Finite Group Databasebeta

← #37545549

Subgroups — #37545549

Order 23168.

408 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#26041963C211yesmaximal, characteristic, complemented
2#26041917C211yesmaximal, characteristic
3#26041635C211yesmaximal, characteristic, complemented
4#26041608C211yesmaximal, characteristic, complemented
5#26041604C211yesmaximal, characteristic, complemented
6#26041577C211yesmaximal, characteristic, complemented
7#26041517C211yesmaximal, characteristic, complemented
8#26041510C211yesmaximal, characteristic, complemented
9#26041503C211yesmaximal, characteristic, complemented
10#26041428C211yesmaximal, characteristic, complemented
11SD16×Dic181C211yesmaximal, characteristic, complemented
12#26041342C211yesmaximal, characteristic, complemented
13#26041275C211yesmaximal, characteristic, complemented
14#102×D181C211yesmaximal, characteristic, complemented
15#452×C181C211yesmaximal, characteristic, complemented
16C2×D4×D181C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C22×#13089678C2211yescharacteristic, complemented
19C2×C8×D181C2211yescharacteristic
20#19153809—21no—
21#19153807C2211yescharacteristic, complemented
22#19153806—21no—
23#19153805C2211yescharacteristic
24#19153799—21no—
25#19153798—21no—
26#19153792C2211yescharacteristic
27#19153792C2211yescharacteristic
28#19153790C2211yescharacteristic
29#19153788—21no—
30#19153784—21no—
31Q8×Dic181C2211yescharacteristic, complemented
32D4×Dic181C2211yescharacteristic
33C4×C22⋊D181C2211yescharacteristic
34#19153775C2211yescharacteristic, complemented
35#19153774—21no—
36C4⋊C4×D181C2211yescharacteristic
37C2×#15668426—21no—
38#19153750C2211yescharacteristic, complemented
39#19153748C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153740C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153740C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic, complemented
52C2×#41×C181C2211yescharacteristic
53C2×Q16×C181C2211yescharacteristic, complemented
54C2×SD16×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic, complemented
56#103×C181C2211yescharacteristic, complemented
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C2×D724C2311yescharacteristic
61C2×C22⋊D181—21no—
62C2×C22⋊D181—21no—
63C2×C22⋊D181D411yescharacteristic
64D4×D181—22no—
65D4×D181—41no—
66C22×Dic181—21no—
67C2×C4×D181—21no—
68C2×C4×D181C2311yescharacteristic
69#15668436—41no—
70C2×#13089678—41no—
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C8×D181—41no—
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181D411yescharacteristic
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426D411yescharacteristic
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96#41×C181—22no—
97#41×C181—41no—
98C2×Q8×C181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181—21no—
101C2×D4×C181C2311yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181C2311yescharacteristic
104Q16×C181—41no—
105SD16×C181—22no—
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181—21no—
112C22×D181—42no—
113C22×D181—42no—
114C22×D181—81no—
115C22×D181C2×D411yescharacteristic
116D724—21no—
117D724—21no—
118C22⋊D181—22no—
119C22⋊D181—42no—
120C22⋊D181—81no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—81no—
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C4×D181—21no—
129C4×D181—21no—
130C4×D181—42no—
131#13089678—21no—
132#13089678—21no—
133C23×C181—21no—
134C23×C181C2×D411yescharacteristic
135Q8×C181—21no—
136Q8×C181—21no—
137Q8×C181—21no—
138Q8×C181—21no—
139D4×C181—21no—
140D4×C181—42no—
141D4×C181#4112yes—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—42no—
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×C181C22×D411yescharacteristic
159C2×D181—21no—
160C2×D181—21no—
161C2×D181—41no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—84no—
165C2×D181#13512yes—
166C4×C181—21no—
167C4×C181—41no—
168C4×C181—41no—
169C4×C181—41no—
170C4×C181C2×#4111yescharacteristic
171C4×C181H(5,𝔽2)11yescharacteristic
172Dic181—21no—
173Dic181—21no—
174Dic181—21no—
175Dic181—41no—
176Dic181—41no—
177Dic181—41no—
178C2×C181—42no—
179C2×C181—41no—
180C2×C181#54611yescharacteristic
181C2×C181C2×#13511yescharacteristic
182C2×C181#57711yescharacteristic
183D181—22no—
184D181—81no—
185C181#329611yescharacteristic, complemented, Sylow
186#3296—1811nomaximal, Sylow
187#552—1811no—
188#546—1811no—
189#488—1811no—
190#483—1811no—
191#482—1811no—
192#478—1811no—
193#467—1811no—
194#465—1811no—
195#464—1811no—
196#452D18111yescharacteristic, complemented
197#450—1811no—
198C4×SD16—1811no—
199C8×D4—1811no—
200#416—1811no—
201C2×#102—1811no—
202C2×#41C2×D18111yescharacteristic
203C22×D4—1811no—
204C2×Q16C2×D18111yescharacteristic, complemented
205C2×SD16—1811no—
206C2×SD16C2×D18111yescharacteristic
207C22×C8—1811no—
208C22×C8—1811no—
209#126—3621no—
210#125—3621no—
211#125—1811no—
212#124—1811no—
213#123—3621no—
214#123—3621no—
215C22⋊Q8—1811no—
216C22⋊Q8—1811no—
217C22⋊Q8C2×D18111yescharacteristic, complemented
218#121—3621no—
219#121—1811no—
220#120—3621no—
221C4×Q8—1811no—
222C4×D4—1811no—
223C4×D4—1811no—
224#117—3621no—
225#117—1811no—
226C2×C4⋊C4—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#103C2×D18111yescharacteristic, complemented
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102C2×D18111yescharacteristic
242C22⋊C8—1811no—
243C22⋊C8C2×D18111yescharacteristic, complemented
244C4×C8—1811no—
245C24—3621no—
246#41—22no—
247#41—41no—
248C2×Q8C22×D18111yescharacteristic
249C2×Q8C22×D18111yescharacteristic
250C2×D4—21no—
251C2×D4—3622no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—1811no—
255C2×D4—1811no—
256C2×D4—7241no—
257C2×D4C22×D18111yescharacteristic
258C22×C4—21no—
259C22×C4—3621no—
260C22×C4—3621no—
261C22×C4—1811no—
262C22×C4C22×D18111yescharacteristic
263Q16—41no—
264SD16—22no—
265SD16—7241no—
266C2×C8—1811no—
267C2×C8—1811no—
268C2×C8—7241no—
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—3621no—
278C4⋊C4—3621no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4—1811no—
282C4⋊C4—1811no—
283C4⋊C4C22×D18111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—1811no—
293C22⋊C4—1811no—
294C42—3621no—
295C42—3621no—
296C42—1811no—
297C23—21no—
298C23—3621no—
299C23—14481no—
300C23—1811no—
301C23—7242no—
302C23—7242no—
303C23D4×D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—42no—
310D4—3622no—
311D4—3621no—
312D4—3621no—
313D4—14481no—
314D4—7242no—
315D4#1566844812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
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—3621no—
345C22—3621no—
346C22—14484no—
347C22—1812no—
348C22—7241no—
349C22—7241no—
350C22—7241no—
351C22C2×D4×D18111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3621no—
357C4—3621no—
358C4—3621no—
359C4—7241no—
360C4—7241no—
361C4—7241no—
362C4C2×#1566844811yescharacteristic
363C4#1915386111yescharacteristic
364C2—41no—
365C2—42no—
366C2—3622no—
367C2—14481no—
368C2#2604191611yescharacteristic
369C2#135×D18111yescharacteristic
370C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#452×C181C2
socleC22×C181C22×D4
prime radical#26041917C2
hypercenter#452D181
hypocenterC181#3296
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #26041963C2semidirect
3. #26041635C2semidirect
4. #26041608C2semidirect
5. #26041604C2semidirect
6. #26041577C2semidirect
7. #26041517C2semidirect
8. #26041510C2semidirect
9. #26041503C2semidirect
10. #26041428C2semidirect
11. SD16×Dic181C2semidirect
12. #26041342C2semidirect
13. #26041275C2semidirect
14. #102×D181C2semidirect
15. #452×C181C2semidirect
18. C22×#13089678C22semidirect
21. #19153807C22semidirect
31. Q8×Dic181C22semidirect
34. #19153775C22semidirect
38. #19153750C22semidirect
39. #19153748C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153740C22semidirect
43. #19153740C22semidirect
44. #19153740C22semidirect
51. C8×Dic181C22semidirect
53. C2×Q16×C181C22semidirect
55. C22⋊Q8×C181C22semidirect
56. #103×C181C22semidirect
58. C22⋊C8×C181C22semidirect
185. C181#3296semidirect
196. #452D181semidirect
204. C2×Q16C2×D181semidirect
217. C22⋊Q8C2×D181semidirect
235. #103C2×D181semidirect
243. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419632
#260419172
#260416352
#260416082
#260416042
#260415772
#260415172
#260415102
#260415032
#260414282
SD16×Dic1812
#260413422
#260412752
#102×D1812
#452×C1812
#3296181