Finite Group Databasebeta

← #37543859

Subgroups — #37543859

Order 23168.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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#26041813C211yesmaximal, characteristic, complemented
2#26041609C211yesmaximal, characteristic, complemented
3#26041592C211yesmaximal, characteristic, complemented
4#26041579C212yesmaximal, complemented
5#26041494C212yesmaximal, complemented
6#26041430C211yesmaximal, characteristic, complemented
7#26041413C211yesmaximal, characteristic
8#26041389C212yesmaximal, complemented
9#26041379C212yesmaximal, complemented
10#26041336C211yesmaximal, characteristic, complemented
11C2×C4×D4×C181C211yesmaximal, characteristic, complemented
12C2×#15668437C2212yes—
13C2×#15668434C2212yes—
14#19153810—21no—
15#19153808C2211yescharacteristic, complemented
16#19153804C2211yescharacteristic, complemented
17#19153799—21no—
18#19153795C2211yescharacteristic
19#19153790C2211yescharacteristic
20C4×Dic362C2211yescharacteristic, complemented
21C4×D724C2211yescharacteristic, complemented
22C4×C22⋊D181—21no—
23#19153773—21no—
24#19153749C2212yescomplemented
25#19153747C2212yescomplemented
26C4⋊#13089678C2212yescomplemented
27#19153739C2212yescomplemented
28#19153737C2212yes—
29#19153737C2212yes—
30#19153736C2212yescomplemented
31C22⋊#13089678C2211yescharacteristic
32C22⋊#13089678C2211yescharacteristic
33#19153727C2212yescomplemented
34C22×D4×C181C2211yescharacteristic
35C23×C4×C181—21no—
36C4×D4×C181—21no—
37C4×D4×C181—21no—
38C4×D4×C181C2212yes—
39C4×D4×C181C2212yes—
40C2×C4⋊C4×C181C2211yescharacteristic
41C2×C22⋊C4×C181—21no—
42C2×C42×C181C2211yescharacteristic
43C2×Dic362C2311yescharacteristic
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C4×D181—21no—
47#15668437—24no—
48#15668434—24no—
49C2×#13089678C2312yes—
50C2×#13089678C2312yes—
51C4⋊Dic181C2311yescharacteristic
52C4⋊Dic181C2311yescharacteristic
53#15668428—21no—
54#15668428—21no—
55#15668428—21no—
56#15668428—21no—
57C22⋊Dic181—21no—
58#15668426—21no—
59#15668426—21no—
60#15668426—21no—
61#15668426—21no—
62C4×Dic181—21no—
63C24×C181—21no—
64C2×D4×C181—24no—
65C2×D4×C181—21no—
66C2×D4×C181C2312yes—
67C22×C4×C181—21no—
68C22×C4×C181—22no—
69C22×C4×C181—22no—
70C22×C4×C181—44no—
71C22×C4×C181D411yescharacteristic
72C22×C4×C181D411yescharacteristic
73C22×C4×C181C2311yescharacteristic
74C4⋊C4×C181—21no—
75C4⋊C4×C181C2312yes—
76C22⋊C4×C181—22no—
77C22⋊C4×C181—22no—
78C42×C181D411yescharacteristic
79C42×C181D411yescharacteristic
80C42×C181C2311yescharacteristic
81C42×C181C2311yescharacteristic
82C22×D181—21no—
83Dic362—22no—
84D724—22no—
85C22⋊D181—42no—
86C2×Dic181—21no—
87C2×Dic181—21no—
88C2×Dic181—21no—
89C4×D181—42no—
90#13089678—22no—
91#13089678—22no—
92C23×C181—24no—
93C23×C181—22no—
94C23×C181—42no—
95C23×C181—42no—
96C23×C181C2×D411yescharacteristic
97D4×C181—24no—
98D4×C181—22no—
99D4×C181#4114yes—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—41no—
104C2×C4×C181—44no—
105C2×C4×C181—44no—
106C2×C4×C181—42no—
107C2×C4×C181C2×D411yescharacteristic
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2×D411yescharacteristic
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2×D411yescharacteristic
112C2×C4×C181C2411yescharacteristic
113C22×C181—21no—
114C22×C181—21no—
115C22×C181—24no—
116C22×C181—48no—
117C22×C181—42no—
118C22×C181—42no—
119C22×C181AGL(1,ℤ/8)12yes—
120C22×C181C22×D411yescharacteristic
121C2×D181—41no—
122C2×D181—41no—
123C2×D181—41no—
124C4×C181—21no—
125C4×C181—21no—
126C4×C181—21no—
127C4×C181—21no—
128C4×C181—21no—
129C4×C181—42no—
130C4×C181C2×#4112yes—
131Dic181—41no—
132Dic181—41no—
133Dic181—41no—
134C2×C181—22no—
135C2×C181—44no—
136C2×C181#52511yescharacteristic
137C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C181#57511yescharacteristic
139D181—81no—
140C181#311411yescharacteristic, complemented, Sylow
141#3114—1811nomaximal, Sylow
142#529—1811no—
143C2×C4×D4D18111yescharacteristic, complemented
144#483—1811no—
145#480—1811no—
146#478—1812no—
147#463—1812no—
148#450—1811no—
149#447—1811no—
150#442—1812no—
151#440—1812no—
152#432—1811no—
153C22×D4C2×D18111yescharacteristic
154C23×C4—21no—
155C2×SD16—1812no—
156C2×D8—1812no—
157#126—3621no—
158#125—1811no—
159#124—1811no—
160#123—3621no—
161C22⋊Q8—1811no—
162#121—1811no—
163C4×Q8—1811no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—3621no—
167C4×D4—1811no—
168C4×D4C2×D18112yes—
169C4×D4C2×D18112yes—
170#117—3621no—
171C2×C4⋊C4C2×D18111yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D18111yescharacteristic
174#107—1812no—
175#106—1812no—
176C4⋊C8—1812no—
177#103—1812no—
178#102—1812no—
179#102—1812no—
180#102—1812no—
181C22⋊C8—1811no—
182C22⋊C8—1811no—
183#97—1812no—
184C24—21no—
185C2×Q8—1811no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—3621no—
189C2×D4—1811no—
190C2×D4C22×D18112yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—3621no—
196C22×C4C22⋊D18111yescharacteristic
197C22×C4C22⋊D18111yescharacteristic
198C22×C4C22×D18111yescharacteristic
199SD16—3624no—
200D8—3624no—
201C2×C8—1812no—
202C2×C8—1812no—
203C4⋊C4—21no—
204C4⋊C4—3621no—
205C4⋊C4—3621no—
206C4⋊C4—3621no—
207C4⋊C4—3621no—
208C4⋊C4—1811no—
209C4⋊C4—1811no—
210C4⋊C4C22×D18112yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—3621no—
214C22⋊C4—3621no—
215C22⋊C4—3621no—
216C22⋊C4—3621no—
217C22⋊C4—3621no—
218C42—3621no—
219C42C22⋊D18111yescharacteristic
220C42C22⋊D18111yescharacteristic
221C42C22×D18111yescharacteristic
222C42C22×D18111yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—3621no—
228C23C2×C22⋊D18111yescharacteristic
229Q8—3622no—
230D4—24no—
231D4—22no—
232D4—3622no—
233D4—7242no—
234D4#1566844914yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—3621no—
243C2×C4—3621no—
244C2×C4—3621no—
245C2×C4—7242no—
246C2×C4C2×C22⋊D18111yescharacteristic
247C2×C4C2×C22⋊D18111yescharacteristic
248C2×C4C2×C22⋊D18111yescharacteristic
249C2×C4C2×C22⋊D18111yescharacteristic
250C2×C4C2×C22⋊D18111yescharacteristic
251C2×C4C23×D18111yescharacteristic
252C8—3622no—
253C8—3622no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—7241no—
261C22—7241no—
262C22—7241no—
263C22#1915384212yes—
264C22C22×C22⋊D18111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—7241no—
272C4—7241no—
273C4—7241no—
274C4C2×#1566844912yes—
275C2—22no—
276C2—44no—
277C2—14481no—
278C2#2604179011yescharacteristic
279C2C2×#1915384211yescharacteristic
280C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×C4×D4×C181C2
socleC22×C181C22×D4
prime radical#26041413C2
hypercenterC2×C4×D4D181
hypocenterC181#3114
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #26041813C2semidirect
2. #26041609C2semidirect
3. #26041592C2semidirect
4. #26041579C2semidirect
5. #26041494C2semidirect
6. #26041430C2semidirect
8. #26041389C2semidirect
9. #26041379C2semidirect
10. #26041336C2semidirect
11. C2×C4×D4×C181C2semidirect
15. #19153808C22semidirect
16. #19153804C22semidirect
20. C4×Dic362C22semidirect
21. C4×D724C22semidirect
24. #19153749C22semidirect
25. #19153747C22semidirect
26. C4⋊#13089678C22semidirect
27. #19153739C22semidirect
30. #19153736C22semidirect
33. #19153727C22semidirect
140. C181#3114semidirect
143. C2×C4×D4D181semidirect

Maximal subgroups

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

subgroupindex
#260418132
#260416092
#260415922
#260415792
#260414942
#260414302
#260414132
#260413892
#260413792
#260413362
C2×C4×D4×C1812
#3114181