Finite Group Databasebeta

← #36365393

Subgroups — #36365393

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#26012175C211yesmaximal, characteristic, complemented
2#26012138C211yesmaximal, characteristic
3#26011844C211yesmaximal, characteristic, complemented
4#26011823C211yesmaximal, characteristic, complemented
5#26011812C211yesmaximal, characteristic, complemented
6#26011790C211yesmaximal, characteristic, complemented
7#26011731C211yesmaximal, characteristic, complemented
8#26011711C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011576C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583813C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149672—21no—
22#19149670C2211yescharacteristic, complemented
23#19149669—21no—
24#19149666C2211yescharacteristic
25#19149661—21no—
26#19149661—21no—
27#19149659C2211yescharacteristic
28#19149655C2211yescharacteristic
29#19149650—21no—
30#19149647—21no—
31Q8×Dic173C2211yescharacteristic, complemented
32#19149641C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149638C2211yescharacteristic, complemented
35#19149637—21no—
36C4⋊C4×D173C2211yescharacteristic
37C22⋊C4×D173—21no—
38#19149612C2211yescharacteristic, complemented
39#19149610C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149604C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173D411yescharacteristic
67D4×D173—22no—
68D4×D173—41no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583816—41no—
74#14583813—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—41no—
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102SD16×C173—41no—
103C2×C8×C173—41no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—42no—
113C22×D173—42no—
114C22×D173—81no—
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118D692—21no—
119D692#4112yes—
120C22⋊D173—22no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132C4×D173—42no—
133C4×D173—81no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137C23×C173C2×D411yescharacteristic
138Q8×C173—21no—
139Q8×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—81no—
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—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—22no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—84no—
165C2×D173#13512yes—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173C2×#4111yescharacteristic
171C4×C173H(5,𝔽2)11yescharacteristic
172Dic173—21no—
173Dic173—21no—
174Dic173—21no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—41no—
179C2×C173—81no—
180C2×C173#54611yescharacteristic
181C2×C173C2×#13511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—22no—
184D173—42no—
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#465D17311yescharacteristic, complemented
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×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16C2×D17311yescharacteristic
207C22×C8—1731no—
208C22×C8C2×D17311yescharacteristic, complemented
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, complemented
218#121—3461no—
219#121C2×D17311yescharacteristic
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#106C2×D17311yescharacteristic, complemented
231C4⋊C8—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103C2×D17311yescharacteristic, complemented
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102C2×D17311yescharacteristic
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244C4×C8—1731no—
245C24—3461no—
246#41—3462no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—21no—
251C2×D4—3462no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1731no—
256C2×D4—6921no—
257C2×D4C22×D17311yescharacteristic
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4C22×D17311yescharacteristic
263Q16—6921no—
264SD16—41no—
265SD16—3462no—
266C2×C8—41no—
267C2×C8—1731no—
268C2×C8—1731no—
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⋊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—21no—
298C23—3461no—
299C23—13841no—
300C23—1731no—
301C23—6922no—
302C23—6922no—
303C23D4×D17311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3461no—
307Q8—3461no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—3462no—
312D4—3461no—
313D4—1732no—
314D4—6922no—
315D4—6922no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
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—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—3461no—
345C22—3461no—
346C22—3462no—
347C22—13844no—
348C22—1732no—
349C22—6921no—
350C22—6921no—
351C22C2×D4×D17311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3461no—
357C4—3461no—
358C4—3461no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4#41×D17311yescharacteristic
363C4#1914972411yescharacteristic
364C2—41no—
365C2—81no—
366C2—3462no—
367C2—6922no—
368C2#2601213011yescharacteristic
369C2#135×D17311yescharacteristic
370C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
prime radical#26012138C2
hypercenter#465D173
hypocenterC173#3296
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #26012175C2semidirect
3. #26011844C2semidirect
4. #26011823C2semidirect
5. #26011812C2semidirect
6. #26011790C2semidirect
7. #26011731C2semidirect
8. #26011711C2semidirect
9. #26011651C2semidirect
10. #26011640C2semidirect
11. #26011576C2semidirect
12. C8×C22⋊D173C2semidirect
13. #26011488C2semidirect
14. #102×D173C2semidirect
15. #465×C173C2semidirect
18. C2×#14583816C22semidirect
22. #19149670C22semidirect
31. Q8×Dic173C22semidirect
34. #19149638C22semidirect
38. #19149612C22semidirect
39. #19149610C22semidirect
40. #19149608C22semidirect
41. #19149604C22semidirect
42. #19149604C22semidirect
43. #19149602C22semidirect
49. C22⋊#13063436C22semidirect
51. C8×Dic173C22semidirect
53. C22×C8×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
56. #106×C173C22semidirect
57. #103×C173C22semidirect
185. C173#3296semidirect
194. #465D173semidirect
208. C22×C8C2×D173semidirect
217. C22⋊Q8C2×D173semidirect
230. #106C2×D173semidirect
235. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121752
#260121382
#260118442
#260118232
#260118122
#260117902
#260117312
#260117112
#260116512
#260116402
#260115762
C8×C22⋊D1732
#260114882
#102×D1732
#465×C1732
#3296173