Finite Group Databasebeta

← #36365404

Subgroups — #36365404

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#26012132C211yesmaximal, characteristic
3#26011847C211yesmaximal, characteristic, complemented
4#26011819C211yesmaximal, characteristic, complemented
5#26011818C211yesmaximal, characteristic, complemented
6#26011794C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011715C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011641C211yesmaximal, characteristic, complemented
11#26011576C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15#467×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic, complemented
21#19149674—21no—
22#19149670—21no—
23#19149670C2211yescharacteristic, complemented
24#19149668C2211yescharacteristic
25#19149664—21no—
26#19149663—21no—
27#19149655C2211yescharacteristic, complemented
28#19149654—21no—
29#19149654C2211yescharacteristic
30#19149649—21no—
31Q8×Dic173C2211yescharacteristic, complemented
32#19149641C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149637C2211yescharacteristic, complemented
35#19149635—21no—
36C2×C4⋊Dic173C2211yescharacteristic
37C22⋊C4×D173—21no—
38#19149610C2211yescharacteristic, complemented
39#19149610C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×Q16×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#103×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—22no—
61#14583827—41no—
62C2×D692—22no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—41no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173D411yescharacteristic
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583813—22no—
75#14583813—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C8×D173—41no—
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173D411yescharacteristic
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101Q16×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117D692—21no—
118D692—21no—
119D692—21no—
120D692—42no—
121D692—42no—
122D692—81no—
123D692#4112yes—
124C22⋊D173—22no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—42no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—42no—
134C4×D173—81no—
135#13063436—21no—
136#13063436—21no—
137C23×C173C2×D411yescharacteristic
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173—21no—
141Q8×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
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#13512yes—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—22no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—84no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173C2×#4111yescharacteristic
172C4×C173H(5,𝔽2)11yescharacteristic
173Dic173—21no—
174Dic173—21no—
175Dic173—21no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—22no—
179C2×C173#54611yescharacteristic
180C2×C173C2×#13511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—41no—
183D173—42no—
184D173—81no—
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#467D17311yescharacteristic, complemented
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450—1731no—
198C4×SD16—1731no—
199C8×D4—1731no—
200#416—1731no—
201C2×#102—1731no—
202C2×#41—1731no—
203C22×D4—1731no—
204C2×Q16C2×D17311yescharacteristic, complemented
205C2×SD16—1731no—
206C2×SD16—1731no—
207C22×C8—1731no—
208C22×C8C2×D17311yescharacteristic
209#126—3461no—
210#125—3461no—
211#125—1731no—
212#124—1731no—
213#123—3461no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8C2×D17311yescharacteristic
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#107C2×D17311yescharacteristic, complemented
229#106—1731no—
230#106—1731no—
231C4⋊C8—1731no—
232#103—1731no—
233#103—1731no—
234#103C2×D17311yescharacteristic, complemented
235#103C2×D17311yescharacteristic, complemented
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244C4×C8—1731no—
245C24—3461no—
246#41—3462no—
247#41—6921no—
248C2×Q8C22×D17311yescharacteristic
249C2×Q8C22×D17311yescharacteristic
250C2×D4—3462no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1731no—
256C2×D4—1731no—
257C2×D4—6921no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4C22×D17311yescharacteristic
263Q16—41no—
264SD16—3462no—
265SD16—6921no—
266C2×C8—41no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—6921no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
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⋊C4C22×D17311yescharacteristic
283C4⋊C4C22×D17311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
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—3461no—
298C23—3461no—
299C23—13841no—
300C23—1731no—
301C23—6922no—
302C23—6922no—
303C23D4×D17311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—3462no—
309D4—3461no—
310D4—3461no—
311D4—3461no—
312D4—13841no—
313D4—1732no—
314D4—6922no—
315D4—6922no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
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—3461no—
341C22—3461no—
342C22—3461no—
343C22—3462no—
344C22—13844no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22—6921no—
350C22#1914970212yes—
351C22C2×D4×D17311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3461no—
358C4—3461no—
359C4—3461no—
360C4—6921no—
361C4—6921no—
362C4#41×D17311yescharacteristic
363C4#1914972411yescharacteristic
364C2—22no—
365C2—13841no—
366C2—6921no—
367C2—6922no—
368C2#2601213011yescharacteristic
369C2C2×#1914970211yescharacteristic
370C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radical#26012132C2
hypercenter#467D173
hypocenterC173#3296
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #26012183C2semidirect
3. #26011847C2semidirect
4. #26011819C2semidirect
5. #26011818C2semidirect
6. #26011794C2semidirect
7. #26011723C2semidirect
8. #26011715C2semidirect
9. #26011652C2semidirect
10. #26011641C2semidirect
11. #26011576C2semidirect
12. C8×C22⋊D173C2semidirect
13. #26011486C2semidirect
14. C2×#19149601C2semidirect
15. #467×C173C2semidirect
20. C2×C8×D173C22semidirect
23. #19149670C22semidirect
27. #19149655C22semidirect
31. Q8×Dic173C22semidirect
34. #19149637C22semidirect
38. #19149610C22semidirect
39. #19149610C22semidirect
40. #19149608C22semidirect
41. #19149603C22semidirect
42. #19149603C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
52. C2×Q16×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
58. #103×C173C22semidirect
185. C173#3296semidirect
193. #467D173semidirect
204. C2×Q16C2×D173semidirect
228. #107C2×D173semidirect
234. #103C2×D173semidirect
235. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121832
#260121322
#260118472
#260118192
#260118182
#260117942
#260117232
#260117152
#260116522
#260116412
#260115762
C8×C22⋊D1732
#260114862
C2×#191496012
#467×C1732
#3296173