Finite Group Databasebeta

← #37542940

Subgroups — #37542940

Order 23168.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15668448C211yesmaximal, characteristic, complemented
2C2×#19153837C212yesmaximal, complemented
3#26041792C211yesmaximal, characteristic, complemented
4#26041438C212yesmaximal, complemented
5#26041434C211yesmaximal, characteristic, complemented
6#26041425C212yesmaximal, complemented
7#26041420C211yesmaximal, characteristic, complemented
8#26041413C211yesmaximal, characteristic, complemented
9#26041272C212yesmaximal, complemented
10C22⋊C8×D181C211yesmaximal, characteristic, complemented
11#450×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic, complemented
13C2×#15668448—28no—
14C2×#15668448—21no—
15C2×#15668448—21no—
16C2×#15668448C2212yes—
17C2×#15668448C2212yes—
18C22×Dic362C2211yescharacteristic
19C22×C22⋊D181—21no—
20C23×Dic181—21no—
21C22×C4×D181C2211yescharacteristic
22#19153837—28no—
23C2×#15668439C2212yescomplemented
24C2×#15668438C2212yescomplemented
25C2×#15668434C2212yescomplemented
26C2×C8×D181C2212yescomplemented
27#19153801—21no—
28#19153792C2211yescharacteristic, complemented
29#19153791C2211yescharacteristic, complemented
30#19153790C2211yescharacteristic
31#19153786—21no—
32#19153778—21no—
33#19153776—21no—
34#19153774C2211yescharacteristic, complemented
35#19153741C2212yescomplemented
36#19153739C2212yescomplemented
37#19153737C2212yescomplemented
38C22⋊#13089678C2211yescharacteristic
39#19153728C2212yescomplemented
40C22×D4×C181C2211yescharacteristic
41C2×SD16×C181C2212yescomplemented
42C22⋊Q8×C181C2211yescharacteristic
43#102×C181C2212yescomplemented
44C22⋊C8×C181C2211yescharacteristic
45C23×D181D411yescharacteristic
46#15668450—24no—
47#15668448—28no—
48#15668448—28no—
49#15668448—24no—
50#15668448—24no—
51#15668448—24no—
52#15668448D418yescomplemented
53C2×Dic362—24no—
54C2×Dic362—21no—
55C2×Dic362C2312yes—
56C2×D724—21no—
57C2×D724—21no—
58C2×D724C2311yescharacteristic
59C2×C22⋊D181—28no—
60C2×C22⋊D181—22no—
61C2×C22⋊D181—22no—
62C2×C22⋊D181—21no—
63C22×Dic181—28no—
64C22×Dic181—22no—
65C22×Dic181—22no—
66C22×Dic181—21no—
67C22×Dic181D411yescharacteristic
68C2×C4×D181—24no—
69C2×C4×D181—21no—
70C2×C4×D181D411yescharacteristic
71C2×C4×D181D411yescharacteristic
72C2×C4×D181C2311yescharacteristic
73C2×C4×D181C2311yescharacteristic
74#15668439—24no—
75#15668438—24no—
76#15668434—24no—
77C2×#13089678C2312yes—
78C8×D181—24no—
79C4⋊Dic181C2311yescharacteristic
80#15668428—21no—
81#15668426—21no—
82#15668426—21no—
83#15668426—21no—
84#15668426—21no—
85C4×Dic181—21no—
86C24×C181—21no—
87C2×Q8×C181C2311yescharacteristic
88C2×D4×C181—24no—
89C2×D4×C181—21no—
90C2×D4×C181C2312yes—
91C22×C4×C181C2311yescharacteristic
92SD16×C181—24no—
93C2×C8×C181C2312yes—
94C4⋊C4×C181—21no—
95C4⋊C4×C181C2311yescharacteristic
96C22⋊C4×C181—21no—
97C22×D181—22no—
98C22×D181—22no—
99C22×D181—21no—
100C22×D181—41no—
101C22×D181C2×D411yescharacteristic
102C22×D181C2×D411yescharacteristic
103Dic362—24no—
104Dic362—22no—
105Dic362C2×D414yes—
106D724—22no—
107D724—42no—
108D724—42no—
109C22⋊D181—28no—
110C22⋊D181—28no—
111C22⋊D181—42no—
112C2×Dic181—24no—
113C2×Dic181—28no—
114C2×Dic181—22no—
115C2×Dic181—22no—
116C2×Dic181—28no—
117C2×Dic181—21no—
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120C4×D181—22no—
121C4×D181—22no—
122C4×D181—22no—
123C4×D181—42no—
124C4×D181C2×D414yes—
125#13089678—22no—
126C23×C181—24no—
127C23×C181—22no—
128C23×C181—42no—
129C23×C181—42no—
130C23×C181C2×D411yescharacteristic
131Q8×C181—22no—
132D4×C181—24no—
133D4×C181—22no—
134D4×C181C2×D414yes—
135C2×C4×C181—22no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2411yescharacteristic
140C8×C181—22no—
141C22×C181—21no—
142C22×C181—21no—
143C22×C181—24no—
144C22×C181—48no—
145C22×C181—42no—
146C22×C181—42no—
147C22×C181#13512yescomplemented
148C22×C181C22×D411yescharacteristic
149C2×D181—21no—
150C2×D181—21no—
151C2×D181—21no—
152C2×D181—21no—
153C2×D181—21no—
154C2×D181—44no—
155C2×D181—41no—
156C2×D181—41no—
157C2×D181—41no—
158C2×D181#12012yes—
159C4×C181—21no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181C22×D412yes—
163Dic181—21no—
164Dic181—24no—
165Dic181—41no—
166Dic181#12012yes—
167C2×C181—22no—
168C2×C181—44no—
169C2×C181C2×#12011yescharacteristic
170C2×C181C2×#13511yescharacteristic
171C2×C181#57511yescharacteristic
172D181—22no—
173D181—41no—
174D181—81no—
175C181#300211yescharacteristic, complemented, Sylow
176#3002—1811nomaximal, Sylow
177C22×#41—1811no—
178C2×#135—1812no—
179#525—1811no—
180#452—1812no—
181#451—1811no—
182#450D18111yescharacteristic, complemented
183#449—1812no—
184#448—1811no—
185#447—1811no—
186#416—1812no—
187C2×C22⋊C8—1811no—
188C2×#41—3628no—
189C2×#41—3621no—
190C2×#41—3621no—
191C2×#41—1811no—
192C2×#41—1812no—
193C2×#41—1812no—
194C22×Q8—1811no—
195C22×D4—3621no—
196C22×D4C2×D18111yescharacteristic
197C23×C4—3621no—
198C23×C4—1811no—
199#135—3628no—
200C2×Q16—1812no—
201C2×SD16—1812no—
202C2×SD16C2×D18112yescomplemented
203C2×D8—1812no—
204C22×C8—1812no—
205#123—3621no—
206C22⋊Q8—1811no—
207C22⋊Q8C2×D18111yescharacteristic
208#121—1811no—
209#121—1811no—
210#120—3621no—
211C4×D4—3621no—
212C4×D4—3621no—
213#117—1811no—
214#103—1812no—
215#103—1812no—
216#102—1812no—
217#102C2×D18112yescomplemented
218C22⋊C8—1811no—
219C22⋊C8—1812no—
220C22⋊C8C2×D18111yescharacteristic
221C24—21no—
222C24—1811no—
223#41—3628no—
224#41—3628no—
225#41—3624no—
226#41—3624no—
227#41—3624no—
228#41—3624no—
229#41—1818no—
230C2×Q8—3624no—
231C2×Q8—3621no—
232C2×Q8—1812no—
233C2×Q8C22×D18111yescharacteristic
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—3621no—
237C2×D4—3621no—
238C2×D4—3628no—
239C2×D4—3622no—
240C2×D4—3622no—
241C2×D4—3621no—
242C2×D4—1811no—
243C2×D4C22×D18112yes—
244C22×C4—3624no—
245C22×C4—3621no—
246C22×C4—3628no—
247C22×C4—3621no—
248C22×C4—3622no—
249C22×C4—3622no—
250C22×C4—1811no—
251C22×C4—1811no—
252C22×C4—1811no—
253C22×C4—1811no—
254C22×C4—1811no—
255C22×C4C22×D18111yescharacteristic
256Q16—3624no—
257SD16—24no—
258SD16—3624no—
259D8—3624no—
260C2×C8—3624no—
261C2×C8—1812no—
262C2×C8C22×D18112yes—
263C4⋊C4—21no—
264C4⋊C4—3621no—
265C4⋊C4—1811no—
266C4⋊C4C22×D18111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C22⋊C4—3621no—
272C42—3621no—
273C23—24no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—3622no—
278C23—3622no—
279C23—3621no—
280C23—1811no—
281C23—1811no—
282C23—7241no—
283C23D4×D18111yescharacteristic
284Q8—22no—
285Q8—3624no—
286Q8—3622no—
287Q8—1814no—
288D4—24no—
289D4—22no—
290D4—3628no—
291D4—3628no—
292D4—3622no—
293D4—7242no—
294D4—7242no—
295D4—7242no—
296D4D4×D18114yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—3622no—
301C2×C4—3622no—
302C2×C4—3624no—
303C2×C4—3628no—
304C2×C4—3622no—
305C2×C4—3622no—
306C2×C4—3628no—
307C2×C4—3622no—
308C2×C4—3621no—
309C2×C4—1811no—
310C2×C4—1811no—
311C2×C4—1814no—
312C2×C4—7242no—
313C2×C4D4×D18111yescharacteristic
314C2×C4C23×D18111yescharacteristic
315C8—22no—
316C8—3622no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—3621no—
324C22—3621no—
325C22—3621no—
326C22—3621no—
327C22—3621no—
328C22—1812no—
329C22—7244no—
330C22—7241no—
331C22—7241no—
332C22—7241no—
333C22#1915383712yescomplemented
334C22C2×D4×D18111yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—3621no—
339C4—3624no—
340C4—1812no—
341C4—7241no—
342C4C2×D4×D18112yes—
343C2—22no—
344C2—44no—
345C2—3622no—
346C2—14481no—
347C2—7241no—
348C2#120×D18111yescharacteristic
349C2C2×#1915383711yescharacteristic
350C2#2604207511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
hypercenter#450D181
hypocenterC181#3002
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#15668448C2semidirect
2. C2×#19153837C2semidirect
3. #26041792C2semidirect
4. #26041438C2semidirect
5. #26041434C2semidirect
6. #26041425C2semidirect
7. #26041420C2semidirect
8. #26041413C2semidirect
9. #26041272C2semidirect
10. C22⋊C8×D181C2semidirect
11. #450×C181C2semidirect
12. C2×#15668450C22semidirect
23. C2×#15668439C22semidirect
24. C2×#15668438C22semidirect
25. C2×#15668434C22semidirect
26. C2×C8×D181C22semidirect
28. #19153792C22semidirect
29. #19153791C22semidirect
34. #19153774C22semidirect
35. #19153741C22semidirect
36. #19153739C22semidirect
37. #19153737C22semidirect
39. #19153728C22semidirect
41. C2×SD16×C181C22semidirect
43. #102×C181C22semidirect
52. #15668448D4semidirect
147. C22×C181#135semidirect
175. C181#3002semidirect
182. #450D181semidirect
202. C2×SD16C2×D181semidirect
217. #102C2×D181semidirect
333. C22#19153837semidirect

Maximal subgroups

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

subgroupindex
C22×#156684482
C2×#191538372
#260417922
#260414382
#260414342
#260414252
#260414202
#260414132
#260412722
C22⋊C8×D1812
#450×C1812
#3002181