Finite Group Databasebeta

← #37543634

Subgroups — #37543634

Order 23168.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#26041790C211yesmaximal, characteristic
2#26041620C212yesmaximal, complemented
3#26041614C211yesmaximal, characteristic, complemented
4#26041609C211yesmaximal, characteristic, complemented
5#26041603C212yesmaximal, complemented
6#26041597C211yesmaximal, characteristic, complemented
7#26041592C211yesmaximal, characteristic, complemented
8#26041315C212yesmaximal, complemented
9#26041273C212yesmaximal, complemented
10C2×C22⋊#13089678C211yesmaximal, characteristic, complemented
11C2×#117×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×#13089678C2212yes—
14#19153799—21no—
15#19153795C2211yescharacteristic
16#19153795C2211yescharacteristic
17#19153790C2211yescharacteristic
18#19153790C2211yescharacteristic
19#19153785—21no—
20C4×C22⋊D181—21no—
21C4×C22⋊D181—21no—
22#19153775C2211yescharacteristic
23#19153749C2212yescomplemented
24#19153749C2212yescomplemented
25#19153747C2212yescomplemented
26#19153747C2212yescomplemented
27#19153739C2212yescomplemented
28#19153739C2212yescomplemented
29#19153736C2212yescomplemented
30#19153736C2212yescomplemented
31C22⋊#13089678C2212yes—
32C22⋊#13089678C2211yescharacteristic
33C22⋊#13089678C2211yescharacteristic
34C23×C4×C181C2211yescharacteristic
35#117×C181—21no—
36#117×C181—21no—
37#117×C181C2212yes—
38#117×C181C2212yes—
39C2×C4⋊C4×C181C2211yescharacteristic
40C2×C4⋊C4×C181C2211yescharacteristic
41C2×C22⋊C4×C181—21no—
42C2×C42×C181—21no—
43#15668449—24no—
44C2×Dic362C2311yescharacteristic
45C2×D724C2311yescharacteristic
46C2×C22⋊D181—21no—
47C2×C22⋊D181—21no—
48C2×C22⋊D181—21no—
49C2×C4×D181—21no—
50C2×#13089678—24no—
51C2×#13089678C2312yes—
52C2×#13089678C2312yes—
53C4⋊Dic181C2311yescharacteristic
54C4⋊Dic181C2311yescharacteristic
55#15668428—21no—
56#15668428—21no—
57C22⋊Dic181—21no—
58C22⋊Dic181—21no—
59C22⋊Dic181—21no—
60#15668426—21no—
61#15668426—21no—
62C4×Dic181—21no—
63C24×C181D411yescharacteristic
64C22×C4×C181—24no—
65C22×C4×C181—21no—
66C22×C4×C181—21no—
67C22×C4×C181D411yescharacteristic
68C22×C4×C181D411yescharacteristic
69C22×C4×C181D411yescharacteristic
70C22×C4×C181C2311yescharacteristic
71C22×C4×C181C2311yescharacteristic
72C22×C4×C181C2311yescharacteristic
73C4⋊C4×C181—21no—
74C4⋊C4×C181—21no—
75C4⋊C4×C181C2312yes—
76C4⋊C4×C181C2312yes—
77C22⋊C4×C181—22no—
78C22⋊C4×C181—22no—
79C42×C181—22no—
80C42×C181—22no—
81C22×D181—21no—
82Dic362—22no—
83D724—22no—
84C22⋊D181—42no—
85C22⋊D181—42no—
86C22⋊D181—42no—
87C2×Dic181—21no—
88C2×Dic181—21no—
89C2×Dic181—21no—
90C4×D181—42no—
91#13089678—22no—
92#13089678—22no—
93C23×C181—22no—
94C23×C181—22no—
95C23×C181—41no—
96C23×C181C2×D411yescharacteristic
97C23×C181C2×D411yescharacteristic
98C23×C181C2×D411yescharacteristic
99C2×C4×C181—22no—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—22no—
104C2×C4×C181—22no—
105C2×C4×C181—22no—
106C2×C4×C181—42no—
107C2×C4×C181—42no—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2×D411yescharacteristic
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181#4114yes—
112C2×C4×C181C2411yescharacteristic
113C22×C181—21no—
114C22×C181—21no—
115C22×C181—21no—
116C22×C181—21no—
117C22×C181—21no—
118C22×C181—21no—
119C22×C181—21no—
120C22×C181—44no—
121C22×C181#12312yes—
122C22×C181#13512yes—
123C22×C181C22×D411yescharacteristic
124C2×D181—41no—
125C2×D181—41no—
126C2×D181—41no—
127C4×C181—21no—
128C4×C181—21no—
129C4×C181—42no—
130C4×C181—42no—
131C4×C181#12312yes—
132C4×C181C2×#4112yes—
133Dic181—41no—
134Dic181—41no—
135Dic181—41no—
136C2×C181—22no—
137C2×C181—22no—
138C2×C181—41no—
139C2×C181C2×#12311yescharacteristic
140C2×C181C2×#13511yescharacteristic
141C2×C181#57511yescharacteristic
142D181—81no—
143C181#309111yescharacteristic, complemented, Sylow
144#3091—1811nomaximal, Sylow
145#525—1811no—
146C2×#117D18111yescharacteristic, complemented
147#485—1812no—
148#484—1811no—
149#483—1811no—
150#482—1812no—
151#481—1811no—
152#480—1811no—
153#427—1812no—
154#416—1812no—
155C2×C22⋊C8—1811no—
156C2×#41—1811no—
157C23×C4C2×D18111yescharacteristic
158C22×C8—1812no—
159#123—3621no—
160C22⋊Q8—1811no—
161C22⋊Q8—1811no—
162#121—1811no—
163#121—1811no—
164#120—3621no—
165C4×D4—3621no—
166C4×D4—3621no—
167#117—21no—
168#117—21no—
169#117—1811no—
170#117C2×D18112yes—
171#117C2×D18112yes—
172C2×C4⋊C4C2×D18111yescharacteristic
173C2×C4⋊C4C2×D18111yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
176#107—1812no—
177#107—1812no—
178#106—1812no—
179#106—1812no—
180#103—1812no—
181#103—1812no—
182#102—1812no—
183#102—1812no—
184C22⋊C8—1811no—
185C22⋊C8—1811no—
186C22⋊C8—1812no—
187C24C22⋊D18111yescharacteristic
188#41—3624no—
189C2×Q8—1811no—
190C2×D4—3621no—
191C2×D4—3621no—
192C2×D4—3621no—
193C2×D4—1811no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—3621no—
198C22×C4C22⋊D18111yescharacteristic
199C22×C4C22⋊D18111yescharacteristic
200C22×C4C22⋊D18111yescharacteristic
201C22×C4C22×D18111yescharacteristic
202C22×C4C22×D18111yescharacteristic
203C22×C4C22×D18111yescharacteristic
204C2×C8—3624no—
205C2×C8—1812no—
206C2×C8—1812no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—3621no—
210C4⋊C4—3621no—
211C4⋊C4—1811no—
212C4⋊C4—1811no—
213C4⋊C4C22×D18112yes—
214C4⋊C4C22×D18112yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—3621no—
218C22⋊C4—3621no—
219C22⋊C4—3621no—
220C22⋊C4—3621no—
221C22⋊C4—3621no—
222C42—22no—
223C42—22no—
224C42—3621no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3621no—
229C23C2×C22⋊D18111yescharacteristic
230C23C2×C22⋊D18111yescharacteristic
231C23C2×C22⋊D18111yescharacteristic
232Q8—3622no—
233D4—3622no—
234D4—7242no—
235D4—7242no—
236D4—7242no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—3621no—
247C2×C4—3621no—
248C2×C4—3621no—
249C2×C4—7242no—
250C2×C4C2×C22⋊D18111yescharacteristic
251C2×C4C2×C22⋊D18111yescharacteristic
252C2×C4C2×C22⋊D18111yescharacteristic
253C2×C4#1566844914yes—
254C2×C4C23×D18111yescharacteristic
255C8—3622no—
256C8—3622no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—7241no—
266C22—7241no—
267C22—7241no—
268C22#1915379912yes—
269C22#1915383612yes—
270C22C22×C22⋊D18111yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—7241no—
276C4—7241no—
277C4—7241no—
278C4#1915379912yes—
279C4C2×#1566844912yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—14481no—
284C2C2×#1915379911yescharacteristic
285C2C2×#1915383611yescharacteristic
286C2#2604207711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#117×C181C2
socleC22×C181C22×D4
prime radical#26041790C2
hypercenterC2×#117D181
hypocenterC181#3091
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #26041620C2semidirect
3. #26041614C2semidirect
4. #26041609C2semidirect
5. #26041603C2semidirect
6. #26041597C2semidirect
7. #26041592C2semidirect
8. #26041315C2semidirect
9. #26041273C2semidirect
10. C2×C22⋊#13089678C2semidirect
11. C2×#117×C181C2semidirect
23. #19153749C22semidirect
24. #19153749C22semidirect
25. #19153747C22semidirect
26. #19153747C22semidirect
27. #19153739C22semidirect
28. #19153739C22semidirect
29. #19153736C22semidirect
30. #19153736C22semidirect
143. C181#3091semidirect
146. C2×#117D181semidirect

Maximal subgroups

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

subgroupindex
#260417902
#260416202
#260416142
#260416092
#260416032
#260415972
#260415922
#260413152
#260412732
C2×C22⋊#130896782
C2×#117×C1812
#3091181