Finite Group Databasebeta

← #37545682

Subgroups — #37545682

Order 23168.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041933C211yesmaximal, characteristic, complemented
2#26041638C211yesmaximal, characteristic, complemented
3#26041607C211yesmaximal, characteristic, complemented
4#26041595C211yesmaximal, characteristic, complemented
5#26041569C211yesmaximal, characteristic, complemented
6#26041530C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041424C211yesmaximal, characteristic, complemented
10#26041413C211yesmaximal, characteristic, complemented
11#26041389C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041280C211yesmaximal, characteristic, complemented
14C2×#19153737C211yesmaximal, characteristic, complemented
15#546×C181C211yesmaximal, characteristic, complemented
16C2×#15668448—21no—
17C2×#15668436C2211yescharacteristic, complemented
18C2×#15668434C2211yescharacteristic
19C2×#15668434C2211yescharacteristic
20C2×#15668433C2211yescharacteristic, complemented
21C22×#13089678C2211yescharacteristic
22#19153816C2211yescharacteristic, complemented
23#19153800—21no—
24#19153800—21no—
25#19153796—21no—
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic, complemented
28C4×D724C2211yescharacteristic, complemented
29C2×C4⋊Dic181C2211yescharacteristic
30C2×C4⋊Dic181C2211yescharacteristic, complemented
31#19153747C2211yescharacteristic, complemented
32#19153747C2211yescharacteristic, complemented
33#19153747C2211yescharacteristic, complemented
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153740C2211yescharacteristic, complemented
36#19153740C2211yescharacteristic, complemented
37#19153740C2211yescharacteristic, complemented
38#19153737C2211yescharacteristic
39#19153737C2211yescharacteristic
40#19153737C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153736C2211yescharacteristic, complemented
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic, complemented
47#19153727C2211yescharacteristic, complemented
48C2×#41×C181C2211yescharacteristic
49C22×D4×C181C2211yescharacteristic
50#124×C181C2211yescharacteristic
51#123×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53#121×C181—21no—
54#121×C181C2211yescharacteristic
55#120×C181—21no—
56C4×D4×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic
58C2×C22⋊C4×C181—21no—
59#15668448—44no—
60C2×Dic362—21no—
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64C22×Dic181—21no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67#15668436—41no—
68#15668434—22no—
69#15668434—41no—
70#15668433—41no—
71C2×#13089678—41no—
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181—21no—
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82#15668426—21no—
83#15668426—21no—
84#15668426—21no—
85C24×C181—21no—
86#41×C181—22no—
87#41×C181—41no—
88C2×Q8×C181C2311yescharacteristic
89C2×D4×C181—22no—
90C2×D4×C181—21no—
91C2×D4×C181—21no—
92C2×D4×C181—21no—
93C2×D4×C181—41no—
94C2×D4×C181D411yescharacteristic
95C2×D4×C181C2311yescharacteristic
96C2×D4×C181C2311yescharacteristic
97C22×C4×C181—21no—
98C22×C4×C181—21no—
99C22×C4×C181C2311yescharacteristic
100C22×C4×C181C2311yescharacteristic
101C4⋊C4×C181—21no—
102C4⋊C4×C181D411yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181—21no—
108C22⋊C4×C181—21no—
109C22⋊C4×C181D411yescharacteristic
110C22⋊C4×C181D411yescharacteristic
111C42×C181C2311yescharacteristic
112C22×D181—21no—
113Dic362—42no—
114D724—21no—
115D724—21no—
116C22⋊D181—42no—
117C22⋊D181—42no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—42no—
122C2×Dic181—42no—
123C4×D181—42no—
124#13089678—21no—
125#13089678—21no—
126#13089678—21no—
127#13089678—21no—
128C23×C181—21no—
129C23×C181—21no—
130C23×C181—42no—
131C23×C181—42no—
132C23×C181—81no—
133C23×C181C2×D411yescharacteristic
134C23×C181C2×D411yescharacteristic
135Q8×C181—21no—
136Q8×C181—21no—
137D4×C181—22no—
138D4×C181—21no—
139D4×C181—21no—
140D4×C181—21no—
141D4×C181—42no—
142D4×C181—42no—
143D4×C181—81no—
144D4×C181#4112yescomplemented
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—42no—
152C2×C4×C181—81no—
153C2×C4×C181C2×D411yescharacteristic
154C2×C4×C181C2×D411yescharacteristic
155C2×C4×C181C2×D411yescharacteristic
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2411yescharacteristic
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—21no—
161C22×C181—22no—
162C22×C181—21no—
163C22×C181—21no—
164C22×C181—41no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181—41no—
169C22×C181—84no—
170C22×C181AGL(1,ℤ/8)12yes—
171C22×C181C22×D411yescharacteristic
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—41no—
175C4×C181—21no—
176C4×C181—21no—
177C4×C181—21no—
178C4×C181—21no—
179C4×C181—41no—
180C4×C181—41no—
181C4×C181C2×#4111yescharacteristic
182C4×C181H(5,𝔽2)11yescharacteristic
183Dic181—41no—
184Dic181—41no—
185Dic181—41no—
186C2×C181—22no—
187C2×C181—41no—
188C2×C181—42no—
189C2×C181—81no—
190C2×C181#54611yescharacteristic
191C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C181#57711yescharacteristic
193D181—81no—
194C181#330611yescharacteristic, complemented, Sylow
195#3306—1811nomaximal, Sylow
196#547—1811no—
197#546D18111yescharacteristic, complemented
198#489—1811no—
199#483—1811no—
200#481—1811no—
201#476—1811no—
202#469—1811no—
203#465—1811no—
204#463—1811no—
205#449—1811no—
206#447—1811no—
207#442—1811no—
208#435—1811no—
209#417—1811no—
210C2×#102—1811no—
211C2×#41—3621no—
212C2×#41C2×D18111yescharacteristic
213C22×D4C2×D18111yescharacteristic
214C2×SD16—1811no—
215C2×D8—1811no—
216C2×D8—1811no—
217C2×M16—1811no—
218C22×C8—1811no—
219C4⋊Q8—1811no—
220#124C2×D18111yescharacteristic
221#123—21no—
222#123—3621no—
223#123—3621no—
224C22⋊Q8—3621no—
225C22⋊Q8C2×D18111yescharacteristic
226#121—21no—
227#121—1811no—
228#121—1811no—
229#121C2×D18111yescharacteristic
230#120—21no—
231C4×D4—1811no—
232C4×D4C2×D18111yescharacteristic
233C4×D4C2×D18111yescharacteristic
234C2×C4⋊C4—1811no—
235C2×C4⋊C4—1811no—
236C2×C22⋊C4—21no—
237#106—1811no—
238#106—1811no—
239#106—1811no—
240C4⋊C8—1811no—
241#103—1811no—
242#103—1811no—
243#103—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248#102—1811no—
249#102—1811no—
250#102—1811no—
251C22⋊C8—1811no—
252C22⋊C8—1811no—
253#97—1811no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—7244no—
258C2×Q8—3621no—
259C2×Q8C22×D18111yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—3621no—
266C2×D4—3621no—
267C2×D4—1811no—
268C2×D4C22⋊D18111yescharacteristic
269C2×D4C22×D18111yescharacteristic
270C2×D4C22×D18111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3621no—
274C22×C4—3621no—
275C22×C4—3621no—
276C22×C4C22×D18111yescharacteristic
277C22×C4C22×D18111yescharacteristic
278SD16—7241no—
279D8—3622no—
280D8—7241no—
281M16—7241no—
282C2×C8—1811no—
283C2×C8—1811no—
284C2×C8—1811no—
285C2×C8—1811no—
286C2×C8—7241no—
287C4⋊C4—21no—
288C4⋊C4—3621no—
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—1811no—
292C4⋊C4—1811no—
293C4⋊C4—1811no—
294C4⋊C4C22⋊D18111yescharacteristic
295C4⋊C4C22×D18111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4C22⋊D18111yescharacteristic
305C22⋊C4C22⋊D18111yescharacteristic
306C42C22×D18111yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—3621no—
313C23C2×C22⋊D18111yescharacteristic
314C23C2×C22⋊D18111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—7242no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—3621no—
326D4—3621no—
327D4—7242no—
328D4—7242no—
329D4#1566844812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—3621no—
339C2×C4—3621no—
340C2×C4—3621no—
341C2×C4—7242no—
342C2×C4—7242no—
343C2×C4—7242no—
344C2×C4C2×C22⋊D18111yescharacteristic
345C2×C4C2×C22⋊D18111yescharacteristic
346C2×C4C2×C22⋊D18111yescharacteristic
347C2×C4C2×C22⋊D18111yescharacteristic
348C2×C4C23×D18111yescharacteristic
349C8—3621no—
350C8—3621no—
351C8—3621no—
352C8—3621no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—7241no—
366C22—7241no—
367C22—7241no—
368C22#1915384112yes—
369C22C22×C22⋊D18111yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—7241no—
377C4—7241no—
378C4—7241no—
379C4C2×#1566844811yescharacteristic
380C4#1915386111yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—14481no—
386C2#2604192311yescharacteristic
387C2C2×#1915384111yescharacteristic
388C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#546×C181C2
socleC22×C181C22×D4
hypercenter#546D181
hypocenterC181#3306
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #26041933C2semidirect
2. #26041638C2semidirect
3. #26041607C2semidirect
4. #26041595C2semidirect
5. #26041569C2semidirect
6. #26041530C2semidirect
7. #26041510C2semidirect
8. #26041491C2semidirect
9. #26041424C2semidirect
10. #26041413C2semidirect
11. #26041389C2semidirect
12. #26041351C2semidirect
13. #26041280C2semidirect
14. C2×#19153737C2semidirect
15. #546×C181C2semidirect
17. C2×#15668436C22semidirect
20. C2×#15668433C22semidirect
22. #19153816C22semidirect
27. #19153790C22semidirect
28. C4×D724C22semidirect
30. C2×C4⋊Dic181C22semidirect
31. #19153747C22semidirect
32. #19153747C22semidirect
33. #19153747C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153740C22semidirect
36. #19153740C22semidirect
37. #19153740C22semidirect
44. #19153736C22semidirect
46. C22⋊#13089678C22semidirect
47. #19153727C22semidirect
144. D4×C181#41semidirect
194. C181#3306semidirect
197. #546D181semidirect
329. D4#15668448semidirect

Maximal subgroups

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

subgroupindex
#260419332
#260416382
#260416072
#260415952
#260415692
#260415302
#260415102
#260414912
#260414242
#260414132
#260413892
#260413512
#260412802
C2×#191537372
#546×C1812
#3306181