Finite Group Databasebeta

← #37545750

Subgroups — #37545750

Order 23168.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042000C211yesmaximal, characteristic, complemented
2#26041923C211yesmaximal, characteristic
3#26041637C211yesmaximal, characteristic, complemented
4#26041621C211yesmaximal, characteristic, complemented
5#26041601C211yesmaximal, characteristic, complemented
6#26041567C211yesmaximal, characteristic, complemented
7#26041535C211yesmaximal, characteristic, complemented
8#26041505C211yesmaximal, characteristic, complemented
9#26041499C211yesmaximal, characteristic, complemented
10#26041438C211yesmaximal, characteristic, complemented
11#26041380C211yesmaximal, characteristic, complemented
12#26041353C211yesmaximal, characteristic, complemented
13#26041284C211yesmaximal, characteristic, complemented
14#26041272C211yesmaximal, characteristic, complemented
15#450×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668439C2211yescharacteristic, complemented
18C2×#15668433C2211yescharacteristic
19C2×C8×D181C2211yescharacteristic, complemented
20#19153815C2211yescharacteristic, complemented
21#19153811—21no—
22#19153809—21no—
23#19153803C2211yescharacteristic
24#19153802—21no—
25#19153797—21no—
26#19153795C2211yescharacteristic
27#19153792C2211yescharacteristic, complemented
28#19153788—21no—
29#19153787C2211yescharacteristic
30#19153785—21no—
31Q8×Dic181C2211yescharacteristic, complemented
32#19153781—21no—
33D4×Dic181C2211yescharacteristic
34C4×C22⋊D181C2211yescharacteristic
35#19153775C2211yescharacteristic
36#19153774C2211yescharacteristic, complemented
37C2×C22⋊Dic181—21no—
38#19153750C2211yescharacteristic, complemented
39#19153748C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153740C2211yescharacteristic, complemented
45#19153739C2211yescharacteristic, complemented
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153728C2211yescharacteristic, complemented
51#19153726C2211yescharacteristic, complemented
52C22×D4×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59#15668448—22no—
60#15668448—41no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181D411yescharacteristic
66C22×Dic181—21no—
67C22×Dic181—21no—
68C2×C4×D181C2311yescharacteristic
69#15668439—41no—
70#15668433—41no—
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C8×D181—41no—
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82C22⋊Dic181—21no—
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88C22⋊Dic181D411yescharacteristic
89#15668426—21no—
90#15668426—21no—
91#15668426D411yescharacteristic
92C4×Dic181—21no—
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C24×C181—21no—
97C2×Q8×C181C2311yescharacteristic
98C2×D4×C181—22no—
99C2×D4×C181—21no—
100C2×D4×C181—41no—
101C2×D4×C181C2311yescharacteristic
102C2×D4×C181C2311yescharacteristic
103C22×C4×C181C2311yescharacteristic
104SD16×C181—22no—
105SD16×C181—41no—
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181C2×D411yescharacteristic
112Dic362—21no—
113Dic362—21no—
114Dic362—42no—
115C22⋊D181—41no—
116C22⋊D181—42no—
117C22⋊D181—81no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181—42no—
123C2×Dic181—81no—
124C2×Dic181C2×D411yescharacteristic
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C4×D181—21no—
128C4×D181—21no—
129#13089678—21no—
130#13089678—21no—
131C23×C181—21no—
132C23×C181—21no—
133C23×C181—42no—
134C23×C181—42no—
135C23×C181—81no—
136C23×C181C2×D411yescharacteristic
137Q8×C181—21no—
138Q8×C181—21no—
139D4×C181—21no—
140D4×C181—21no—
141D4×C181—21no—
142D4×C181—42no—
143D4×C181#4112yes—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2411yescharacteristic
150C8×C181—21no—
151C8×C181—21no—
152C22×C181—21no—
153C22×C181—22no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181—41no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181—84no—
162C22×C181C22×D411yescharacteristic
163C2×D181—21no—
164C2×D181—21no—
165C2×D181—21no—
166C4×C181—21no—
167C4×C181—41no—
168C4×C181—41no—
169C4×C181C2×#4111yescharacteristic
170C4×C181H(5,𝔽2)11yescharacteristic
171Dic181—21no—
172Dic181—21no—
173Dic181—21no—
174Dic181—41no—
175Dic181—41no—
176Dic181—41no—
177Dic181—41no—
178C2×C181—41no—
179C2×C181—42no—
180C2×C181—81no—
181C2×C181#54611yescharacteristic
182C2×C181#57511yescharacteristic
183C2×C181#57711yescharacteristic
184D181—41no—
185C181#331111yescharacteristic, complemented, Sylow
186#3311—1811nomaximal, Sylow
187#556—1811no—
188#546—1811no—
189#489—1811no—
190#485—1811no—
191#482—1811no—
192#476—1811no—
193#470—1811no—
194#465—1811no—
195#464—1811no—
196#452—1811no—
197#450D18111yescharacteristic, complemented
198#440—1811no—
199#435—1811no—
200#418—1811no—
201#416—1811no—
202C2×#41—1811no—
203C22×D4C2×D18111yescharacteristic
204C2×Q16—1811no—
205C2×SD16C2×D18111yescharacteristic
206C2×SD16C2×D18111yescharacteristic
207C2×M16—1811no—
208C22×C8—1811no—
209C4⋊Q8—1811no—
210#126—3621no—
211#126—3621no—
212#124—3621no—
213#124—1811no—
214#123—3621no—
215C22⋊Q8—1811no—
216C22⋊Q8—1811no—
217C22⋊Q8C2×D18111yescharacteristic
218#121—3621no—
219#121—1811no—
220#120—3621no—
221C4×Q8—3621no—
222C4×Q8—1811no—
223C4×D4—1811no—
224C4×D4—1811no—
225#117—1811no—
226#117—1811no—
227C2×C22⋊C4—3621no—
228#107—1811no—
229#106—1811no—
230#106—1811no—
231C4⋊C8—1811no—
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103—1811no—
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102C2×D18111yescharacteristic
241#102C2×D18111yescharacteristic
242C22⋊C8—1811no—
243C22⋊C8C2×D18111yescharacteristic
244#97—1811no—
245C24—21no—
246#41—3622no—
247#41—7241no—
248C2×Q8—3621no—
249C2×Q8—1811no—
250C2×Q8C22×D18111yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—3621no—
255C2×D4—3621no—
256C2×D4—1811no—
257C2×D4C22×D18111yescharacteristic
258C2×D4C22×D18111yescharacteristic
259C22×C4—3621no—
260C22×C4—3621no—
261C22×C4—1811no—
262C22×C4C22×D18111yescharacteristic
263Q16—7241no—
264SD16—22no—
265SD16—41no—
266M16—7241no—
267C2×C8—1811no—
268C2×C8—1811no—
269C2×C8—7241no—
270C2×C8C22×D18111yescharacteristic
271C2×C8C22×D18111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—1811no—
278C4⋊C4—1811no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4C22×D18111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—1811no—
292C22⋊C4—1811no—
293C42—3621no—
294C42—3621no—
295C42—3621no—
296C42—1811no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1811no—
303C23D4×D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3621no—
307Q8—3621no—
308Q8—7242no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—14481no—
314D4—7241no—
315D4—7242no—
316D4#1566844812yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—3621no—
322C2×C4—3621no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—14481no—
328C2×C4—1811no—
329C2×C4—1811no—
330C2×C4—1811no—
331C2×C4—7242no—
332C2×C4D4×D18111yescharacteristic
333C2×C4C23×D18111yescharacteristic
334C8—21no—
335C8—21no—
336C8—3621no—
337C8—3621no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—3621no—
349C22—3621no—
350C22—3621no—
351C22C2×D4×D18111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—3621no—
356C4—3621no—
357C4—3621no—
358C4—7241no—
359C4—7241no—
360C4—7241no—
361C4—7241no—
362C4C2×#1566844811yescharacteristic
363C4#1915386111yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—7241no—
368C2#2604191611yescharacteristic
369C2#2604207511yescharacteristic
370C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
prime radical#26041923C2
hypercenter#450D181
hypocenterC181#3311
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. #26042000C2semidirect
3. #26041637C2semidirect
4. #26041621C2semidirect
5. #26041601C2semidirect
6. #26041567C2semidirect
7. #26041535C2semidirect
8. #26041505C2semidirect
9. #26041499C2semidirect
10. #26041438C2semidirect
11. #26041380C2semidirect
12. #26041353C2semidirect
13. #26041284C2semidirect
14. #26041272C2semidirect
15. #450×C181C2semidirect
17. C2×#15668439C22semidirect
19. C2×C8×D181C22semidirect
20. #19153815C22semidirect
27. #19153792C22semidirect
31. Q8×Dic181C22semidirect
36. #19153774C22semidirect
38. #19153750C22semidirect
39. #19153748C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153741C22semidirect
43. #19153741C22semidirect
44. #19153740C22semidirect
45. #19153739C22semidirect
50. #19153728C22semidirect
51. #19153726C22semidirect
185. C181#3311semidirect
197. #450D181semidirect

Maximal subgroups

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

subgroupindex
#260420002
#260419232
#260416372
#260416212
#260416012
#260415672
#260415352
#260415052
#260414992
#260414382
#260413802
#260413532
#260412842
#260412722
#450×C1812
#3311181