Finite Group Databasebeta

← #37542996

Subgroups — #37542996

Order 23168.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19153862C211yesmaximal, characteristic
2C2×#19153847C211yesmaximal, characteristic, complemented
3C22×#15668436C211yesmaximal, characteristic, complemented
4#26041439C214yesmaximal, complemented
5#26041429C214yesmaximal, complemented
6#26041283C211yesmaximal, characteristic, complemented
7#26041273C211yesmaximal, characteristic, complemented
8#26041246C211yesmaximal, characteristic, complemented
9#536×C181C211yesmaximal, characteristic, complemented
10#19153862—24no—
11#19153862—24no—
12C2×#15668450—21no—
13C2×#15668449C2212yes—
14C2×#15668449C2211yescharacteristic
15C22×D724C2212yes—
16C22×D724C2211yescharacteristic
17C2×D4×D181—22no—
18C2×D4×D181—21no—
19#19153847—24no—
20C2×#15668439C2212yescomplemented
21C2×#15668436—24no—
22C2×#15668436C2212yescomplemented
23C2×#15668436C2214yescomplemented
24C2×#15668433C2211yescharacteristic, complemented
25C22×#13089678C2211yescharacteristic, complemented
26#19153739C2212yescomplemented
27#19153736C2212yescomplemented
28#19153736C2214yescomplemented
29C22⋊#13089678C2214yescomplemented
30C2×#41×C181C2211yescharacteristic
31C22×Q8×C181C2211yescharacteristic, complemented
32C4⋊Q8×C181—21no—
33#124×C181—21no—
34#123×C181—22no—
35C22⋊Q8×C181C2214yescomplemented
36#117×C181C2211yescharacteristic, complemented
37C23×D181—22no—
38C23×D181—21no—
39#15668450—28no—
40#15668449—24no—
41#15668449—24no—
42#15668449—22no—
43#15668449D414yes—
44C2×Dic362C2311yescharacteristic
45C2×D724—24no—
46C2×D724—24no—
47C2×D724—22no—
48C2×D724—21no—
49C2×D724—22no—
50C2×D724D414yes—
51C2×D724C2311yescharacteristic
52C2×D724C2312yes—
53C2×C22⋊D181—22no—
54C2×C22⋊D181—21no—
55D4×D181—216no—
56D4×D181—28no—
57C2×C4×D181—22no—
58C2×C4×D181—21no—
59#15668439—24no—
60#15668436—28no—
61#15668436—24no—
62#15668433—22no—
63C2×#13089678—22no—
64C2×#13089678C2312yes—
65C2×#13089678C2312yes—
66#41×C181—22no—
67#41×C181—22no—
68C2×Q8×C181—21no—
69C2×Q8×C181—22no—
70C2×Q8×C181—22no—
71C2×Q8×C181D411yescharacteristic
72C2×Q8×C181C2312yes—
73C2×D4×C181D412yes—
74C2×D4×C181D411yescharacteristic
75C22×C4×C181—21no—
76C22×C4×C181C2312yes—
77C22×C4×C181C2311yescharacteristic
78C4⋊C4×C181—24no—
79C4⋊C4×C181C2312yes—
80C22⋊C4×C181—21no—
81C22⋊C4×C181—24no—
82C42×C181—21no—
83C22×D181—22no—
84C22×D181—21no—
85C22×D181—28no—
86C22×D181—24no—
87C22×D181—82no—
88C22×D181—84no—
89Dic362—21no—
90Dic362C2×D412yes—
91D724—28no—
92D724—24no—
93D724—22no—
94D724—21no—
95D724C2×D412yes—
96D724C2×D414yes—
97C22⋊D181—28no—
98C22⋊D181—24no—
99C2×Dic181—21no—
100C4×D181—28no—
101C4×D181—24no—
102#13089678—22no—
103#13089678—22no—
104C23×C181C2×D412yes—
105C23×C181C2×D411yescharacteristic
106Q8×C181—22no—
107Q8×C181—22no—
108Q8×C181—22no—
109Q8×C181—42no—
110D4×C181—24no—
111D4×C181—22no—
112C2×C4×C181—22no—
113C2×C4×C181—22no—
114C2×C4×C181—22no—
115C2×C4×C181—22no—
116C2×C4×C181—42no—
117C2×C4×C181C2×D412yes—
118C2×C4×C181C2×D412yes—
119C2×C4×C181C2×D412yes—
120C2×C4×C181C2×D411yescharacteristic
121C2×C4×C181C2411yescharacteristic
122C22×C181—22no—
123C22×C181—22no—
124C22×C181—24no—
125C22×C181#12012yes—
126C22×C181C22×D411yescharacteristic
127C2×D181—24no—
128C2×D181—22no—
129C2×D181—44no—
130C2×D181—42no—
131C2×D181—84no—
132C2×D181—88no—
133C4×C181—22no—
134C4×C181—42no—
135C4×C181—42no—
136C4×C181#12012yes—
137C4×C181C22×D411yescharacteristic
138C4×C181C22×D411yescharacteristic
139Dic181—22no—
140C2×C181—22no—
141C2×C181—42no—
142C2×C181C2×#12011yescharacteristic
143C2×C181#57712yes—
144D181—44no—
145D181—42no—
146C181#300911yescharacteristic, complemented, Sylow
147#3009—1811nomaximal, Sylow
148C2×H(5,𝔽2)—1811no—
149C2×#137—1811no—
150C22×SD16—1811no—
151#536D18111yescharacteristic, complemented
152#452—1814no—
153#450—1814no—
154#418—1811no—
155#416—1811no—
156#408—1811no—
157H(5,𝔽2)—3624no—
158H(5,𝔽2)—3624no—
159C2×#41—3621no—
160C2×#41—1812no—
161C2×#41—1811no—
162C2×#41C2×D18111yescharacteristic
163C22×Q8C2×D18111yescharacteristic, complemented
164C22×D4—3622no—
165C22×D4—3621no—
166C22×D4—1812no—
167C22×D4—1811no—
168#137—3624no—
169C2×Q16—1812no—
170C2×SD16—3624no—
171C2×SD16—1814no—
172C2×SD16—1812no—
173C2×M16—1811no—
174C22×C8—1811no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D18114yescomplemented
179#117C2×D18111yescharacteristic, complemented
180#103—1812no—
181#102—1812no—
182#102—1814no—
183C22⋊C8—1814no—
184C24—3622no—
185C24—3621no—
186#41—22no—
187#41—22no—
188#41—3624no—
189#41—3624no—
190#41—3628no—
191#41—3622no—
192#41—1814no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1811no—
197C2×Q8C22⋊D18111yescharacteristic
198C2×Q8C22×D18112yes—
199C2×D4—3624no—
200C2×D4—3624no—
201C2×D4—36216no—
202C2×D4—3628no—
203C2×D4—3622no—
204C2×D4—3621no—
205C2×D4—3622no—
206C2×D4—3622no—
207C2×D4—3621no—
208C2×D4—1814no—
209C2×D4—1811no—
210C2×D4—1812no—
211C2×D4C22⋊D18112yes—
212C2×D4C22⋊D18111yescharacteristic
213C22×C4—21no—
214C22×C4—3622no—
215C22×C4—3621no—
216C22×C4C22×D18112yes—
217C22×C4C22×D18111yescharacteristic
218Q16—3624no—
219SD16—3628no—
220SD16—3624no—
221M16—3622no—
222C2×C8—3622no—
223C2×C8—1812no—
224C2×C8—1812no—
225C4⋊C4—24no—
226C4⋊C4C22×D18112yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—3628no—
231C23—3624no—
232C23—3622no—
233C23—3621no—
234C23—14482no—
235C23—14484no—
236C23C2×C22⋊D18112yes—
237C23C2×C22⋊D18111yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—3621no—
243Q8—1812no—
244D4—24no—
245D4—22no—
246D4—3628no—
247D4—3628no—
248D4—3624no—
249D4—3624no—
250D4—3622no—
251D4—3621no—
252D4—1812no—
253D4—1814no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—3628no—
260C2×C4—3621no—
261C2×C4—3624no—
262C2×C4D4×D18112yes—
263C2×C4D4×D18112yes—
264C2×C4C2×C22⋊D18112yes—
265C2×C4C2×C22⋊D18111yescharacteristic
266C2×C4C23×D18111yescharacteristic
267C8—3622no—
268C8—3622no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—3624no—
273C22—3622no—
274C22—14484no—
275C22—14488no—
276C22—7244no—
277C22—7242no—
278C22#1915378412yes—
279C22C22×C22⋊D18111yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—3622no—
284C4#1915378412yes—
285C4C2×D4×D18111yescharacteristic
286C4C2×D4×D18111yescharacteristic
287C2—22no—
288C2—42no—
289C2—7244no—
290C2—7242no—
291C2C2×#1915378411yescharacteristic
292C2#2604208712yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#536×C181C2
socleC22×C181C22×D4
prime radicalC2×#19153862C2
hypercenter#536D181
hypocenterC181#3009
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#19153847C2semidirect
3. C22×#15668436C2semidirect
4. #26041439C2semidirect
5. #26041429C2semidirect
6. #26041283C2semidirect
7. #26041273C2semidirect
8. #26041246C2semidirect
9. #536×C181C2semidirect
20. C2×#15668439C22semidirect
22. C2×#15668436C22semidirect
23. C2×#15668436C22semidirect
24. C2×#15668433C22semidirect
25. C22×#13089678C22semidirect
26. #19153739C22semidirect
27. #19153736C22semidirect
28. #19153736C22semidirect
29. C22⋊#13089678C22semidirect
31. C22×Q8×C181C22semidirect
35. C22⋊Q8×C181C22semidirect
36. #117×C181C22semidirect
146. C181#3009semidirect
151. #536D181semidirect
163. C22×Q8C2×D181semidirect
178. C22⋊Q8C2×D181semidirect
179. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538622
C2×#191538472
C22×#156684362
#260414392
#260414292
#260412832
#260412732
#260412462
#536×C1812
#3009181