD16×C181C2212yes—
38C4⋊Q8×C181C2211yescharacteristic, complemented
39#124×C181C2211yescharacteristic
40#97×C181C2211yescharacteristic, complemented
41C23×D181—21no—
42C2×Dic362—21no—
43C2×Dic362—22no—
44C2×Dic362C2311yescharacteristic
45C2×Dic362C2312yes—
46Q8×D181—24no—
47Q8×D181—24no—
48Q8×D181—24no—
49C2×D724C2311yescharacteristic
50C2×C22⋊D181—21no—
51D4×D181—24no—
52C2×C4×D181—21no—
53C2×C4×D181—22no—
54C2×C4×D181D412yes—
55C2×C4×D181C2311yescharacteristic
56Dic724—24no—
57#15668439—24no—
58#15668439—24no—
59#15668438—24no—
60#15668437—24no—
61#15668436—24no—
62C2×#13089678C2312yes—
63C8×D181D418yes—
64C4⋊Dic181—22no—
65#15668428—24no—
66C22⋊Dic181—22no—
67#15668426—22no—
68#15668426—22no—
69C4×Dic181D412yes—
70C4×Dic181C2311yescharacteristic
71C2×Q8×C181C2311yescharacteristic
72C2×Q8×C181C2312yes—
73C2×D4×C181C2311yescharacteristic
74Q16×C181—24no—
75SD16×C181—24no—
76C2×C8×C181C2312yes—
77C4⋊C4×C181—22no—
78C22⋊C4×C181—22no—
79C42×C181C2311yescharacteristic
80C22×D181—21no—
81C22×D181—42no—
82C22×D181—44no—
83C22×D181C2×D411yescharacteristic
84Dic362—22no—
85Dic362—22no—
86Dic362—22no—
87Dic362—44no—
88Dic362—42no—
89D724—22no—
90C22⋊D181—42no—
91C2×Dic181—21no—
92C2×Dic181—22no—
93C2×Dic181C2×D412yes—
94C2×Dic181C2×D411yescharacteristic
95C4×D181—24no—
96C4×D181—42no—
97C4×D181—44no—
98C4×D181C2×D414yes—
99#13089678C2×D414yes—
100C23×C181—21no—
101Q8×C181—22no—
102Q8×C181—22no—
103Q8×C181—22no—
104D4×C181—22no—
105C2×C4×C181—21no—
106C2×C4×C181—22no—
107C2×C4×C181C2×D412yes—
108C2×C4×C181C2411yescharacteristic
109C8×C181C2×D414yes—
110C22×C181—41no—
111C22×C181—42no—
112C22×C181C22×D411yescharacteristic
113C2×D181—41no—
114C2×D181—42no—
115C2×D181—84no—
116C2×D181#12712yes—
117C2×D181#13714yes—
118C4×C181—22no—
119C4×C181—41no—
120C4×C181—42no—
121C4×C181C22×D412yes—
122Dic181—22no—
123Dic181—41no—
124Dic181—42no—
125Dic181#12712yes—
126C2×C181—81no—
127C2×C181C2×#12711yescharacteristic
128C2×C181C2×#13712yes—
129D181—81no—
130D181#49714yesdirect product factor
131C181C2×#49711yescharacteristic, complemented, Sylow
132C2×#497—1811nomaximal, Sylow
133C22×Q16—1812no—
134C22×SD16—1812no—
135C2×C4⋊Q8—1811no—
136C2×#124—1811no—
137#497—1812no—
138#497—1812no—
139#497—1811no—
140#497—1811no—
141#497—1811no—
142#497D18111yescharacteristic, direct product factor
143C2×#97—1811no—
144C22×Q8—1811no—
145C22×Q8—1812no—
146C22×D4—1811no—
147C2×Q16—3628no—
148C2×Q16—1812no—
149C2×Q16—1812no—
150C2×Q16—1812no—
151C2×Q16C2×D18112yescomplemented
152C2×SD16—3628no—
153C2×SD16—1812no—
154C2×SD16—1812no—
155C2×SD16—1812no—
156C2×SD16C2×D18112yes—
157C22×C8—1812no—
158C4⋊Q8—3622no—
159C4⋊Q8—1812no—
160C4⋊Q8—1811no—
161C4⋊Q8C2×D18111yescharacteristic, complemented
162#124—3622no—
163#124—1811no—
164#124—1811no—
165#124—1811no—
166#124C2×D18111yescharacteristic
167C2×C4⋊C4—3622no—
168C2×C22⋊C4—3622no—
169C2×C42—1811no—
170#97—1812no—
171#97—1811no—
172#97C2×D18111yescharacteristic, complemented
173C24—3621no—
174C2×Q8—3621no—
175C2×Q8—3622no—
176C2×Q8—3624no—
177C2×Q8—3624no—
178C2×Q8—3624no—
179C2×Q8—1811no—
180C2×Q8—1812no—
181C2×Q8C22×D18111yescharacteristic
182C2×Q8C22×D18112yes—
183C2×D4—3624no—
184C2×D4—3621no—
185C2×D4—1811no—
186C2×D4C22×D18111yescharacteristic
187C22×C4—3621no—
188C22×C4—3622no—
189C22×C4—1812no—
190C22×C4—1811no—
191Q16—24no—
192Q16—3624no—
193Q16—3624no—
194Q16—3624no—
195SD16—24no—
196SD16—3624no—
197SD16—3624no—
198SD16—3624no—
199C2×C8—1812no—
200C2×C8—1818no—
201C2×C8C22×D18112yes—
202C4⋊C4—22no—
203C4⋊C4—3624no—
204C4⋊C4—3622no—
205C22⋊C4—22no—
206C22⋊C4—3622no—
207C22⋊C4—3622no—
208C22⋊C4—3622no—
209C42—1812no—
210C42—1811no—
211C42C22×D18111yescharacteristic
212C23—21no—
213C23—3621no—
214C23—1811no—
215C23—7242no—
216C23—7244no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—3622no—
221Q8—3622no—
222Q8—3622no—
223Q8—7242no—
224Q8—7244no—
225D4—22no—
226D4—3622no—
227D4—7242no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—3624no—
231C2×C4—3621no—
232C2×C4—3622no—
233C2×C4—1812no—
234C2×C4—1811no—
235C2×C4—1814no—
236C2×C4—7242no—
237C2×C4—7244no—
238C2×C4D4×D18112yes—
239C2×C4C23×D18111yescharacteristic
240C8—1814no—
241C8D4×D18114yes—
242C22—42no—
243C22—41no—
244C22—14484no—
245C22—1814no—
246C22—1812no—
247C22—7242no—
248C22—7241no—
249C22C2×D4×D18111yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—3622no—
254C4—1812no—
255C4—7241no—
256C4—7242no—
257C4C2×D4×D18112yes—
258C2—81no—
259C2—14481no—
260C2—1814no—
261C2#127×D18111yescharacteristic
262C2#137×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#497×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×D181C22
hypercenter#497D181
hypocenterC181C2×#497
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D181C2semidirect
3. C4⋊Q8×D181C2semidirect
5. #26041674C2semidirect
6. #26041673C2semidirect
7. #26041672C2semidirect
8. #26041671C2semidirect
9. #26041670C2semidirect
10. #97×D181C2semidirect
11. #497×C181C2semidirect
15. C2×Dic724C22semidirect
16. C2×#15668439C22semidirect
17. C2×#15668439C22semidirect
24. #19153816C22semidirect
26. #19153814C22semidirect
34. #19153727C22semidirect
35. #19153726C22semidirect
36. C2×Q16×C181C22semidirect
38. C4⋊Q8×C181C22semidirect
40. #97×C181C22semidirect
130. D181#497direct (N × H)
131. C181C2×#497semidirect
142. #497D181direct (N × H)
151. C2×Q16C2×D181semidirect
161. C4⋊Q8C2×D181semidirect
172. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1812
C2×SD16×D1812
C4⋊Q8×D1812
#124×D1812
#260416742
#260416732
#260416722
#260416712
#260416702
#97×D1812
#497×C1812
C2×#497181