Finite Group Databasebeta

← #37543677

Subgroups — #37543677

Order 23168.

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

normal subgroups only

This group has 418 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#26041846C211yesmaximal, characteristic
2#26041759C211yesmaximal, characteristic, complemented
3#26041602C212yesmaximal, complemented
4#26041601C212yesmaximal, complemented
5#26041596C212yesmaximal, complemented
6#26041591C212yesmaximal, complemented
7#26041321C211yesmaximal, characteristic, complemented
8#26041315C211yesmaximal, characteristic, complemented
9C2×#19153737C211yesmaximal, characteristic, complemented
10#26041247C211yesmaximal, characteristic, complemented
11#418×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×#13089678C2211yescharacteristic
14#19153812—21no—
15#19153803—21no—
16#19153787C2212yes—
17#19153787C2212yes—
18#19153785—22no—
19C4×Dic362—21no—
20C4×D724—21no—
21#19153776—22no—
22#19153775C2211yescharacteristic
23#19153774C2212yescomplemented
24C2×C4⋊Dic181C2211yescharacteristic
25C4⋊C4×D181C2212yescomplemented
26#19153750C2212yescomplemented
27#19153749C2211yescharacteristic, complemented
28#19153749C2211yescharacteristic, complemented
29#19153748C2212yescomplemented
30#19153747C2211yescharacteristic, complemented
31#19153747C2211yescharacteristic, complemented
32#19153737C2212yes—
33#19153737C2212yes—
34#19153728C2212yescomplemented
35#19153728C2212yescomplemented
36C22×D4×C181C2211yescharacteristic
37C2×M16×C181C2211yescharacteristic
38#117×C181C2211yescharacteristic
39#102×C181C2212yes—
40#102×C181C2212yes—
41#15668449—22no—
42#15668449—22no—
43C2×Dic362D411yescharacteristic
44C2×D724D411yescharacteristic
45C2×C22⋊D181—22no—
46C2×C22⋊D181—22no—
47C2×C22⋊D181D412yes—
48C22×Dic181—21no—
49C2×C4×D181—22no—
50C2×C4×D181C2312yes—
51C2×#13089678—22no—
52C2×#13089678C2311yescharacteristic
53C2×#13089678C2311yescharacteristic
54C4⋊Dic181—21no—
55C4⋊Dic181C2312yes—
56C4⋊Dic181C2311yescharacteristic
57C4⋊Dic181C2311yescharacteristic
58#15668428—22no—
59C22⋊Dic181—22no—
60C22⋊Dic181—22no—
61C22⋊Dic181—21no—
62#15668426—22no—
63C4×Dic181—22no—
64C4×Dic181—21no—
65C24×C181—21no—
66C2×D4×C181—21no—
67C2×D4×C181—22no—
68C2×D4×C181—22no—
69C2×D4×C181C2312yes—
70C22×C4×C181C2311yescharacteristic
71M16×C181—22no—
72C2×C8×C181C2312yes—
73C4⋊C4×C181C2311yescharacteristic
74C4⋊C4×C181C2311yescharacteristic
75C22⋊C4×C181—21no—
76C42×C181—21no—
77C22×D181C2×D412yes—
78Dic362—22no—
79D724—22no—
80C22⋊D181—24no—
81C22⋊D181—82no—
82C22⋊D181—82no—
83C2×Dic181—22no—
84C2×Dic181—21no—
85C2×Dic181—21no—
86C2×Dic181—81no—
87C2×Dic181C2×D412yes—
88C4×D181—22no—
89C4×D181—22no—
90C4×D181—82no—
91#13089678—21no—
92#13089678—21no—
93C23×C181—22no—
94C23×C181—42no—
95C23×C181—81no—
96C23×C181—81no—
97C23×C181C2×D411yescharacteristic
98D4×C181—22no—
99D4×C181—22no—
100D4×C181—42no—
101C2×C4×C181—21no—
102C2×C4×C181—21no—
103C2×C4×C181C2×D411yescharacteristic
104C2×C4×C181#4112yes—
105C2×C4×C181#4112yes—
106C2×C4×C181C2411yescharacteristic
107C8×C181—22no—
108C22×C181—21no—
109C22×C181—21no—
110C22×C181—42no—
111C22×C181—42no—
112C22×C181—42no—
113C22×C181—84no—
114C22×C181#12312yes—
115C22×C181C22×D411yescharacteristic
116C2×D181—22no—
117C2×D181—22no—
118C2×D181—22no—
119C4×C181—41no—
120C4×C181—41no—
121C4×C181#12312yes—
122C4×C181C2×#4111yescharacteristic
123C4×C181C2×#4111yescharacteristic
124Dic181—22no—
125Dic181—42no—
126Dic181—41no—
127Dic181—41no—
128C2×C181—22no—
129C2×C181—82no—
130C2×C181C2×#12311yescharacteristic
131C2×C181#57611yescharacteristic
132C2×C181#57711yescharacteristic
133D181—42no—
134C181#309611yescharacteristic, complemented, Sylow
135#3096—1811nomaximal, Sylow
136#535—1811no—
137#520—1811no—
138#482—1812no—
139#482—1812no—
140#481—1812no—
141#480—1812no—
142#428—1811no—
143#427—1811no—
144#418D18111yescharacteristic, complemented
145C2×#102—1811no—
146#408—1811no—
147C2×#41—1811no—
148C22×D4C2×D18111yescharacteristic
149C2×M16C2×D18111yescharacteristic
150C22×C8—1811no—
151#127—3621no—
152#124—3621no—
153#121—1812no—
154#121—1812no—
155#120—3622no—
156C4×Q8—3621no—
157C4×D4—3621no—
158C4×D4—3622no—
159#117—1811no—
160#117—1812no—
161#117C2×D18111yescharacteristic
162C2×C4⋊C4—1811no—
163C2×C4⋊C4—1812no—
164#107—1812no—
165#107—1811no—
166#107—1811no—
167#106—1812no—
168#106—1811no—
169#106—1811no—
170#102—1812no—
171#102—1812no—
172#102C2×D18112yes—
173#102C2×D18112yes—
174C22⋊C8—1812no—
175C22⋊C8—1812no—
176C24—21no—
177#41—3622no—
178#41—3622no—
179C2×Q8—1811no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—3622no—
184C2×D4—3622no—
185C2×D4—1812no—
186C2×D4—1811no—
187C2×D4C22×D18112yes—
188C22×C4—3622no—
189C22×C4—3621no—
190C22×C4—1812no—
191C22×C4C22×D18111yescharacteristic
192M16—22no—
193C2×C8—3622no—
194C2×C8—1811no—
195C2×C8—1811no—
196C2×C8C22×D18112yes—
197C4⋊C4—3622no—
198C4⋊C4—3621no—
199C4⋊C4—1812no—
200C4⋊C4—1811no—
201C4⋊C4—1811no—
202C4⋊C4C22×D18111yescharacteristic
203C4⋊C4C22×D18111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—3622no—
206C22⋊C4—3621no—
207C22⋊C4—3622no—
208C22⋊C4—3622no—
209C42—21no—
210C42—3622no—
211C42—3621no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1812no—
217C23D4×D18111yescharacteristic
218Q8—3622no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—3624no—
223D4—3622no—
224D4—14482no—
225D4—14482no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—3622no—
229C2×C4—3622no—
230C2×C4—3621no—
231C2×C4—3621no—
232C2×C4—3622no—
233C2×C4—14482no—
234C2×C4—14481no—
235C2×C4—1812no—
236C2×C4D4×D18111yescharacteristic
237C2×C4#1566844812yes—
238C2×C4#1566844912yes—
239C2×C4C23×D18111yescharacteristic
240C8—22no—
241C8—3621no—
242C8—3621no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—3622no—
250C22—3622no—
251C22—3622no—
252C22#1915379812yes—
253C22C2×D4×D18111yescharacteristic
254C4—41no—
255C4—41no—
256C4—3622no—
257C4—7242no—
258C4—7241no—
259C4—7241no—
260C4#1915379812yes—
261C4C2×#1566844811yescharacteristic
262C4C2×#1566844911yescharacteristic
263C2—22no—
264C2—82no—
265C2—7242no—
266C2C2×#1915379811yescharacteristic
267C2#2604208311yescharacteristic
268C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#418×C181C2
socleC22×C181C22×D4
prime radical#26041846C2
hypercenter#418D181
hypocenterC181#3096
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #26041759C2semidirect
3. #26041602C2semidirect
4. #26041601C2semidirect
5. #26041596C2semidirect
6. #26041591C2semidirect
7. #26041321C2semidirect
8. #26041315C2semidirect
9. C2×#19153737C2semidirect
10. #26041247C2semidirect
11. #418×C181C2semidirect
23. #19153774C22semidirect
25. C4⋊C4×D181C22semidirect
26. #19153750C22semidirect
27. #19153749C22semidirect
28. #19153749C22semidirect
29. #19153748C22semidirect
30. #19153747C22semidirect
31. #19153747C22semidirect
34. #19153728C22semidirect
35. #19153728C22semidirect
134. C181#3096semidirect
144. #418D181semidirect

Maximal subgroups

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

subgroupindex
#260418462
#260417592
#260416022
#260416012
#260415962
#260415912
#260413212
#260413152
C2×#191537372
#260412472
#418×C1812
#3096181