Finite Group Databasebeta

← #37542987

Subgroups — #37542987

Order 23168.

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

normal subgroups only

This group has 784 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, complemented
2C2×#19153842C211yesmaximal, characteristic, complemented
3C22×#15668434C211yesmaximal, characteristic, complemented
4#26041422C214yesmaximal, complemented
5#26041412C214yesmaximal, complemented
6#26041283C211yesmaximal, characteristic, complemented
7#26041273C211yesmaximal, characteristic, complemented
8#26041246C211yesmaximal, characteristic, complemented
9#535×C181C211yesmaximal, characteristic, complemented
10#19153862—24no—
11#19153862—24no—
12C2×#15668450—21no—
13C2×#15668449C2211yescharacteristic
14C2×#15668449C2212yescomplemented
15C22×D724C2211yescharacteristic
16C22×D724C2212yescomplemented
17C2×D4×D181—22no—
18C2×D4×D181—21no—
19#19153842—24no—
20C2×#15668437C2212yescomplemented
21C2×#15668434—24no—
22C2×#15668434C2212yescomplemented
23C2×#15668434C2214yescomplemented
24C2×#15668433C2211yescharacteristic, complemented
25C22×#13089678C2211yescharacteristic, complemented
26#19153739C2212yescomplemented
27#19153736C2212yescomplemented
28#19153736C2214yescomplemented
29C22⋊#13089678C2214yescomplemented
30C2×#41×C181C2211yescharacteristic
31C22×D4×C181C2211yescharacteristic, complemented
32#127×C181—21no—
33#124×C181—21no—
34#121×C181C2214yescomplemented
35#120×C181—22no—
36#117×C181C2211yescharacteristic, complemented
37C23×D181—21no—
38C23×D181—22no—
39#15668450—28no—
40#15668449—24no—
41#15668449—24no—
42#15668449—22no—
43#15668449D414yescomplemented
44C2×Dic362C2311yescharacteristic
45C2×D724—24no—
46C2×D724—24no—
47C2×D724—21no—
48C2×D724—22no—
49C2×D724—22no—
50C2×D724D414yescomplemented
51C2×D724C2312yes—
52C2×D724C2311yescharacteristic
53C2×C22⋊D181—21no—
54C2×C22⋊D181—22no—
55D4×D181—216no—
56D4×D181—28no—
57C2×C4×D181—21no—
58C2×C4×D181—22no—
59#15668437—24no—
60#15668434—28no—
61#15668434—24no—
62#15668433—22no—
63C2×#13089678—22no—
64C2×#13089678C2312yes—
65C2×#13089678C2312yes—
66C24×C181—21no—
67#41×C181—22no—
68#41×C181—22no—
69C2×Q8×C181D411yescharacteristic
70C2×D4×C181—24no—
71C2×D4×C181—22no—
72C2×D4×C181—22no—
73C2×D4×C181—21no—
74C2×D4×C181D412yes—
75C2×D4×C181D411yescharacteristic
76C2×D4×C181C2312yes—
77C22×C4×C181C2312yes—
78C22×C4×C181C2311yescharacteristic
79C4⋊C4×C181C2312yes—
80C22⋊C4×C181—21no—
81C22⋊C4×C181—24no—
82C42×C181—21no—
83C22×D181—21no—
84C22×D181—22no—
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×C181—22no—
105C23×C181—42no—
106C23×C181—82no—
107C23×C181C2×D412yes—
108C23×C181C2×D411yescharacteristic
109Q8×C181—22no—
110D4×C181—24no—
111D4×C181—22no—
112D4×C181—22no—
113D4×C181—22no—
114D4×C181—42no—
115D4×C181—84no—
116C2×C4×C181—22no—
117C2×C4×C181—22no—
118C2×C4×C181—22no—
119C2×C4×C181C2×D412yes—
120C2×C4×C181C2×D412yes—
121C2×C4×C181C2×D412yes—
122C2×C4×C181C2×D411yescharacteristic
123C2×C4×C181C2411yescharacteristic
124C22×C181—22no—
125C22×C181—22no—
126C22×C181—24no—
127C22×C181—42no—
128C22×C181—44no—
129C22×C181—84no—
130C22×C181#12012yes—
131C22×C181C22×D411yescharacteristic
132C2×D181—24no—
133C2×D181—22no—
134C2×D181—44no—
135C2×D181—42no—
136C2×D181—84no—
137C2×D181—88no—
138C4×C181—22no—
139C4×C181—42no—
140C4×C181#12012yes—
141C4×C181C22×D411yescharacteristic
142C4×C181C22×D411yescharacteristic
143Dic181—22no—
144C2×C181—22no—
145C2×C181—42no—
146C2×C181—82no—
147C2×C181C2×#12011yescharacteristic
148C2×C181#57612yes—
149D181—44no—
150D181—42no—
151C181#300811yescharacteristic, complemented, Sylow
152#3008—1811nomaximal, Sylow
153C2×H(5,𝔽2)—1811no—
154C2×AGL(1,ℤ/8)—1811no—
155C22×D8—1811no—
156#535D18111yescharacteristic, complemented
157#449—1814no—
158#447—1814no—
159#418—1811no—
160#416—1811no—
161#408—1811no—
162H(5,𝔽2)—3624no—
163H(5,𝔽2)—3624no—
164C2×#41—3621no—
165C2×#41—1812no—
166C2×#41—1811no—
167C2×#41C2×D18111yescharacteristic
168C22×D4—3622no—
169C22×D4—3621no—
170C22×D4—1812no—
171C22×D4—1811no—
172C22×D4C2×D18111yescharacteristic, complemented
173AGL(1,ℤ/8)—3624no—
174C2×SD16—1812no—
175C2×D8—3624no—
176C2×D8—1814no—
177C2×D8—1812no—
178C2×M16—1811no—
179C22×C8—1811no—
180#127—21no—
181#124—21no—
182#121C2×D18114yescomplemented
183#120—22no—
184#117C2×D18111yescharacteristic, complemented
185#103—1812no—
186#102—1812no—
187#102—1814no—
188C22⋊C8—1814no—
189C24—21no—
190C24—3622no—
191C24—3621no—
192#41—22no—
193#41—22no—
194#41—3624no—
195#41—3624no—
196#41—3628no—
197#41—3622no—
198#41—1814no—
199C2×Q8—1811no—
200C2×Q8C22⋊D18111yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—3624no—
206C2×D4—3624no—
207C2×D4—36216no—
208C2×D4—3628no—
209C2×D4—3622no—
210C2×D4—3621no—
211C2×D4—3622no—
212C2×D4—3622no—
213C2×D4—3621no—
214C2×D4—1814no—
215C2×D4—1811no—
216C2×D4—1812no—
217C2×D4C22⋊D18112yes—
218C2×D4C22⋊D18111yescharacteristic
219C2×D4C22×D18112yes—
220C22×C4—3622no—
221C22×C4—3621no—
222C22×C4C22×D18112yes—
223C22×C4C22×D18111yescharacteristic
224SD16—3624no—
225D8—3628no—
226D8—3624no—
227M16—3622no—
228C2×C8—3622no—
229C2×C8—1812no—
230C2×C8—1812no—
231C4⋊C4C22×D18112yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—3628no—
239C23—3624no—
240C23—3622no—
241C23—3621no—
242C23—14482no—
243C23—14484no—
244C23C2×C22⋊D18112yes—
245C23C2×C22⋊D18111yescharacteristic
246Q8—22no—
247Q8—3621no—
248Q8—1812no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—3628no—
256D4—3628no—
257D4—3624no—
258D4—3624no—
259D4—3622no—
260D4—3621no—
261D4—1812no—
262D4—1814no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—3628no—
267C2×C4—3621no—
268C2×C4—3624no—
269C2×C4D4×D18112yes—
270C2×C4D4×D18112yes—
271C2×C4C2×C22⋊D18112yes—
272C2×C4C2×C22⋊D18111yescharacteristic
273C2×C4C23×D18111yescharacteristic
274C8—3622no—
275C8—3622no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—3624no—
283C22—3622no—
284C22—14484no—
285C22—14488no—
286C22—7244no—
287C22—7242no—
288C22#1915378412yes—
289C22C22×C22⋊D18111yescharacteristic
290C4—22no—
291C4—42no—
292C4—3622no—
293C4#1915378412yes—
294C4C2×D4×D18111yescharacteristic
295C4C2×D4×D18111yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—7244no—
300C2—7242no—
301C2C2×#1915378411yescharacteristic
302C2#2604208212yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#535×C181C2
socleC22×C181C22×D4
hypercenter#535D181
hypocenterC181#3008
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2semidirect
2. C2×#19153842C2semidirect
3. C22×#15668434C2semidirect
4. #26041422C2semidirect
5. #26041412C2semidirect
6. #26041283C2semidirect
7. #26041273C2semidirect
8. #26041246C2semidirect
9. #535×C181C2semidirect
14. C2×#15668449C22semidirect
16. C22×D724C22semidirect
20. C2×#15668437C22semidirect
22. C2×#15668434C22semidirect
23. C2×#15668434C22semidirect
24. C2×#15668433C22semidirect
25. C22×#13089678C22semidirect
26. #19153739C22semidirect
27. #19153736C22semidirect
28. #19153736C22semidirect
29. C22⋊#13089678C22semidirect
31. C22×D4×C181C22semidirect
34. #121×C181C22semidirect
36. #117×C181C22semidirect
43. #15668449D4semidirect
50. C2×D724D4semidirect
151. C181#3008semidirect
156. #535D181semidirect
172. C22×D4C2×D181semidirect
182. #121C2×D181semidirect
184. #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×#191538422
C22×#156684342
#260414222
#260414122
#260412832
#260412732
#260412462
#535×C1812
#3008181