Finite Group Databasebeta

← #37543679

Subgroups — #37543679

Order 23168.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#26041843C211yesmaximal, characteristic
2#26041603C212yesmaximal, complemented
3#26041603C212yesmaximal, complemented
4#26041597C212yesmaximal, complemented
5#26041592C212yesmaximal, complemented
6#26041319C211yesmaximal, characteristic, complemented
7#26041315C211yesmaximal, characteristic, complemented
8#26041283C211yesmaximal, characteristic, complemented
9C2×#19153736C211yesmaximal, characteristic, complemented
10#26041246C211yesmaximal, characteristic, complemented
11#520×C181C211yesmaximal, characteristic, complemented
12C22×D724C2211yescharacteristic
13C2×#15668433C2211yescharacteristic
14C22×#13089678C2211yescharacteristic
15#19153812—21no—
16#19153805—21no—
17#19153790C2212yes—
18#19153790C2212yes—
19#19153784—22no—
20#19153775C2211yescharacteristic
21#19153749C2211yescharacteristic, complemented
22#19153749C2212yescomplemented
23#19153749C2211yescharacteristic, complemented
24#19153747C2211yescharacteristic, complemented
25#19153747C2212yescomplemented
26#19153747C2211yescharacteristic, complemented
27#19153736C2212yes—
28#19153736C2212yes—
29#19153736C2212yes—
30#19153736C2212yes—
31C22⋊#13089678C2212yescomplemented
32C22⋊#13089678C2212yescomplemented
33C2×#41×C181C2211yescharacteristic
34C4×Q8×C181—21no—
35C4×D4×C181—21no—
36C4×D4×C181—22no—
37#117×C181C2211yescharacteristic
38#117×C181C2212yescomplemented
39C2×C4⋊C4×C181C2211yescharacteristic
40C2×C4⋊C4×C181C2212yescomplemented
41C23×D181—21no—
42C2×D724—21no—
43C2×D724—22no—
44C2×D724—22no—
45C2×D724C2312yes—
46C2×C22⋊D181—22no—
47C2×C22⋊D181—22no—
48#15668433—22no—
49C2×#13089678—22no—
50C2×#13089678C2311yescharacteristic
51C2×#13089678C2312yes—
52C2×#13089678C2311yescharacteristic
53C4⋊Dic181C2311yescharacteristic
54C4⋊Dic181C2311yescharacteristic
55C22⋊Dic181—21no—
56#15668426—22no—
57#15668426—22no—
58C4×Dic181—21no—
59#41×C181—22no—
60#41×C181—22no—
61C2×Q8×C181D411yescharacteristic
62C2×D4×C181D412yes—
63C2×D4×C181D411yescharacteristic
64C22×C4×C181—21no—
65C22×C4×C181—22no—
66C22×C4×C181C2312yes—
67C22×C4×C181C2311yescharacteristic
68C4⋊C4×C181—21no—
69C4⋊C4×C181—22no—
70C4⋊C4×C181C2312yes—
71C4⋊C4×C181C2311yescharacteristic
72C4⋊C4×C181C2311yescharacteristic
73C22⋊C4×C181—21no—
74C22⋊C4×C181—22no—
75C42×C181—21no—
76C42×C181—22no—
77C22×D181—22no—
78C22×D181—42no—
79C22×D181—81no—
80C22×D181—81no—
81D724—22no—
82D724—22no—
83D724—42no—
84C22⋊D181—82no—
85C22⋊D181—82no—
86C2×Dic181—21no—
87C2×Dic181—21no—
88#13089678—21no—
89#13089678—21no—
90#13089678—22no—
91C23×C181C2×D412yes—
92C23×C181C2×D411yescharacteristic
93Q8×C181—22no—
94D4×C181—24no—
95D4×C181—22no—
96C2×C4×C181—22no—
97C2×C4×C181—22no—
98C2×C4×C181—21no—
99C2×C4×C181—21no—
100C2×C4×C181—22no—
101C2×C4×C181—81no—
102C2×C4×C181—82no—
103C2×C4×C181C2×D412yes—
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181#4112yes—
106C2×C4×C181#4112yes—
107C2×C4×C181C2411yescharacteristic
108C22×C181—21no—
109C22×C181—21no—
110C22×C181—22no—
111C22×C181—22no—
112C22×C181—22no—
113C22×C181#12312yes—
114C22×C181C22×D411yescharacteristic
115C2×D181—42no—
116C2×D181—42no—
117C2×D181—42no—
118C2×D181—84no—
119C4×C181—22no—
120C4×C181—41no—
121C4×C181—42no—
122C4×C181—41no—
123C4×C181#12312yes—
124C4×C181C2×#4111yescharacteristic
125C4×C181C2×#4111yescharacteristic
126Dic181—41no—
127Dic181—41no—
128C2×C181—22no—
129C2×C181—42no—
130C2×C181C2×#12311yescharacteristic
131C2×C181#57611yescharacteristic
132C2×C181#57711yescharacteristic
133D181—82no—
134C181#309611yescharacteristic, complemented, Sylow
135#3096—1811nomaximal, Sylow
136#535—1811no—
137#520D18111yescharacteristic, complemented
138#482—1812no—
139#482—1812no—
140#481—1812no—
141#480—1812no—
142#428—1811no—
143#427—1811no—
144#418—1811no—
145C2×#102—1811no—
146#408—1811no—
147C2×#41C2×D18111yescharacteristic
148C22×D4—1811no—
149C2×M16—1811no—
150C22×C8—1811no—
151#127—3621no—
152#124—3621no—
153#121—1812no—
154#121—1812no—
155#120—3622no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—22no—
159#117—1811no—
160#117C2×D18111yescharacteristic
161#117C2×D18112yescomplemented
162C2×C4⋊C4C2×D18111yescharacteristic
163C2×C4⋊C4C2×D18112yescomplemented
164#107—1811no—
165#107—1811no—
166#107—1812no—
167#106—1811no—
168#106—1811no—
169#106—1812no—
170#102—1812no—
171#102—1812no—
172#102—1812no—
173#102—1812no—
174C22⋊C8—1812no—
175C22⋊C8—1812no—
176C24—3621no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D18111yescharacteristic
180C2×D4—3622no—
181C2×D4—3622no—
182C2×D4—3621no—
183C2×D4—3622no—
184C2×D4—3622no—
185C2×D4—1812no—
186C2×D4C22⋊D18112yes—
187C2×D4C22⋊D18111yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4C22×D18112yes—
191C22×C4C22×D18111yescharacteristic
192M16—3622no—
193C2×C8—3622no—
194C2×C8—1811no—
195C2×C8—1812no—
196C2×C8—1811no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—1811no—
200C4⋊C4—1811no—
201C4⋊C4C22×D18112yes—
202C4⋊C4C22×D18111yescharacteristic
203C4⋊C4C22×D18111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—3621no—
207C22⋊C4—3622no—
208C22⋊C4—3622no—
209C42—21no—
210C42—22no—
211C42—3621no—
212C23—3622no—
213C23—14481no—
214C23—14481no—
215C23—7242no—
216C23C2×C22⋊D18112yes—
217C23C2×C22⋊D18111yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—3622no—
222D4—3622no—
223D4—14482no—
224D4—14482no—
225D4—7242no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—3621no—
234C2×C4—3621no—
235C2×C4C2×C22⋊D18112yes—
236C2×C4C2×C22⋊D18111yescharacteristic
237C2×C4#1566844912yes—
238C2×C4#1566844912yes—
239C2×C4C23×D18111yescharacteristic
240C8—3621no—
241C8—3621no—
242C8—3622no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—14484no—
249C22—7242no—
250C22—7242no—
251C22—7242no—
252C22#1915379912yes—
253C22C22×C22⋊D18111yescharacteristic
254C4—22no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—7241no—
259C4—7241no—
260C4#1915379912yes—
261C4C2×#1566844911yescharacteristic
262C4C2×#1566844911yescharacteristic
263C2—22no—
264C2—42no—
265C2—14482no—
266C2C2×#1915379911yescharacteristic
267C2#2604208211yescharacteristic
268C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#520×C181C2
socleC22×C181C22×D4
prime radical#26041843C2
hypercenter#520D181
hypocenterC181#3096
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #26041603C2semidirect
3. #26041603C2semidirect
4. #26041597C2semidirect
5. #26041592C2semidirect
6. #26041319C2semidirect
7. #26041315C2semidirect
8. #26041283C2semidirect
9. C2×#19153736C2semidirect
10. #26041246C2semidirect
11. #520×C181C2semidirect
21. #19153749C22semidirect
22. #19153749C22semidirect
23. #19153749C22semidirect
24. #19153747C22semidirect
25. #19153747C22semidirect
26. #19153747C22semidirect
31. C22⋊#13089678C22semidirect
32. C22⋊#13089678C22semidirect
38. #117×C181C22semidirect
40. C2×C4⋊C4×C181C22semidirect
134. C181#3096semidirect
137. #520D181semidirect
161. #117C2×D181semidirect
163. C2×C4⋊C4C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418432
#260416032
#260415972
#260415922
#260413192
#260413152
#260412832
C2×#191537362
#260412462
#520×C1812
#3096181