Finite Group Databasebeta

← #37545341

Subgroups — #37545341

Order 23168.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19153841C212yesmaximal, complemented
2D4×D724C211yesmaximal, characteristic, complemented
3#26041655C211yesmaximal, characteristic, complemented
4#26041530C212yesmaximal, complemented
5#26041479C211yesmaximal, characteristic, complemented
6#26041471C211yesmaximal, characteristic, complemented
7#26041424C212yesmaximal, complemented
8#26041412C212yesmaximal, complemented
9C4×#15668436C211yesmaximal, characteristic, complemented
10#26041357C211yesmaximal, characteristic, complemented
11#550×C181C211yesmaximal, characteristic, complemented
12C22×D724C2212yes—
13C2×D4×D181—21no—
14#19153841—28no—
15C2×#15668436C2211yescharacteristic
16C2×#15668436C2212yescomplemented
17C2×#15668434C2212yes—
18C2×#15668434C2212yescomplemented
19C2×#15668434C2212yescomplemented
20C2×#15668433C2212yescomplemented
21#19153813C2211yescharacteristic
22#19153791—21no—
23#19153790C2212yes—
24#19153786—22no—
25C4×D724C2211yescharacteristic
26#19153747C2211yescharacteristic, complemented
27C4⋊#13089678C2211yescharacteristic, complemented
28#19153740C2211yescharacteristic, complemented
29#19153737C2212yescomplemented
30#19153736C2211yescharacteristic
31#19153736C2212yescomplemented
32C22⋊#13089678C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
34C2×#41×C181C2212yescomplemented
35#127×C181—21no—
36#127×C181C2211yescharacteristic, complemented
37#121×C181—22no—
38#121×C181C2212yescomplemented
39C4×Q8×C181C2211yescharacteristic, complemented
40C4×D4×C181—21no—
41C4×D4×C181C2211yescharacteristic
42C23×D181—22no—
43C2×D724—24no—
44C2×D724—22no—
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—42no—
48C2×D724C2312yes—
49C2×D724C2311yescharacteristic
50C2×C22⋊D181—22no—
51D4×D181—44no—
52C2×C4×D181—21no—
53#15668436—24no—
54#15668436D414yescomplemented
55#15668434—24no—
56#15668434—24no—
57#15668434—42no—
58#15668433—24no—
59C2×#13089678C2311yescharacteristic
60C2×#13089678C2312yes—
61C2×#13089678C2311yescharacteristic
62C4⋊Dic181C2311yescharacteristic
63#15668426—21no—
64#15668426—22no—
65#41×C181—24no—
66#41×C181—42no—
67C2×Q8×C181C2311yescharacteristic
68C2×D4×C181—22no—
69C2×D4×C181—22no—
70C2×D4×C181—21no—
71C2×D4×C181—21no—
72C2×D4×C181D411yescharacteristic
73C2×D4×C181C2312yes—
74C22×C4×C181—22no—
75C22×C4×C181C2312yes—
76C4⋊C4×C181—21no—
77C4⋊C4×C181D411yescharacteristic
78C4⋊C4×C181C2311yescharacteristic
79C22⋊C4×C181—22no—
80C22⋊C4×C181D412yes—
81C42×C181—21no—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—22no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88D724—22no—
89D724—21no—
90D724—22no—
91D724—44no—
92D724—42no—
93D724—81no—
94D724C2×D412yes—
95C22⋊D181—44no—
96C2×Dic181—21no—
97C4×D181—42no—
98#13089678—22no—
99#13089678—21no—
100#13089678C2×D412yescomplemented
101C23×C181—22no—
102C23×C181C2×D412yes—
103Q8×C181—21no—
104Q8×C181C2×D412yescomplemented
105D4×C181—22no—
106D4×C181—22no—
107D4×C181—22no—
108D4×C181—44no—
109D4×C181—42no—
110D4×C181—81no—
111D4×C181—82no—
112C2×C4×C181—21no—
113C2×C4×C181—22no—
114C2×C4×C181—22no—
115C2×C4×C181—21no—
116C2×C4×C181—44no—
117C2×C4×C181C2×D411yescharacteristic
118C2×C4×C181C2×D412yes—
119C2×C4×C181C2×D411yescharacteristic
120C2×C4×C181C2411yescharacteristic
121C22×C181—22no—
122C22×C181—22no—
123C22×C181—22no—
124C22×C181—42no—
125C22×C181—42no—
126C22×C181—42no—
127C22×C181C22×D411yescharacteristic
128C2×D181—22no—
129C2×D181—41no—
130C2×D181—41no—
131C2×D181—42no—
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—88no—
135C4×C181—22no—
136C4×C181—22no—
137C4×C181—21no—
138C4×C181—41no—
139C4×C181AGL(1,ℤ/8)12yescomplemented
140C4×C181C22×D411yescharacteristic
141C4×C181H(5,𝔽2)11yescharacteristic
142Dic181—41no—
143C2×C181—42no—
144C2×C181—82no—
145C2×C181D4211yescharacteristic
146C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C181#57611yescharacteristic
148D181—42no—
149D181—82no—
150C181#327411yescharacteristic, complemented, Sylow
151#3274—1811nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1812no—
153#550D18111yescharacteristic, complemented
154D42—1811no—
155#493—1811no—
156#469—1812no—
157#460—1811no—
158#459—1811no—
159#449—1812no—
160#447—1812no—
161C4×SD16—1811no—
162#436—1811no—
163C2×#41C2×D18112yescomplemented
164C22×D4—3621no—
165C22×D4—1812no—
166AGL(1,ℤ/8)—3628no—
167C2×SD16—1812no—
168C2×SD16—1811no—
169C2×D8—1812no—
170C2×D8—1812no—
171C2×D8—1812no—
172C2×M16—1812no—
173#127—21no—
174#127—1811no—
175#127C2×D18111yescharacteristic, complemented
176#121—22no—
177#121—3621no—
178#121—1812no—
179#121C2×D18112yescomplemented
180#120—3622no—
181C4×Q8C2×D18111yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—1811no—
184C4×D4C2×D18111yescharacteristic
185#106—1811no—
186C4⋊C8—1811no—
187#103—1811no—
188#102—1812no—
189#102—1811no—
190#102—1812no—
191C22⋊C8—1812no—
192C4×C8—1811no—
193C24—3622no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D18111yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3624no—
202C2×D4—3622no—
203C2×D4—3621no—
204C2×D4—3622no—
205C2×D4—3621no—
206C2×D4—1812no—
207C2×D4—1811no—
208C2×D4—7242no—
209C2×D4—7244no—
210C2×D4C22⋊D18111yescharacteristic
211C2×D4C22×D18112yes—
212C22×C4—22no—
213C22×C4—3621no—
214C22×C4C22×D18112yes—
215SD16—3624no—
216SD16—1814no—
217D8—3624no—
218D8—3624no—
219D8—7242no—
220M16—3624no—
221C2×C8—1811no—
222C2×C8—1812no—
223C2×C8—1811no—
224C4⋊C4—21no—
225C4⋊C4—1811no—
226C4⋊C4C22⋊D18111yescharacteristic
227C4⋊C4C22×D18111yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—3621no—
230C22⋊C4—3622no—
231C22⋊C4C22⋊D18112yes—
232C42—21no—
233C42C22×D18111yescharacteristic
234C23—22no—
235C23—3621no—
236C23—3622no—
237C23—14482no—
238C23—7244no—
239C23—7244no—
240C23C2×C22⋊D18112yes—
241Q8—21no—
242Q8D4×D18112yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—3622no—
251D4—3621no—
252D4—3622no—
253D4—14481no—
254D4—1812no—
255D4—7244no—
256D4—7244no—
257D4—7242no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—3621no—
264C2×C4—7242no—
265C2×C4C2×C22⋊D18111yescharacteristic
266C2×C4C2×C22⋊D18112yes—
267C2×C4C2×C22⋊D18111yescharacteristic
268C2×C4C23×D18111yescharacteristic
269C8—3622no—
270C8—3621no—
271C8—1812no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3622no—
279C22—14488no—
280C22—7242no—
281C22—7242no—
282C22—7242no—
283C22—7241no—
284C22—7241no—
285C22C22×C22⋊D18111yescharacteristic
286C4—22no—
287C4—22no—
288C4—21no—
289C4—41no—
290C4—7241no—
291C4#1915384112yescomplemented
292C4C2×D4×D18111yescharacteristic
293C4#1915386111yescharacteristic
294C2—42no—
295C2—82no—
296C2—14482no—
297C2—7242no—
298C2D4×C22⋊D18111yescharacteristic
299C2C2×#1915384111yescharacteristic
300C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#550×C181C2
socleC22×C181C22×D4
hypercenter#550D181
hypocenterC181#3274
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#19153841C2semidirect
2. D4×D724C2semidirect
3. #26041655C2semidirect
4. #26041530C2semidirect
5. #26041479C2semidirect
6. #26041471C2semidirect
7. #26041424C2semidirect
8. #26041412C2semidirect
9. C4×#15668436C2semidirect
10. #26041357C2semidirect
11. #550×C181C2semidirect
16. C2×#15668436C22semidirect
18. C2×#15668434C22semidirect
19. C2×#15668434C22semidirect
20. C2×#15668433C22semidirect
26. #19153747C22semidirect
27. C4⋊#13089678C22semidirect
28. #19153740C22semidirect
29. #19153737C22semidirect
31. #19153736C22semidirect
32. C22⋊#13089678C22semidirect
33. C4×#13089678C22semidirect
34. C2×#41×C181C22semidirect
36. #127×C181C22semidirect
38. #121×C181C22semidirect
39. C4×Q8×C181C22semidirect
54. #15668436D4semidirect
100. #13089678C2×D4semidirect
104. Q8×C181C2×D4semidirect
139. C4×C181AGL(1,ℤ/8)semidirect
150. C181#3274semidirect
153. #550D181semidirect
163. C2×#41C2×D181semidirect
175. #127C2×D181semidirect
179. #121C2×D181semidirect
181. C4×Q8C2×D181semidirect
242. Q8D4×D181semidirect
291. C4#19153841semidirect

Maximal subgroups

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

subgroupindex
C2×#191538412
D4×D7242
#260416552
#260415302
#260414792
#260414712
#260414242
#260414122
C4×#156684362
#260413572
#550×C1812
#3274181