Finite Group Databasebeta

← #36365584

Subgroups — #36365584

Order 22144.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012181C211yesmaximal, characteristic, complemented
2#26011855C211yesmaximal, characteristic, complemented
3#26011812C211yesmaximal, characteristic, complemented
4#26011802C211yesmaximal, characteristic, complemented
5#26011791C211yesmaximal, characteristic, complemented
6#26011735C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011688C211yesmaximal, characteristic, complemented
9#26011642C211yesmaximal, characteristic, complemented
10#26011642C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011496C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15D42×C173C211yesmaximal, characteristic
16C2×#14583814C2211yescharacteristic
17C2×#14583814C2211yescharacteristic, complemented
18C2×#14583814C2211yescharacteristic, complemented
19C2×#14583810C2211yescharacteristic, complemented
20C22×#13063436C2211yescharacteristic, complemented
21#19149672—21no—
22#19149671C2211yescharacteristic, complemented
23#19149669—21no—
24#19149660—21no—
25#19149658C2211yescharacteristic, complemented
26#19149658C2211yescharacteristic, complemented
27C4×Dic346C2211yescharacteristic, complemented
28#19149638C2211yescharacteristic, complemented
29#19149635—21no—
30C2×C4⋊Dic173C2211yescharacteristic, complemented
31#19149612C2211yescharacteristic, complemented
32#19149612C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149602C2211yescharacteristic, complemented
36#19149600C2211yescharacteristic
37#19149600C2211yescharacteristic
38#19149600C2211yescharacteristic
39#19149600C2211yescharacteristic, complemented
40#19149600C2211yescharacteristic, complemented
41#19149600C2211yescharacteristic, complemented
42#19149600C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic, complemented
44#19149600C2211yescharacteristic, complemented
45C22⋊#13063436C2211yescharacteristic, complemented
46C22⋊#13063436C2211yescharacteristic, complemented
47#19149590C2211yescharacteristic, complemented
48C22×D4×C173—21no—
49C22×D4×C173C2211yescharacteristic
50C22×D4×C173C2211yescharacteristic
51#127×C173C2211yescharacteristic
52#121×C173—21no—
53#121×C173C2211yescharacteristic
54#121×C173C2211yescharacteristic
55#120×C173—21no—
56#120×C173—21no—
57C4×D4×C173C2211yescharacteristic
58C4×D4×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C22×Dic173—21no—
61#14583814—22no—
62#14583814—41no—
63#14583814—41no—
64#14583810—41no—
65C2×#13063436—41no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C4⋊Dic173—21no—
71C4⋊Dic173C2311yescharacteristic
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74#14583805—21no—
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80C22⋊Dic173—21no—
81C22⋊Dic173—21no—
82C4×Dic173—21no—
83C4×Dic173—21no—
84C24×C173—21no—
85C24×C173—21no—
86C2×D4×C173—22no—
87C2×D4×C173—22no—
88C2×D4×C173—21no—
89C2×D4×C173—21no—
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173—21no—
93C2×D4×C173—21no—
94C2×D4×C173—41no—
95C2×D4×C173—41no—
96C2×D4×C173—44no—
97C2×D4×C173D411yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C4⋊C4×C173D411yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C42×C173C2311yescharacteristic
112Dic346—21no—
113Dic346—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—42no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122#13063436—21no—
123C23×C173—21no—
124C23×C173—21no—
125C23×C173—21no—
126C23×C173—42no—
127C23×C173—42no—
128C23×C173—42no—
129C23×C173—42no—
130C23×C173—81no—
131C23×C173—81no—
132C23×C173C2×D411yescharacteristic
133C23×C173C2×D411yescharacteristic
134D4×C173—22no—
135D4×C173—21no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—42no—
141D4×C173—42no—
142D4×C173—42no—
143D4×C173—42no—
144D4×C173—42no—
145D4×C173—81no—
146D4×C173#4112yes—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—42no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—22no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—84no—
173C22×C173—84no—
174C22×C173AGL(1,ℤ/8)12yes—
175C22×C173C22×D411yescharacteristic
176C4×C173—21no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—41no—
181C4×C173C2×#4111yescharacteristic
182C4×C173H(5,𝔽2)11yescharacteristic
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187C2×C173—22no—
188C2×C173—41no—
189C2×C173—42no—
190C2×C173—81no—
191C2×C173—81no—
192C2×C173#54611yescharacteristic
193C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
194C2×C173#57711yescharacteristic
195C173#331011yescharacteristic, complemented, Sylow
196#3310—1731nomaximal, Sylow
197#552—1731no—
198D42D17311yescharacteristic
199#490—1731no—
200#482—1731no—
201#480—1731no—
202#478—1731no—
203#468—1731no—
204#465—1731no—
205#460—1731no—
206#450—1731no—
207#450—1731no—
208#440—1731no—
209#435—1731no—
210#418—1731no—
211C2×#102—1731no—
212C22×D4—21no—
213C22×D4C2×D17311yescharacteristic
214C22×D4C2×D17311yescharacteristic
215C2×SD16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×M16—1731no—
219C22×C8—1731no—
220#127C2×D17311yescharacteristic
221#126—3461no—
222#125—3461no—
223#125—1731no—
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227#121—21no—
228#121C2×D17311yescharacteristic
229#121C2×D17311yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8—1731no—
233C4×D4C2×D17311yescharacteristic
234C4×D4C2×D17311yescharacteristic
235#117—3461no—
236#117—1731no—
237C2×C4⋊C4—1731no—
238#107—1731no—
239#107—1731no—
240#106—1731no—
241C4⋊C8—1731no—
242#103—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97—1731no—
255C24—21no—
256C24—21no—
257C2×Q8—1731no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D17311yescharacteristic
270C2×D4C22×D17311yescharacteristic
271C2×D4C22×D17311yescharacteristic
272C2×D4C22×D17311yescharacteristic
273C22×C4—21no—
274C22×C4—3461no—
275C22×C4C22×D17311yescharacteristic
276C22×C4C22×D17311yescharacteristic
277SD16—3462no—
278SD16—6921no—
279SD16—6921no—
280M16—6921no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8—6921no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4C22⋊D17311yescharacteristic
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4C22⋊D17311yescharacteristic
304C22⋊C4C22⋊D17311yescharacteristic
305C42—3461no—
306C42—3461no—
307C42C22×D17311yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D17311yescharacteristic
318C23C2×C22⋊D17311yescharacteristic
319Q8—3461no—
320Q8—3461no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#1458382512yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—6922no—
345C2×C4C2×C22⋊D17311yescharacteristic
346C2×C4C2×C22⋊D17311yescharacteristic
347C2×C4C2×C22⋊D17311yescharacteristic
348C2×C4C2×C22⋊D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—3461no—
351C8—3461no—
352C8—3461no—
353C8—3461no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#1914970512yes—
371C22C22×C22⋊D17311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—6921no—
378C4—6921no—
379C4—6921no—
380C4—6921no—
381C4C2×#1458382511yescharacteristic
382C4#1914972411yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#2601213511yescharacteristic
389C2C2×#1914970511yescharacteristic
390C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radicalD42×C173C2
socleC22×C173C22×D4
hypercenterD42D173
hypocenterC173#3310
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #26012181C2semidirect
2. #26011855C2semidirect
3. #26011812C2semidirect
4. #26011802C2semidirect
5. #26011791C2semidirect
6. #26011735C2semidirect
7. #26011718C2semidirect
8. #26011688C2semidirect
9. #26011642C2semidirect
10. #26011642C2semidirect
11. #26011591C2semidirect
12. #26011563C2semidirect
13. #26011496C2semidirect
14. C2×#19149600C2semidirect
17. C2×#14583814C22semidirect
18. C2×#14583814C22semidirect
19. C2×#14583810C22semidirect
20. C22×#13063436C22semidirect
22. #19149671C22semidirect
25. #19149658C22semidirect
26. #19149658C22semidirect
27. C4×Dic346C22semidirect
28. #19149638C22semidirect
30. C2×C4⋊Dic173C22semidirect
31. #19149612C22semidirect
32. #19149612C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149602C22semidirect
39. #19149600C22semidirect
40. #19149600C22semidirect
41. #19149600C22semidirect
42. #19149600C22semidirect
43. #19149600C22semidirect
44. #19149600C22semidirect
45. C22⋊#13063436C22semidirect
46. C22⋊#13063436C22semidirect
47. #19149590C22semidirect
195. C173#3310semidirect

Maximal subgroups

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

subgroupindex
#260121812
#260118552
#260118122
#260118022
#260117912
#260117352
#260117182
#260116882
#260116422
#260115912
#260115632
#260114962
C2×#191496002
D42×C1732
#3310173