Finite Group Databasebeta

← #14539805

Subgroups — #14539805

Order 2176.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587626C211yesmaximal, characteristic, complemented
2#11763345C211yesmaximal, characteristic, complemented
3#11763195C212yesmaximal, complemented
4#11763172C212yesmaximal, complemented
5#11763007C212yesmaximal, complemented
6#11763003C212yesmaximal, complemented
7#11762959C211yesmaximal, characteristic, complemented
8#11762945C211yesmaximal, characteristic, complemented
9#11762943C211yesmaximal, characteristic, complemented
10#11762863C211yesmaximal, characteristic, complemented
11C17×#450C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17—21no—
14C22×C22⋊D17—21no—
15C22×D68C2211yescharacteristic
16C2×SD16×C17C2212yescomplemented
17C17×C22⋊Q8C2211yescharacteristic
18#10587633C2211yescharacteristic, complemented
19Q8×Dic17C2211yescharacteristic, complemented
20C2×#63230C2212yescomplemented
21#10587626—21no—
22#10587626—21no—
23#10587626C2212yes—
24#10587626C2212yes—
25C2×#63228C2212yescomplemented
26#10587616C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28C2×D136C2212yescomplemented
29#10587584—21no—
30#10587579C2211yescharacteristic, complemented
31#10587578—21no—
32#10587576C2211yescharacteristic, complemented
33#10587572—21no—
34#10587567—21no—
35C17×#102C2212yescomplemented
36C17×C22⋊C8C2211yescharacteristic
37#10587503C2212yescomplemented
38C22⋊#3622C2211yescharacteristic
39#10587492C2212yescomplemented
40#10587486C2212yescomplemented
41#10587485C2212yescomplemented
42#10587480C2212yescomplemented
43C24×C17—21no—
44C23×D17—21no—
45C2×Q8×C17C2311yescharacteristic
46C2×D4×C17—21no—
47C2×D4×C17—42no—
48C2×D4×C17C2312yes—
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—21no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C2×C22⋊D17—22no—
54C2×C22⋊D17—22no—
55C2×C22⋊D17—44no—
56C2×C22⋊D17—44no—
57C22×Dic17D411yescharacteristic
58C22×Dic17D411yescharacteristic
59C2×D68—21no—
60C2×D68—42no—
61C2×D68C2312yes—
62C2×D68C2311yescharacteristic
63C2×C4×D17—21no—
64SD16×C17—42no—
65C2×C8×C17C2312yes—
66C4⋊C4×C17—21no—
67C4⋊C4×C17C2311yescharacteristic
68C22⋊C4×C17—21no—
69#63230—42no—
70#63228—42no—
71#63227—21no—
72#63227—21no—
73#63227—21no—
74#63227—21no—
75C4⋊Dic17—21no—
76C4⋊Dic17—21no—
77C4⋊Dic17C2311yescharacteristic
78#63225—21no—
79C4×Dic17—21no—
80C4×Dic17D411yescharacteristic
81C4×Dic17D411yescharacteristic
82C4×Dic17C2311yescharacteristic
83C4×Dic17C2311yescharacteristic
84C2×#3622C2312yes—
85D136—42no—
86C23×C17—22no—
87C23×C17—42no—
88C23×C17—42no—
89C23×C17—42no—
90C23×C17C2×D411yescharacteristic
91C22×D17—21no—
92C22×D17—22no—
93C22×D17—42no—
94C22×D17—42no—
95C22×D17—42no—
96Q8×C17—22no—
97D4×C17—22no—
98D4×C17—22no—
99D4×C17—42no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17C2×D411yescharacteristic
104C2×C4×C17C2411yescharacteristic
105C22⋊D17—44no—
106C22⋊D17—44no—
107C22⋊D17—44no—
108C22⋊D17—42no—
109C22⋊D17—44no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—21no—
113C2×Dic17C2×D411yescharacteristic
114C2×Dic17C2×D411yescharacteristic
115C2×Dic17C2×D411yescharacteristic
116C2×Dic17C2×D411yescharacteristic
117D68—22no—
118D68—22no—
119D68—22no—
120D68—42no—
121C4×D17—42no—
122C8×C17—22no—
123#3622—22no—
124#3217—171nomaximal, Sylow
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—84no—
131C22×C17AGL(1,ℤ/8)12yes—
132C22×C17C22×D411yescharacteristic
133C2×D17—42no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—41no—
139C2×D17—84no—
140C4×C17—21no—
141C4×C17—41no—
142C4×C17—41no—
143C4×C17H(5,𝔽2)12yes—
144Dic17—21no—
145Dic17—21no—
146Dic17—21no—
147Dic17—41no—
148Dic17AGL(1,ℤ/8)12yes—
149C2×#127—171no—
150#529—171no—
151#496—172no—
152#490—172no—
153#460—172no—
154#459—172no—
155#450—171no—
156#450D1711yescharacteristic, complemented
157#447—171no—
158#447—171no—
159#432—171no—
160C2×C17—22no—
161C2×C17—82no—
162C2×C17#53511yescharacteristic
163C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
165D17—81no—
166D17—82no—
167C22×D4—171no—
168C22×D4—341no—
169C22×D4—341no—
170C22×D4C2×D1711yescharacteristic
171C2×SD16—172no—
172C2×SD16C2×D1712yescomplemented
173C2×D8—172no—
174C2×D8—172no—
175#127—172no—
176#127—172no—
177#127—341no—
178#127—341no—
179#126—341no—
180#125—171no—
181#124—171no—
182#123—341no—
183C22⋊Q8C2×D1711yescharacteristic
184#121—171no—
185C4×Q8—171no—
186C4×D4—171no—
187C4×D4—341no—
188#117—341no—
189C2×C42—171no—
190C4⋊C8—172no—
191#102—172no—
192#102—172no—
193#102—172no—
194#102C2×D1712yescomplemented
195C22⋊C8—171no—
196C22⋊C8C2×D1711yescharacteristic
197#97—172no—
198C17#321711yescharacteristic, complemented, Sylow
199C24—21no—
200C24—341no—
201C2×Q8C22×D1711yescharacteristic
202C2×D4—21no—
203C2×D4—42no—
204C2×D4—172no—
205C2×D4—171no—
206C2×D4—342no—
207C2×D4—342no—
208C2×D4—342no—
209C2×D4—342no—
210C2×D4—341no—
211C2×D4—341no—
212C2×D4—682no—
213C2×D4—684no—
214C2×D4—684no—
215C2×D4C22×D1712yes—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—341no—
219C22×C4C22×D1711yescharacteristic
220SD16—42no—
221SD16—682no—
222D8—682no—
223D8—682no—
224C2×C8—172no—
225C2×C8C22×D1712yes—
226C4⋊C4—21no—
227C4⋊C4—171no—
228C4⋊C4—341no—
229C4⋊C4—341no—
230C4⋊C4—341no—
231C4⋊C4C22×D1711yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—341no—
234C22⋊C4—341no—
235C22⋊C4—341no—
236C22⋊C4—341no—
237C42—171no—
238C42—171no—
239C42—171no—
240C42—171no—
241C42—341no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—341no—
247C23—342no—
248C23—682no—
249C23—682no—
250C23—682no—
251C23D4×D1711yescharacteristic
252Q8—22no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—342no—
257D4—342no—
258D4—342no—
259D4—682no—
260D4—684no—
261D4—684no—
262D4—684no—
263D4—684no—
264D4—682no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—171no—
269C2×C4—171no—
270C2×C4—171no—
271C2×C4—171no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—341no—
275C2×C4—682no—
276C2×C4D4×D1711yescharacteristic
277C2×C4C23×D1711yescharacteristic
278C8—22no—
279C8—342no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—681no—
287C22—682no—
288C22—682no—
289C22—681no—
290C22—682no—
291C22—681no—
292C22—1364no—
293C22#1058760012yes—
294C22C2×D4×D1711yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—172no—
299C4—341no—
300C4—341no—
301C4—341no—
302C4—681no—
303C4#1058769112yes—
304C2—22no—
305C2—82no—
306C2—1362no—
307C2—1361no—
308C2#1176337411yescharacteristic
309C2AGL(1,ℤ/8)×D1711yescharacteristic
310C2C2×#1058760011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#450C2
socleC22×C17C22×D4
hypercenter#450D17
hypocenterC17#3217
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#10587626C2semidirect
2. #11763345C2semidirect
3. #11763195C2semidirect
4. #11763172C2semidirect
5. #11763007C2semidirect
6. #11763003C2semidirect
7. #11762959C2semidirect
8. #11762945C2semidirect
9. #11762943C2semidirect
10. #11762863C2semidirect
11. C17×#450C2semidirect
16. C2×SD16×C17C22semidirect
18. #10587633C22semidirect
19. Q8×Dic17C22semidirect
20. C2×#63230C22semidirect
25. C2×#63228C22semidirect
28. C2×D136C22semidirect
30. #10587579C22semidirect
32. #10587576C22semidirect
35. C17×#102C22semidirect
37. #10587503C22semidirect
39. #10587492C22semidirect
40. #10587486C22semidirect
41. #10587485C22semidirect
42. #10587480C22semidirect
156. #450D17semidirect
172. C2×SD16C2×D17semidirect
194. #102C2×D17semidirect
198. C17#3217semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
#117633452
#117631952
#117631722
#117630072
#117630032
#117629592
#117629452
#117629432
#117628632
C17×#4502
#321717