Finite Group Databasebeta

← #36365402

Subgroups — #36365402

Order 22144.

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#26012183C211yesmaximal, characteristic, complemented
2#26012135C211yesmaximal, characteristic
3#26011844C211yesmaximal, characteristic, complemented
4#26011822C211yesmaximal, characteristic, complemented
5#26011814C211yesmaximal, characteristic, complemented
6#26011793C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011717C211yesmaximal, characteristic, complemented
9#26011711C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11SD16×Dic173C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011484C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583816C2211yescharacteristic, complemented
18C22×#13063436C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic, complemented
20#19149674—21no—
21#19149670—21no—
22#19149670C2211yescharacteristic, complemented
23#19149666C2211yescharacteristic
24#19149663—21no—
25#19149660—21no—
26#19149658C2211yescharacteristic
27#19149655C2211yescharacteristic, complemented
28#19149651—21no—
29#19149650C2211yescharacteristic
30#19149648—21no—
31Q8×Dic173C2211yescharacteristic, complemented
32D4×Dic173C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149637C2211yescharacteristic, complemented
35#19149635—21no—
36C2×C4⋊Dic173C2211yescharacteristic
37C2×C22⋊Dic173—21no—
38#19149612C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173D411yescharacteristic
65C22×Dic173—21no—
66C22×Dic173—21no—
67C22×Dic173—21no—
68C2×C4×D173C2311yescharacteristic
69#14583816—41no—
70C2×#13063436—41no—
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C8×D173—41no—
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173D411yescharacteristic
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C24×C173—21no—
97C2×Q8×C173C2311yescharacteristic
98C2×D4×C173—22no—
99C2×D4×C173—21no—
100C2×D4×C173—41no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104SD16×C173—22no—
105SD16×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114C22⋊D173—22no—
115C22⋊D173—42no—
116C22⋊D173—81no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—42no—
122C2×Dic173—42no—
123C2×Dic173—81no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—21no—
128C4×D173—21no—
129#13063436—21no—
130#13063436—21no—
131C23×C173—21no—
132C23×C173—21no—
133C23×C173—42no—
134C23×C173—42no—
135C23×C173—81no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—42no—
143D4×C173#4112yes—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—22no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—84no—
161C22×C173#13512yes—
162C22×C173C22×D411yescharacteristic
163C2×D173—21no—
164C2×D173—21no—
165C2×D173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—22no—
179C2×C173—42no—
180C2×C173—81no—
181C2×C173#54611yescharacteristic
182C2×C173C2×#13511yescharacteristic
183C2×C173#57711yescharacteristic
184D173—41no—
185C173#329611yescharacteristic, complemented, Sylow
186#3296—1731nomaximal, Sylow
187#552—1731no—
188#546—1731no—
189#488—1731no—
190#483—1731no—
191#482—1731no—
192#478—1731no—
193#467—1731no—
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450D17311yescharacteristic, complemented
198C4×SD16—1731no—
199C8×D4—1731no—
200#416—1731no—
201C2×#102—1731no—
202C2×#41—1731no—
203C22×D4C2×D17311yescharacteristic
204C2×Q16—1731no—
205C2×SD16C2×D17311yescharacteristic
206C2×SD16C2×D17311yescharacteristic
207C22×C8—1731no—
208C22×C8—1731no—
209#126—3461no—
210#125—3461no—
211#125—1731no—
212#124—1731no—
213#123—3461no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8—1731no—
217C22⋊Q8C2×D17311yescharacteristic
218#121—3461no—
219#121—1731no—
220#120—3461no—
221C4×Q8—1731no—
222C4×D4—1731no—
223C4×D4—1731no—
224#117—3461no—
225#117—1731no—
226C2×C4⋊C4—1731no—
227C2×C22⋊C4—3461no—
228#107—1731no—
229#106—1731no—
230#106—1731no—
231C4⋊C8—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102C2×D17311yescharacteristic
241#102C2×D17311yescharacteristic
242C22⋊C8—1731no—
243C22⋊C8C2×D17311yescharacteristic
244C4×C8—1731no—
245C24—21no—
246#41—3462no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4C22×D17311yescharacteristic
257C2×D4C22×D17311yescharacteristic
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4C22×D17311yescharacteristic
263Q16—6921no—
264SD16—22no—
265SD16—41no—
266C2×C8—1731no—
267C2×C8—1731no—
268C2×C8—6921no—
269C2×C8—6921no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—1731no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4—1731no—
283C4⋊C4C22×D17311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—1731no—
293C22⋊C4—1731no—
294C42—3461no—
295C42—3461no—
296C42—1731no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1731no—
303C23D4×D17311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3461no—
307Q8—3461no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—3462no—
313D4—13841no—
314D4—6922no—
315D4#1458382512yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—13841no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—1731no—
330C2×C4—6922no—
331C2×C4—6922no—
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—3461no—
348C22—3461no—
349C22—3461no—
350C22#1914970012yes—
351C22C2×D4×D17311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—3461no—
356C4—3461no—
357C4—3461no—
358C4—6921no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4C2×#1458382511yescharacteristic
363C4#1914972411yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—6921no—
368C2#2601212811yescharacteristic
369C2C2×#1914970011yescharacteristic
370C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radical#26012135C2
hypercenter#450D173
hypocenterC173#3296
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012183C2semidirect
3. #26011844C2semidirect
4. #26011822C2semidirect
5. #26011814C2semidirect
6. #26011793C2semidirect
7. #26011730C2semidirect
8. #26011717C2semidirect
9. #26011711C2semidirect
10. #26011650C2semidirect
11. SD16×Dic173C2semidirect
12. #26011554C2semidirect
13. #26011484C2semidirect
14. C2×#19149600C2semidirect
15. #450×C173C2semidirect
17. C2×#14583816C22semidirect
19. C2×C8×D173C22semidirect
22. #19149670C22semidirect
27. #19149655C22semidirect
31. Q8×Dic173C22semidirect
34. #19149637C22semidirect
38. #19149612C22semidirect
39. #19149611C22semidirect
40. #19149611C22semidirect
41. #19149608C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149603C22semidirect
45. #19149602C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
185. C173#3296semidirect
197. #450D173semidirect

Maximal subgroups

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

subgroupindex
#260121832
#260121352
#260118442
#260118222
#260118142
#260117932
#260117302
#260117172
#260117112
#260116502
SD16×Dic1732
#260115542
#260114842
C2×#191496002
#450×C1732
#3296173