Finite Group Databasebeta

← #14543142

Subgroups — #14543142

Order 2176.

1060 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763557C212yesmaximal, complemented
2#11763551C211yesmaximal, characteristic, complemented
3#11763525C212yesmaximal, complemented
4#11763487C212yesmaximal, complemented
5#11763484C214yesmaximal, complemented
6#11763450C214yesmaximal, complemented
7#11763429C214yesmaximal, complemented
8#11763398C214yesmaximal, complemented
9#11763384C212yesmaximal, complemented
10#11763370C212yesmaximal, complemented
11#11763344C212yesmaximal, complemented
12#124×D17C211yesmaximal, characteristic, complemented
13C17×#563C211yesmaximal, characteristic, complemented
14C2×#63256C2212yes—
15C2×Q8×D17C2211yescharacteristic, complemented
16C2×D4×D17C2211yescharacteristic
17C22×D68C2212yescomplemented
18C17×C4⋊Q8C2212yescomplemented
19C17×#126C2214yescomplemented
20C17×#125C2212yescomplemented
21C17×#124C2211yescharacteristic
22C17×#123C2212yes—
23C17×C22⋊Q8C2214yescomplemented
24#10587633C2211yescharacteristic, complemented
25#10587633C2212yescomplemented
26#10587633C2214yescomplemented
27#10587630C2212yescomplemented
28#10587623C2212yes—
29#10587622C2211yescharacteristic
30#10587615C2212yescomplemented
31C4×C22⋊D17C2214yescomplemented
32#10587584C2214yescomplemented
33#10587583C2212yescomplemented
34#10587582C2214yescomplemented
35#10587582C2214yescomplemented
36#10587582C2214yescomplemented
37#10587581C2212yes—
38#10587581C2214yescomplemented
39#10587581C2214yescomplemented
40#10587581C2214yescomplemented
41#10587580C2214yescomplemented
42#10587580C2214yescomplemented
43#10587580C2214yescomplemented
44#10587580C2214yescomplemented
45#10587579C2214yescomplemented
46#10587579C2212yescomplemented
47#10587578C2214yescomplemented
48#10587578C2214yescomplemented
49#10587573C2212yescomplemented
50#10587571C2214yescomplemented
51#10587571C2214yescomplemented
52#10587570C2214yes—
53#10587569C2214yescomplemented
54#10587568C2214yes—
55#10587568C2214yescomplemented
56#10587568C2212yescomplemented
57C22⋊C4×D17C2212yescomplemented
58C22⋊C4×D17C2214yescomplemented
59C22⋊C4×D17C2214yescomplemented
60#10587565C2214yescomplemented
61#10587561C2212yescomplemented
62#10587561C2211yescharacteristic, complemented
63#10587561C2212yescomplemented
64C4×D68C2214yescomplemented
65#10587558C2212yescomplemented
66C42×D17C2211yescharacteristic, complemented
67C23×D17C2312yes—
68C23×D17C2312yes—
69C2×Q8×C17C2312yes—
70C2×Q8×C17C2311yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C22×C4×C17C2312yes—
73C2×C22⋊D17C2312yes—
74C2×C22⋊D17C2314yes—
75#63256—44no—
76Q8×D17—42no—
77D4×D17—42no—
78C2×D68—44no—
79C2×D68C2314yes—
80C2×D68C2314yes—
81C2×D68C2311yescharacteristic
82C2×D68C2312yes—
83C2×D68C2314yes—
84C2×D68C2312yes—
85C2×C4×D17C2311yescharacteristic
86C2×C4×D17C2312yes—
87C2×C4×D17C2314yes—
88C2×C4×D17C2312yes—
89C2×C4×D17C2314yes—
90C2×C4×D17C2314yes—
91C2×Dic34C2312yes—
92C2×Dic34C2311yescharacteristic
93C2×Dic34C2312yes—
94C4⋊C4×C17C2312yes—
95C4⋊C4×C17C2314yes—
96C4⋊C4×C17C2314yes—
97C4⋊C4×C17C2314yes—
98C4⋊C4×C17C2312yes—
99C22⋊C4×C17C2314yes—
100C22⋊C4×C17C2314yes—
101C22⋊C4×C17C2312yes—
102C42×C17C2311yescharacteristic
103C42×C17C2312yes—
104C22⋊Dic17C2312yes—
105C22⋊Dic17C2314yes—
106#63227C2314yes—
107#63227C2314yes—
108#63227C2314yes—
109#63227C2314yes—
110#63227C2314yes—
111#63227C2314yes—
112#63227C2314yes—
113#63227C2314yes—
114#63227C2314yes—
115#63227C2314yes—
116C4⋊Dic17C2314yes—
117#63225C2314yes—
118#63225C2314yes—
119#63225C2314yes—
120C4×Dic17C2312yes—
121C4×Dic17C2311yescharacteristic
122C4×Dic17C2312yes—
123C4×Dic17C2314yes—
124C23×C17C2412yes—
125C22×D17—24no—
126C22×D17—42no—
127C22×D17—44no—
128C22×D17—44no—
129C22×D17C2414yes—
130C22×D17C2412yes—
131C22×D17C2411yescharacteristic
132Q8×C17—41no—
133Q8×C17—42no—
134D4×C17—41no—
135C2×C4×C17—42no—
136C2×C4×C17C2412yes—
137C2×C4×C17C2414yes—
138C2×C4×C17C2412yes—
139C2×C4×C17C2411yescharacteristic
140C2×C4×C17C2414yes—
141C22⋊D17—42no—
142C22⋊D17—44no—
143C2×Dic17C2412yes—
144C2×Dic17C2412yes—
145C2×Dic17C2414yes—
146C2×Dic17C2411yescharacteristic
147D68—28no—
148D68—44no—
149D68—42no—
150D68—44no—
151D68—41no—
152D68—42no—
153C4×D17—22no—
154C4×D17—24no—
155C4×D17—44no—
156C4×D17—42no—
157C4×D17—44no—
158C4×D17—44no—
159Dic34—42no—
160Dic34—41no—
161Dic34—42no—
162#3545—171nomaximal, Sylow
163C22×C17—22no—
164C22×C17—24no—
165C22×C17C2511yescharacteristic
166C2×D17—22no—
167C2×D17—24no—
168C2×D17—22no—
169C2×D17—24no—
170C2×D17—24no—
171C2×D17—24no—
172C2×D17—84no—
173C2×D17—84no—
174C2×D17H(5,𝔽2)12yes—
175C4×C17—21no—
176C4×C17—22no—
177C4×C17—24no—
178C4×C17—22no—
179C4×C17—24no—
180Dic17—22no—
181Dic17—21no—
182Dic17—24no—
183Dic17—22no—
184#563D1711yescharacteristic, complemented
185#562—172no—
186H(3,𝔽4)—171no—
187#555—172no—
188#551—174no—
189#551—172no—
190#546—174no—
191#542—174no—
192#538—174no—
193#536—172no—
194#534—172no—
195#529—172no—
196C2×#124—171no—
197C2×C17—42no—
198C2×C17C2×H(5,𝔽2)11yescharacteristic
199C2×C17#58512yes—
200D17—22no—
201D17—44no—
202D17—42no—
203C2×#41—172no—
204C22×Q8—171no—
205C22×D4—171no—
206C22×D4—172no—
207C4⋊Q8—172no—
208C4⋊Q8C2×D1712yescomplemented
209#126—174no—
210#126—174no—
211#126C2×D1714yescomplemented
212#125C2×D1712yescomplemented
213#124—171no—
214#124—172no—
215#124—174no—
216#124—171no—
217#124—174no—
218#124—174no—
219#124—171no—
220#124—172no—
221#124—172no—
222#124C2×D1711yescharacteristic
223#123—172no—
224#123—174no—
225#123—174no—
226#123—174no—
227#123—174no—
228#123—174no—
229#123C2×D1712yes—
230C22⋊Q8—172no—
231C22⋊Q8—174no—
232C22⋊Q8—174no—
233C22⋊Q8—174no—
234C22⋊Q8C2×D1714yescomplemented
235#121—172no—
236#121—174no—
237#121—174no—
238#121—174no—
239#121—174no—
240#120—172no—
241#120—172no—
242#120—174no—
243#120—174no—
244C4×Q8—172no—
245C4×D4—174no—
246C4×D4—172no—
247C4×D4—174no—
248C4×D4—174no—
249#117—174no—
250#117—174no—
251#117—172no—
252C2×C22⋊C4—174no—
253C2×C22⋊C4—172no—
254C2×C22⋊C4—174no—
255C2×C42—171no—
256C17#354511yescharacteristic, complemented, Sylow
257C24—172no—
258C24—172no—
259#41—684no—
260C2×Q8—172no—
261C2×Q8—171no—
262C2×Q8—172no—
263C2×Q8—682no—
264C2×Q8C22×D1712yes—
265C2×Q8C22×D1711yescharacteristic
266C2×D4—174no—
267C2×D4—172no—
268C2×D4—174no—
269C2×D4—174no—
270C2×D4—172no—
271C2×D4—172no—
272C2×D4—174no—
273C2×D4—171no—
274C2×D4—684no—
275C2×D4—682no—
276C2×D4C22×D1711yescharacteristic
277C22×C4—171no—
278C22×C4—172no—
279C22×C4—172no—
280C22×C4—174no—
281C22×C4—174no—
282C22×C4—174no—
283C22×C4C22×D1712yes—
284C4⋊C4—174no—
285C4⋊C4—174no—
286C4⋊C4—174no—
287C4⋊C4—174no—
288C4⋊C4C22×D1714yes—
289C4⋊C4C22×D1714yes—
290C4⋊C4C22×D1714yes—
291C4⋊C4C22×D1712yes—
292C4⋊C4C22×D1712yes—
293C22⋊C4—174no—
294C22⋊C4—172no—
295C22⋊C4—174no—
296C22⋊C4—174no—
297C22⋊C4—174no—
298C22⋊C4—174no—
299C22⋊C4—174no—
300C22⋊C4—174no—
301C22⋊C4—174no—
302C22⋊C4—174no—
303C22⋊C4—174no—
304C22⋊C4—174no—
305C22⋊C4C22×D1714yes—
306C22⋊C4C22×D1714yes—
307C22⋊C4C22×D1712yes—
308C42—171no—
309C42—172no—
310C42—172no—
311C42—174no—
312C42C22×D1711yescharacteristic
313C42C22×D1712yes—
314C23—171no—
315C23—174no—
316C23—172no—
317C23—344no—
318C23—682no—
319C23—684no—
320C23—684no—
321C23C23×D1712yes—
322Q8—41no—
323Q8—42no—
324Q8—681no—
325Q8—682no—
326Q8—682no—
327D4—41no—
328D4—348no—
329D4—684no—
330D4—681no—
331D4—682no—
332D4—684no—
333D4—682no—
334D4—682no—
335D4—684no—
336C2×C4—42no—
337C2×C4—172no—
338C2×C4—174no—
339C2×C4—171no—
340C2×C4—172no—
341C2×C4—342no—
342C2×C4—344no—
343C2×C4—684no—
344C2×C4—684no—
345C2×C4—684no—
346C2×C4—682no—
347C2×C4C23×D1712yes—
348C2×C4C23×D1712yes—
349C2×C4C23×D1711yescharacteristic
350C2×C4C23×D1714yes—
351C2×C4C23×D1714yes—
352C22—22no—
353C22—24no—
354C22—172no—
355C22—342no—
356C22—344no—
357C22—342no—
358C22—344no—
359C22—344no—
360C22—344no—
361C22—1364no—
362C22—1364no—
363C22C24×D1711yescharacteristic
364C4—24no—
365C4—21no—
366C4—22no—
367C4—24no—
368C4—22no—
369C4—342no—
370C4—341no—
371C4—342no—
372C4—344no—
373C2—42no—
374C2—342no—
375C2—684no—
376C2—682no—
377C2H(5,𝔽2)×D1711yescharacteristic
378C2#1176364612yes—
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#563C2
hypercenter#563D17
hypocenterC17#3545
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #11763557C2semidirect
2. #11763551C2semidirect
3. #11763525C2semidirect
4. #11763487C2semidirect
5. #11763484C2semidirect
6. #11763450C2semidirect
7. #11763429C2semidirect
8. #11763398C2semidirect
9. #11763384C2semidirect
10. #11763370C2semidirect
11. #11763344C2semidirect
12. #124×D17C2semidirect
13. C17×#563C2semidirect
15. C2×Q8×D17C22semidirect
17. C22×D68C22semidirect
18. C17×C4⋊Q8C22semidirect
19. C17×#126C22semidirect
20. C17×#125C22semidirect
23. C17×C22⋊Q8C22semidirect
24. #10587633C22semidirect
25. #10587633C22semidirect
26. #10587633C22semidirect
27. #10587630C22semidirect
30. #10587615C22semidirect
31. C4×C22⋊D17C22semidirect
32. #10587584C22semidirect
33. #10587583C22semidirect
34. #10587582C22semidirect
35. #10587582C22semidirect
36. #10587582C22semidirect
38. #10587581C22semidirect
39. #10587581C22semidirect
40. #10587581C22semidirect
41. #10587580C22semidirect
42. #10587580C22semidirect
43. #10587580C22semidirect
44. #10587580C22semidirect
45. #10587579C22semidirect
46. #10587579C22semidirect
47. #10587578C22semidirect
48. #10587578C22semidirect
49. #10587573C22semidirect
50. #10587571C22semidirect
51. #10587571C22semidirect
53. #10587569C22semidirect
55. #10587568C22semidirect
56. #10587568C22semidirect
57. C22⋊C4×D17C22semidirect
58. C22⋊C4×D17C22semidirect
59. C22⋊C4×D17C22semidirect
60. #10587565C22semidirect
61. #10587561C22semidirect
62. #10587561C22semidirect
63. #10587561C22semidirect
64. C4×D68C22semidirect
65. #10587558C22semidirect
66. C42×D17C22semidirect
184. #563D17semidirect
208. C4⋊Q8C2×D17semidirect
211. #126C2×D17semidirect
212. #125C2×D17semidirect
234. C22⋊Q8C2×D17semidirect
256. C17#3545semidirect

Maximal subgroups

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

subgroupindex
#117635572
#117635512
#117635252
#117634872
#117634842
#117634502
#117634292
#117633982
#117633842
#117633702
#117633442
#124×D172
C17×#5632
#354517