Finite Group Databasebeta

← #37542950

Subgroups — #37542950

Order 23168.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D181C211yesmaximal, characteristic, complemented
2C2×#19153844C211yesmaximal, characteristic, complemented
3C2×#19153842C211yesmaximal, characteristic, complemented
4#26041791C211yesmaximal, characteristic, complemented
5#26041431C214yesmaximal, complemented
6#26041414C214yesmaximal, complemented
7#26041283C211yesmaximal, characteristic, complemented
8#26041242C211yesmaximal, characteristic, complemented
9#418×C181C211yesmaximal, characteristic, complemented
10C24×D181—21no—
11C2×#15668449C2211yescharacteristic, complemented
12C22×D724C2211yescharacteristic
13C22×C22⋊D181—21no—
14C2×D4×D181—24no—
15C2×D4×D181—22no—
16C2×D4×D181—24no—
17C2×D4×D181C2214yes—
18C22×C4×D181C2211yescharacteristic
19#19153844—24no—
20#19153842—24no—
21C2×#15668438C2212yescomplemented
22C2×#15668437C2212yescomplemented
23C2×D1448C2212yescomplemented
24C2×#15668434C2212yescomplemented
25C2×#15668433C2211yescharacteristic, complemented
26#19153800—21no—
27#19153796C2212yescomplemented
28#19153791C2212yescomplemented
29#19153786—21no—
30C4×D724—22no—
31#19153736C2214yescomplemented
32#19153728C2214yescomplemented
33C22×D4×C181C2211yescharacteristic
34C2×M16×C181C2211yescharacteristic, complemented
35#117×C181C2211yescharacteristic, complemented
36#102×C181C2214yescomplemented
37C23×D181—28no—
38C23×D181—24no—
39C23×D181—21no—
40C23×D181—48no—
41C23×D181D411yescharacteristic
42#15668449—22no—
43#15668449—22no—
44C2×Dic362C2311yescharacteristic, complemented
45C2×D724—22no—
46C2×D724—21no—
47C2×D724—22no—
48C2×D724D414yescomplemented
49C2×D724C2312yes—
50C2×D724C2311yescharacteristic, complemented
51C2×C22⋊D181—28no—
52C2×C22⋊D181—24no—
53C2×C22⋊D181—21no—
54D4×D181—216no—
55D4×D181—28no—
56D4×D181—28no—
57C22×Dic181D411yescharacteristic
58C2×C4×D181—22no—
59C2×C4×D181—22no—
60C2×C4×D181—21no—
61C2×C4×D181D412yes—
62C2×C4×D181C2312yes—
63#15668438—24no—
64#15668437—24no—
65D1448—24no—
66#15668434—24no—
67#15668433—22no—
68C2×#13089678C2312yescomplemented
69C4⋊Dic181—22no—
70#15668426—22no—
71#15668426—22no—
72C24×C181—21no—
73C2×D4×C181—21no—
74C2×D4×C181—22no—
75C2×D4×C181D414yescomplemented
76C2×D4×C181C2312yes—
77C22×C4×C181C2311yescharacteristic
78M16×C181—22no—
79C2×C8×C181C2312yescomplemented
80C4⋊C4×C181C2312yescomplemented
81C22⋊C4×C181—21no—
82C42×C181—21no—
83C22×D181—22no—
84C22×D181—24no—
85C22×D181—216no—
86C22×D181—24no—
87C22×D181—22no—
88C22×D181—21no—
89C22×D181—432no—
90C22×D181—48no—
91C22×D181—44no—
92C22×D181—48no—
93C22×D181C2×D412yes—
94Dic362—21no—
95Dic362—21no—
96D724—24no—
97D724—22no—
98D724—21no—
99D724—21no—
100D724—44no—
101D724C2×D414yes—
102C22⋊D181—216no—
103C22⋊D181—42no—
104C2×Dic181—22no—
105C2×Dic181—21no—
106C2×Dic181C2×D412yes—
107C4×D181—24no—
108C4×D181—22no—
109C4×D181—22no—
110C4×D181—42no—
111#13089678—22no—
112C23×C181—24no—
113C23×C181—22no—
114C23×C181—44no—
115C23×C181C2×D411yescharacteristic
116D4×C181—24no—
117D4×C181—22no—
118D4×C181C2×D414yes—
119C2×C4×C181—22no—
120C2×C4×C181C2×D411yescharacteristic
121C2×C4×C181C2×D412yes—
122C2×C4×C181C2×D412yes—
123C2×C4×C181C2411yescharacteristic
124C8×C181—22no—
125C22×C181—22no—
126C22×C181—24no—
127C22×C181—48no—
128C22×C181—44no—
129C22×C181#12012yes—
130C22×C181C22×D411yescharacteristic
131C2×D181—28no—
132C2×D181—22no—
133C2×D181—22no—
134C2×D181—24no—
135C2×D181—432no—
136C2×D181—48no—
137C2×D181—44no—
138C2×D181—41no—
139C2×D181—42no—
140C2×D181AGL(1,ℤ/8)14yescomplemented
141C4×C181—42no—
142C4×C181#12012yes—
143C4×C181C22×D411yescharacteristic
144C4×C181C22×D411yescharacteristic
145Dic181—22no—
146Dic181—41no—
147C2×C181—22no—
148C2×C181—44no—
149C2×C181C2×#12011yescharacteristic
150C2×C181C2×AGL(1,ℤ/8)12yes—
151D181—24no—
152D181—44no—
153D181—81no—
154C181#300311yescharacteristic, complemented, Sylow
155#3003—1811nomaximal, Sylow
156C23×D4—1811no—
157C2×AGL(1,ℤ/8)—1811no—
158C2×AGL(1,ℤ/8)—1811no—
159#525—1811no—
160#450—1814no—
161#447—1814no—
162#418—1811no—
163#418D18111yescharacteristic, complemented
164#407—1811no—
165C25—3621no—
166C2×#41—1811no—
167C22×D4—3624no—
168C22×D4—3624no—
169C22×D4—3622no—
170C22×D4—3621no—
171C22×D4—1814no—
172C22×D4—1811no—
173C22×D4C2×D18111yescharacteristic
174C23×C4—1811no—
175AGL(1,ℤ/8)—3624no—
176AGL(1,ℤ/8)—3624no—
177C2×SD16—1812no—
178C2×SD16—1812no—
179C2×D8—1812no—
180C2×D8—1812no—
181C2×M16—1811no—
182C2×M16C2×D18111yescharacteristic, complemented
183#123—3621no—
184C22⋊Q8—1812no—
185#121—1812no—
186#120—3621no—
187C4×D4—3622no—
188#117C2×D18111yescharacteristic, complemented
189#102—1814no—
190#102C2×D18114yescomplemented
191C22⋊C8—1814no—
192C24—21no—
193C24—3628no—
194C24—3624no—
195C24—3621no—
196C24—1811no—
197C24—7248no—
198#41—3622no—
199#41—3622no—
200C2×Q8—1811no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—3622no—
204C2×D4—36216no—
205C2×D4—3628no—
206C2×D4—3628no—
207C2×D4—3628no—
208C2×D4—3621no—
209C2×D4—3624no—
210C2×D4—3621no—
211C2×D4—3622no—
212C2×D4—1814no—
213C2×D4—1811no—
214C2×D4—1812no—
215C2×D4D72414yescomplemented
216C2×D4C22×D18112yes—
217C22×C4—3622no—
218C22×C4—3622no—
219C22×C4—3621no—
220C22×C4—1811no—
221C22×C4—1812no—
222C22×C4—1812no—
223C22×C4C22×D18111yescharacteristic
224SD16—3624no—
225SD16—3624no—
226D8—3624no—
227D8—3624no—
228M16—22no—
229M16—3622no—
230C2×C8—1812no—
231C2×C8C22×D18112yescomplemented
232C4⋊C4—3622no—
233C4⋊C4C22×D18112yescomplemented
234C22⋊C4—21no—
235C22⋊C4—3622no—
236C22⋊C4—3622no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—3624no—
242C23—36216no—
243C23—3622no—
244C23—3624no—
245C23—3622no—
246C23—3621no—
247C23—1812no—
248C23—7244no—
249C23—72432no—
250C23—7248no—
251C23—7248no—
252C23D4×D18111yescharacteristic
253Q8—3621no—
254Q8—3621no—
255D4—24no—
256D4—22no—
257D4—36216no—
258D4—3624no—
259D4—3621no—
260D4—3622no—
261D4—3621no—
262D4—1814no—
263D4—7244no—
264D4—7242no—
265D4C2×D72414yes—
266C2×C4—22no—
267C2×C4—3622no—
268C2×C4—3624no—
269C2×C4—3622no—
270C2×C4—3622no—
271C2×C4—3621no—
272C2×C4—1812no—
273C2×C4—7242no—
274C2×C4D4×D18111yescharacteristic
275C2×C4D4×D18112yes—
276C2×C4C2×D72412yes—
277C2×C4C23×D18111yescharacteristic
278C8—22no—
279C8—3622no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—3628no—
285C22—3622no—
286C22—3622no—
287C22—3624no—
288C22—1814no—
289C22—7248no—
290C22—72432no—
291C22—7241no—
292C22—7242no—
293C22—7244no—
294C22#1915378612yes—
295C22C2×D4×D18111yescharacteristic
296C4—42no—
297C4—3622no—
298C4—7241no—
299C4#1915378612yes—
300C4C2×D4×D18111yescharacteristic
301C4C22×D72411yescharacteristic
302C2—22no—
303C2—44no—
304C2—3624no—
305C2—14481no—
306C2—7244no—
307C2C2×#1915378611yescharacteristic
308C2AGL(1,ℤ/8)×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#418×C181C2
socleC22×C181C22×D4
hypercenter#418D181
hypocenterC181#3003
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D181C2semidirect
2. C2×#19153844C2semidirect
3. C2×#19153842C2semidirect
4. #26041791C2semidirect
5. #26041431C2semidirect
6. #26041414C2semidirect
7. #26041283C2semidirect
8. #26041242C2semidirect
9. #418×C181C2semidirect
11. C2×#15668449C22semidirect
21. C2×#15668438C22semidirect
22. C2×#15668437C22semidirect
23. C2×D1448C22semidirect
24. C2×#15668434C22semidirect
25. C2×#15668433C22semidirect
27. #19153796C22semidirect
28. #19153791C22semidirect
31. #19153736C22semidirect
32. #19153728C22semidirect
34. C2×M16×C181C22semidirect
35. #117×C181C22semidirect
36. #102×C181C22semidirect
44. C2×Dic362C23semidirect
48. C2×D724D4semidirect
50. C2×D724C23semidirect
68. C2×#13089678C23semidirect
75. C2×D4×C181D4semidirect
79. C2×C8×C181C23semidirect
80. C4⋊C4×C181C23semidirect
140. C2×D181AGL(1,ℤ/8)semidirect
154. C181#3003semidirect
163. #418D181semidirect
182. C2×M16C2×D181semidirect
188. #117C2×D181semidirect
190. #102C2×D181semidirect
215. C2×D4D724semidirect
231. C2×C8C22×D181semidirect
233. C4⋊C4C22×D181semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1812
C2×#191538442
C2×#191538422
#260417912
#260414312
#260414142
#260412832
#260412422
#418×C1812
#3003181