Finite Group Databasebeta

← #37544558

Subgroups — #37544558

Order 23168.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041790C211yesmaximal, characteristic, complemented
2#26041620C211yesmaximal, characteristic, complemented
3#26041617C211yesmaximal, characteristic, complemented
4#26041603C211yesmaximal, characteristic, complemented
5#26041600C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041530C211yesmaximal, characteristic, complemented
8#26041527C211yesmaximal, characteristic, complemented
9#26041523C211yesmaximal, characteristic, complemented
10#26041441C211yesmaximal, characteristic, complemented
11#26041439C211yesmaximal, characteristic, complemented
12#26041424C211yesmaximal, characteristic, complemented
13#26041422C211yesmaximal, characteristic, complemented
14#26041243C211yesmaximal, characteristic, complemented
15#525×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×#15668439C2211yescharacteristic, complemented
18C2×#15668437C2211yescharacteristic, complemented
19C2×#15668436C2211yescharacteristic, complemented
20C2×#15668434C2211yescharacteristic, complemented
21C2×#15668433C2211yescharacteristic
22C2×#15668433C2211yescharacteristic
23#19153799—21no—
24#19153795C2211yescharacteristic
25#19153795C2211yescharacteristic
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic
28#19153785—21no—
29C4×C22⋊D181—21no—
30C4×C22⋊D181—21no—
31#19153775C2211yescharacteristic
32#19153749C2211yescharacteristic, complemented
33#19153749C2211yescharacteristic, complemented
34#19153747C2211yescharacteristic, complemented
35#19153747C2211yescharacteristic, complemented
36#19153740C2211yescharacteristic, complemented
37#19153740C2211yescharacteristic, complemented
38#19153739C2211yescharacteristic, complemented
39#19153739C2211yescharacteristic, complemented
40#19153737C2211yescharacteristic, complemented
41#19153737C2211yescharacteristic, complemented
42#19153736C2211yescharacteristic, complemented
43#19153736C2211yescharacteristic, complemented
44C22⋊#13089678C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C2×#41×C181C2211yescharacteristic
49C23×C4×C181C2211yescharacteristic
50#123×C181—21no—
51C22⋊Q8×C181C2211yescharacteristic
52C22⋊Q8×C181C2211yescharacteristic
53#121×C181C2211yescharacteristic
54#121×C181C2211yescharacteristic
55#120×C181—21no—
56C4×D4×C181—21no—
57C4×D4×C181—21no—
58#117×C181C2211yescharacteristic
59#15668449—41no—
60#15668449—41no—
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C4×D181—21no—
67#15668439—41no—
68#15668437—41no—
69#15668436—41no—
70#15668434—41no—
71#15668433—41no—
72#15668433—41no—
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C2×#13089678C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81C22⋊Dic181—21no—
82C22⋊Dic181—21no—
83C22⋊Dic181—21no—
84#15668426—21no—
85#15668426—21no—
86C4×Dic181—21no—
87C24×C181D411yescharacteristic
88#41×C181—41no—
89#41×C181—41no—
90C2×Q8×C181C2311yescharacteristic
91C2×D4×C181—21no—
92C2×D4×C181—21no—
93C2×D4×C181—21no—
94C2×D4×C181C2311yescharacteristic
95C22×C4×C181—21no—
96C22×C4×C181—41no—
97C22×C4×C181—41no—
98C22×C4×C181D411yescharacteristic
99C22×C4×C181D411yescharacteristic
100C22×C4×C181D411yescharacteristic
101C22×C4×C181C2311yescharacteristic
102C22×C4×C181C2311yescharacteristic
103C22×C4×C181C2311yescharacteristic
104C4⋊C4×C181—21no—
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C22⋊C4×C181—21no—
109C22⋊C4×C181—21no—
110C22⋊C4×C181—21no—
111C22⋊C4×C181—21no—
112C22⋊C4×C181—21no—
113C42×C181—21no—
114C22×D181—21no—
115Dic362—21no—
116Dic362—21no—
117D724—21no—
118D724—21no—
119C22⋊D181—42no—
120C22⋊D181—42no—
121C22⋊D181—42no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C4×D181—42no—
126#13089678—21no—
127#13089678—21no—
128#13089678—21no—
129#13089678—21no—
130C23×C181—21no—
131C23×C181—41no—
132C23×C181—41no—
133C23×C181—41no—
134C23×C181C2×D411yescharacteristic
135C23×C181C2×D411yescharacteristic
136C23×C181C2×D411yescharacteristic
137Q8×C181—21no—
138Q8×C181—21no—
139D4×C181—21no—
140D4×C181—21no—
141D4×C181—42no—
142D4×C181—42no—
143D4×C181—42no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—41no—
154C2×C4×C181—41no—
155C2×C4×C181—41no—
156C2×C4×C181—42no—
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2×D411yescharacteristic
160C2×C4×C181C2411yescharacteristic
161C22×C181—21no—
162C22×C181—21no—
163C22×C181—21no—
164C22×C181—21no—
165C22×C181—21no—
166C22×C181—21no—
167C22×C181—21no—
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—44no—
171C22×C181—41no—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181C22×D411yescharacteristic
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—41no—
178C4×C181—21no—
179C4×C181—21no—
180C4×C181—21no—
181C4×C181—41no—
182C4×C181—41no—
183C4×C181—41no—
184C4×C181H(5,𝔽2)11yescharacteristic
185C4×C181H(5,𝔽2)11yescharacteristic
186Dic181—41no—
187Dic181—41no—
188Dic181—41no—
189C2×C181—41no—
190C2×C181—41no—
191C2×C181—41no—
192C2×C181—81no—
193C2×C181#53411yescharacteristic
194C2×C181#57511yescharacteristic
195C2×C181#57511yescharacteristic
196D181—81no—
197C181#319711yescharacteristic, complemented, Sylow
198#3197—1811nomaximal, Sylow
199#525—1811no—
200#525D18111yescharacteristic, complemented
201#485—1811no—
202#485—1811no—
203#482—1811no—
204#482—1811no—
205#470—1811no—
206#469—1811no—
207#468—1811no—
208#468—1811no—
209#452—1811no—
210#452—1811no—
211#449—1811no—
212#449—1811no—
213#407—1811no—
214C2×#41—1811no—
215C2×#41C2×D18111yescharacteristic
216C23×C4C2×D18111yescharacteristic
217C2×Q16—1811no—
218C2×SD16—1811no—
219C2×SD16—1811no—
220C2×D8—1811no—
221C2×M16—1811no—
222C2×M16—1811no—
223#123—21no—
224#123—3621no—
225C22⋊Q8—1811no—
226C22⋊Q8—1811no—
227C22⋊Q8C2×D18111yescharacteristic
228C22⋊Q8C2×D18111yescharacteristic
229#121—1811no—
230#121—1811no—
231#121C2×D18111yescharacteristic
232#121C2×D18111yescharacteristic
233#120—21no—
234#120—3621no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—3621no—
238C4×D4—3621no—
239#117—1811no—
240#117C2×D18111yescharacteristic
241#107—1811no—
242#107—1811no—
243#106—1811no—
244#106—1811no—
245#103—1811no—
246#103—1811no—
247#103—1811no—
248#103—1811no—
249#102—1811no—
250#102—1811no—
251#102—1811no—
252#102—1811no—
253C22⋊C8—1811no—
254C22⋊C8—1811no—
255C22⋊C8—1811no—
256C22⋊C8—1811no—
257C24C22⋊D18111yescharacteristic
258#41—41no—
259#41—41no—
260#41—7241no—
261#41—7241no—
262C2×Q8—1811no—
263C2×Q8C22×D18111yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—3621no—
268C2×D4—3621no—
269C2×D4—3621no—
270C2×D4—1811no—
271C2×D4C22×D18111yescharacteristic
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—3621no—
276C22×C4C22⋊D18111yescharacteristic
277C22×C4C22⋊D18111yescharacteristic
278C22×C4C22⋊D18111yescharacteristic
279C22×C4C22×D18111yescharacteristic
280C22×C4C22×D18111yescharacteristic
281C22×C4C22×D18111yescharacteristic
282Q16—7241no—
283SD16—7241no—
284SD16—7241no—
285D8—7241no—
286M16—7241no—
287M16—7241no—
288C2×C8—1811no—
289C2×C8—1811no—
290C2×C8—1811no—
291C2×C8—1811no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—3621no—
295C4⋊C4—3621no—
296C4⋊C4—1811no—
297C4⋊C4—1811no—
298C4⋊C4C22×D18111yescharacteristic
299C4⋊C4C22×D18111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—3621no—
306C22⋊C4—3621no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C42—21no—
311C42—3621no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—3621no—
317C23C2×C22⋊D18111yescharacteristic
318C23C2×C22⋊D18111yescharacteristic
319C23C2×C22⋊D18111yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3621no—
323Q8—3621no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—3621no—
330D4—3621no—
331D4—7242no—
332D4—7242no—
333D4—7242no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—3621no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—7242no—
351C2×C4C2×C22⋊D18111yescharacteristic
352C2×C4C2×C22⋊D18111yescharacteristic
353C2×C4C2×C22⋊D18111yescharacteristic
354C2×C4C23×D18111yescharacteristic
355C8—3621no—
356C8—3621no—
357C8—3621no—
358C8—3621no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—7241no—
373C22—7241no—
374C22—7241no—
375C22C22×C22⋊D18111yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—7241no—
383C4—7241no—
384C4—7241no—
385C4#1915386111yescharacteristic
386C4#1915386111yescharacteristic
387C2—41no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—14481no—
392C2#2604183811yescharacteristic
393C2#2604207711yescharacteristic
394C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#525×C181C2
socleC22×C181C22×D4
hypercenter#525D181
hypocenterC181#3197
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #26041790C2semidirect
2. #26041620C2semidirect
3. #26041617C2semidirect
4. #26041603C2semidirect
5. #26041600C2semidirect
6. #26041535C2semidirect
7. #26041530C2semidirect
8. #26041527C2semidirect
9. #26041523C2semidirect
10. #26041441C2semidirect
11. #26041439C2semidirect
12. #26041424C2semidirect
13. #26041422C2semidirect
14. #26041243C2semidirect
15. #525×C181C2semidirect
17. C2×#15668439C22semidirect
18. C2×#15668437C22semidirect
19. C2×#15668436C22semidirect
20. C2×#15668434C22semidirect
32. #19153749C22semidirect
33. #19153749C22semidirect
34. #19153747C22semidirect
35. #19153747C22semidirect
36. #19153740C22semidirect
37. #19153740C22semidirect
38. #19153739C22semidirect
39. #19153739C22semidirect
40. #19153737C22semidirect
41. #19153737C22semidirect
42. #19153736C22semidirect
43. #19153736C22semidirect
197. C181#3197semidirect
200. #525D181semidirect

Maximal subgroups

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

subgroupindex
#260417902
#260416202
#260416172
#260416032
#260416002
#260415352
#260415302
#260415272
#260415232
#260414412
#260414392
#260414242
#260414222
#260412432
#525×C1812
#3197181