Finite Group Databasebeta

← #37543099

Subgroups — #37543099

Order 23168.

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

normal subgroups only

This group has 704 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
1#26042115C211yesmaximal, characteristic
2C2×#19153847C212yesmaximal, complemented
3#26041461C214yesmaximal, complemented
4#26041456C214yesmaximal, complemented
5#26041297C212yesmaximal, complemented
6#26041263C211yesmaximal, characteristic, complemented
7#536×C181C211yesmaximal, characteristic, complemented
8#19153864C2212yes—
9#19153863—22no—
10#19153862—22no—
11#19153861—22no—
12#19153861C2212yes—
13C2×#15668449—22no—
14C2×#15668449C2212yes—
15#41×D181—24no—
16#19153847—24no—
17#19153847C2218yescomplemented
18C2×#15668439—22no—
19C2×#15668436—22no—
20C2×#15668433C2212yescomplemented
21#19153744C2218yescomplemented
22#19153735C2212yescomplemented
23#19153733C2212yescomplemented
24C2×#41×C181C2211yescharacteristic
25C22×Q8×C181C2211yescharacteristic, complemented
26C4⋊Q8×C181C2212yescomplemented
27#124×C181C2212yescomplemented
28#123×C181—22no—
29C22⋊Q8×C181—22no—
30#117×C181C2211yescharacteristic, complemented
31#15668450—24no—
32#15668449—24no—
33#15668449—24no—
34#15668449—22no—
35#15668449—24no—
36#15668449—22no—
37#15668449D414yes—
38#15668449C2314yes—
39#15668448—28no—
40#15668448—24no—
41C2×Dic362—22no—
42C2×Dic362D412yes—
43Q8×D181—24no—
44C2×D724—22no—
45C2×D724D412yes—
46C2×C22⋊D181—24no—
47C2×C22⋊D181—22no—
48D4×D181—28no—
49D4×D181—24no—
50C2×C4×D181—24no—
51C2×C4×D181—22no—
52#15668439—28no—
53#15668436—28no—
54#15668433—22no—
55#15668433C2314yes—
56C2×#13089678—22no—
57#41×C181—22no—
58#41×C181—22no—
59C2×Q8×C181—21no—
60C2×Q8×C181—22no—
61C2×Q8×C181—22no—
62C2×Q8×C181C2312yes—
63C2×Q8×C181C2311yescharacteristic
64C2×D4×C181D412yes—
65C2×D4×C181C2311yescharacteristic
66C22×C4×C181—21no—
67C22×C4×C181D412yes—
68C22×C4×C181C2311yescharacteristic
69C4⋊C4×C181—21no—
70C4⋊C4×C181—24no—
71C22⋊C4×C181—21no—
72C22⋊C4×C181—24no—
73C42×C181C2312yes—
74C22×D181—24no—
75C22×D181—22no—
76Dic362—24no—
77Dic362—22no—
78Dic362C2×D414yes—
79D724—24no—
80D724—22no—
81D724C2×D414yes—
82C22⋊D181—28no—
83C22⋊D181—24no—
84C22⋊D181—24no—
85C2×Dic181—24no—
86C2×Dic181—22no—
87C4×D181—28no—
88C4×D181—24no—
89C4×D181—24no—
90#13089678—24no—
91C23×C181C2×D412yes—
92C23×C181C2×D411yescharacteristic
93Q8×C181—22no—
94Q8×C181—24no—
95Q8×C181—41no—
96Q8×C181—41no—
97D4×C181—24no—
98D4×C181—22no—
99C2×C4×C181—22no—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—42no—
104C2×C4×C181C2×D412yes—
105C2×C4×C181C2×D412yes—
106C2×C4×C181C2×D412yes—
107C2×C4×C181C2×D411yescharacteristic
108C2×C4×C181C2411yescharacteristic
109C22×C181—22no—
110C22×C181—41no—
111C22×C181—42no—
112C22×C181#12012yes—
113C22×C181C22×D411yescharacteristic
114C2×D181—24no—
115C2×D181—82no—
116C2×D181—84no—
117C4×C181—22no—
118C4×C181—42no—
119C4×C181—42no—
120C4×C181#12011yescharacteristic
121C4×C181#12011yescharacteristic
122C4×C181C22×D412yes—
123Dic181—24no—
124C2×C181—21no—
125C2×C181—22no—
126C2×C181—42no—
127C2×C181C2×#12011yescharacteristic
128D181—44no—
129C181#302211yescharacteristic, complemented, Sylow
130#3022—1811nomaximal, Sylow
131#585—1811no—
132C2×#137—1812no—
133#536D18111yescharacteristic, complemented
134#456—1814no—
135#455—1814no—
136#421—1812no—
137#413—1811no—
1382−1+4—3622no—
1392−1+4—1812no—
140H(5,𝔽2)—3622no—
141H(5,𝔽2)—3622no—
142H(5,𝔽2)—1812no—
143C2×#41—3624no—
144C2×#41—3622no—
145C2×#41—1812no—
146C2×#41C2×D18111yescharacteristic
147C22×Q8C2×D18111yescharacteristic, complemented
148#137—3624no—
149#137—1818no—
150C2×Q16—3622no—
151C2×SD16—3622no—
152C2×M16—1812no—
153C4⋊Q8C2×D18112yescomplemented
154#124C2×D18112yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D18111yescharacteristic, complemented
158#104—1818no—
159#101—1812no—
160#100—1812no—
161#41—22no—
162#41—22no—
163#41—3624no—
164#41—3624no—
165#41—3628no—
166#41—3624no—
167#41—3624no—
168#41—3622no—
169#41—3622no—
170#41—3624no—
171#41—1814no—
172#41—1814no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—3622no—
177C2×Q8—3624no—
178C2×Q8—1812no—
179C2×Q8C22×D18112yes—
180C2×Q8C22×D18111yescharacteristic
181C2×D4—3622no—
182C2×D4—3628no—
183C2×D4—3624no—
184C2×D4—3624no—
185C2×D4—3622no—
186C2×D4—1812no—
187C2×D4C22⋊D18112yes—
188C2×D4C22×D18111yescharacteristic
189C22×C4—21no—
190C22×C4—3624no—
191C22×C4—3622no—
192C22×C4C22⋊D18112yes—
193C22×C4C22×D18111yescharacteristic
194Q16—3628no—
195SD16—3628no—
196M16—3622no—
197M16—1814no—
198C2×C8—3622no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D18112yes—
204C23—3624no—
205C23—3622no—
206C23C2×C22⋊D18112yes—
207C23C2×C22⋊D18111yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—3624no—
213Q8—3622no—
214Q8—1814no—
215D4—24no—
216D4—22no—
217D4—3624no—
218D4—3628no—
219D4—3624no—
220D4—3624no—
221D4—3622no—
222D4—1814no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—3628no—
229C2×C4—3624no—
230C2×C4—3622no—
231C2×C4—3624no—
232C2×C4—3624no—
233C2×C4D4×D18112yes—
234C2×C4D4×D18112yes—
235C2×C4C2×C22⋊D18112yes—
236C2×C4C2×C22⋊D18111yescharacteristic
237C2×C4C23×D18111yescharacteristic
238C8—3624no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—3624no—
243C22—14482no—
244C22—14484no—
245C22#1915378412yes—
246C22C22×C22⋊D18111yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—3624no—
251C4#1915378411yescharacteristic
252C4#1915378411yescharacteristic
253C4C2×D4×D18112yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—7244no—
258C2C2×#1915378411yescharacteristic
subgroupG/H
centerC2C2×#19153784
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#536×C181C2
socleC2×C181C2×#120
prime radical#26042115C2
hypercenter#536D181
hypocenterC181#3022
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#19153847C2semidirect
3. #26041461C2semidirect
4. #26041456C2semidirect
5. #26041297C2semidirect
6. #26041263C2semidirect
7. #536×C181C2semidirect
17. #19153847C22semidirect
20. C2×#15668433C22semidirect
21. #19153744C22semidirect
22. #19153735C22semidirect
23. #19153733C22semidirect
25. C22×Q8×C181C22semidirect
26. C4⋊Q8×C181C22semidirect
27. #124×C181C22semidirect
30. #117×C181C22semidirect
129. C181#3022semidirect
133. #536D181semidirect
147. C22×Q8C2×D181semidirect
153. C4⋊Q8C2×D181semidirect
154. #124C2×D181semidirect
157. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421152
C2×#191538472
#260414612
#260414562
#260412972
#260412632
#536×C1812
#3022181