Finite Group Databasebeta

← #14538923

Subgroups — #14538923

Order 2176.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#11763345C211yesmaximal, characteristic, complemented
2C2×D4×Dic17C211yesmaximal, characteristic, complemented
3#11763137C211yesmaximal, characteristic, complemented
4#11763123C211yesmaximal, characteristic, complemented
5#11763107C212yesmaximal, complemented
6#11763026C212yesmaximal, complemented
7#11762943C211yesmaximal, characteristic
8#11762918C212yesmaximal, complemented
9#11762910C212yesmaximal, complemented
10#11762863C211yesmaximal, characteristic, complemented
11C17×#450C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C23×Dic17—21no—
14C2×SD16×C17C2212yes—
15C17×C22⋊Q8C2211yescharacteristic
16C2×C22⋊Dic17—21no—
17#10587633C2211yescharacteristic, complemented
18Q8×Dic17C2211yescharacteristic, complemented
19D4×Dic17—21no—
20D4×Dic17—21no—
21D4×Dic17C2212yes—
22D4×Dic17C2212yes—
23C2×#63228C2212yes—
24#10587616C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4×Dic17C2211yescharacteristic
27#10587584—21no—
28#10587579C2211yescharacteristic, complemented
29#10587578—21no—
30#10587576C2211yescharacteristic, complemented
31#10587572—21no—
32#10587567—21no—
33C17×#102C2212yes—
34C17×C22⋊C8C2211yescharacteristic
35#10587506C2212yescomplemented
36#10587503C2212yes—
37C22⋊#3622C2211yescharacteristic
38#10587492C2212yescomplemented
39#10587489C2212yescomplemented
40#10587486C2212yescomplemented
41#10587485C2212yescomplemented
42#10587479C2212yescomplemented
43C24×C17—21no—
44C2×Q8×C17C2311yescharacteristic
45C2×D4×C17—24no—
46C2×D4×C17—21no—
47C2×D4×C17C2312yes—
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—21no—
50C22×Dic17—21no—
51C22×Dic17—22no—
52C22×Dic17—22no—
53C22×Dic17—44no—
54C22×Dic17D411yescharacteristic
55C22×Dic17D411yescharacteristic
56C2×D68C2311yescharacteristic
57C2×C4×D17—21no—
58SD16×C17—24no—
59C2×C8×C17C2312yes—
60C4⋊C4×C17—21no—
61C4⋊C4×C17C2311yescharacteristic
62C22⋊C4×C17—21no—
63C22⋊Dic17—22no—
64C22⋊Dic17—22no—
65#63228—24no—
66#63227—21no—
67#63227—21no—
68#63227—21no—
69#63227—21no—
70C4⋊Dic17—21no—
71C4⋊Dic17—21no—
72C4⋊Dic17—21no—
73C4⋊Dic17C2312yes—
74C4⋊Dic17C2311yescharacteristic
75#63225—21no—
76C4×Dic17—21no—
77C4×Dic17D411yescharacteristic
78C4×Dic17D411yescharacteristic
79C4×Dic17C2311yescharacteristic
80C4×Dic17C2311yescharacteristic
81C2×#3622C2312yes—
82C23×C17—24no—
83C23×C17—22no—
84C23×C17—42no—
85C23×C17—42no—
86C23×C17C2×D411yescharacteristic
87C22×D17—21no—
88Q8×C17—22no—
89D4×C17—24no—
90D4×C17—22no—
91D4×C17#4114yes—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17C2×D411yescharacteristic
96C2×C4×C17C2411yescharacteristic
97C22⋊D17—42no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—21no—
101C2×Dic17—41no—
102C2×Dic17—44no—
103C2×Dic17—44no—
104C2×Dic17—42no—
105C2×Dic17C2×D411yescharacteristic
106C2×Dic17C2×D411yescharacteristic
107C2×Dic17C2×D411yescharacteristic
108C2×Dic17C2×D411yescharacteristic
109D68—22no—
110C4×D17—42no—
111C8×C17—22no—
112#3622—22no—
113#3114—171nomaximal, Sylow
114C22×C17—21no—
115C22×C17—21no—
116C22×C17—24no—
117C22×C17—48no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17AGL(1,ℤ/8)12yes—
121C22×C17C22×D411yescharacteristic
122C2×D17—41no—
123C2×D17—41no—
124C2×D17—41no—
125C4×C17—21no—
126C4×C17—41no—
127C4×C17—41no—
128C4×C17C2×#4112yes—
129Dic17—21no—
130Dic17—21no—
131Dic17—21no—
132Dic17—21no—
133Dic17—42no—
134Dic17—41no—
135#529—171no—
136C2×C4×D4—171no—
137#483—171no—
138#480—171no—
139#478—172no—
140#463—172no—
141#450D1711yescharacteristic, complemented
142#447—171no—
143#442—172no—
144#440—172no—
145#432—171no—
146C2×C17—22no—
147C2×C17—44no—
148C2×C17#52511yescharacteristic
149C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C17#57511yescharacteristic
151D17—81no—
152C22×D4C2×D1711yescharacteristic
153C23×C4—341no—
154C2×SD16C2×D1712yes—
155C2×D8—172no—
156#126—341no—
157#125—171no—
158#124—171no—
159#123—341no—
160C22⋊Q8C2×D1711yescharacteristic
161#121—171no—
162C4×Q8—171no—
163C4×D4—172no—
164C4×D4—172no—
165C4×D4—171no—
166C4×D4—341no—
167C4×D4—341no—
168C4×D4—341no—
169#117—341no—
170C2×C4⋊C4—171no—
171C2×C22⋊C4—341no—
172C2×C42—171no—
173#107—172no—
174#106—172no—
175C4⋊C8—172no—
176#103—172no—
177#102—172no—
178#102—172no—
179#102C2×D1712yes—
180C22⋊C8—171no—
181C22⋊C8C2×D1711yescharacteristic
182#97—172no—
183C17#311411yescharacteristic, complemented, Sylow
184C24—21no—
185C2×Q8C22×D1711yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—171no—
189C2×D4—341no—
190C2×D4C22×D1712yes—
191C22×C4—171no—
192C22×C4—171no—
193C22×C4—341no—
194C22×C4—342no—
195C22×C4—342no—
196C22×C4—341no—
197C22×C4—684no—
198C22×C4C22×D1711yescharacteristic
199SD16—24no—
200D8—344no—
201C2×C8—172no—
202C2×C8C22×D1712yes—
203C4⋊C4—21no—
204C4⋊C4—171no—
205C4⋊C4—172no—
206C4⋊C4—341no—
207C4⋊C4—341no—
208C4⋊C4—341no—
209C4⋊C4—341no—
210C4⋊C4C22×D1711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—341no—
213C22⋊C4—341no—
214C22⋊C4—341no—
215C22⋊C4—342no—
216C22⋊C4—342no—
217C22⋊C4—341no—
218C42—171no—
219C42—171no—
220C42—171no—
221C42—171no—
222C42—341no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—341no—
228C23D4×D1711yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—342no—
233D4—682no—
234D4#6325414yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—171no—
239C2×C4—171no—
240C2×C4—171no—
241C2×C4—171no—
242C2×C4—342no—
243C2×C4—341no—
244C2×C4—342no—
245C2×C4—682no—
246C2×C4—681no—
247C2×C4—684no—
248C2×C4—684no—
249C2×C4—682no—
250C2×C4D4×D1711yescharacteristic
251C2×C4C23×D1711yescharacteristic
252C8—22no—
253C8—342no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—681no—
261C22—681no—
262C22—681no—
263C22#1058760012yes—
264C22C2×D4×D1711yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—341no—
269C4—341no—
270C4—341no—
271C4—341no—
272C4—682no—
273C4—681no—
274C4C2×#6325412yes—
275C2—22no—
276C2—44no—
277C2—1361no—
278C2#1176331511yescharacteristic
279C2C2×#1058760011yescharacteristic
280C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#450C2
socleC22×C17C22×D4
prime radical#11762943C2
hypercenter#450D17
hypocenterC17#3114
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. #11763345C2semidirect
2. C2×D4×Dic17C2semidirect
3. #11763137C2semidirect
4. #11763123C2semidirect
5. #11763107C2semidirect
6. #11763026C2semidirect
8. #11762918C2semidirect
9. #11762910C2semidirect
10. #11762863C2semidirect
11. C17×#450C2semidirect
17. #10587633C22semidirect
18. Q8×Dic17C22semidirect
28. #10587579C22semidirect
30. #10587576C22semidirect
35. #10587506C22semidirect
38. #10587492C22semidirect
39. #10587489C22semidirect
40. #10587486C22semidirect
41. #10587485C22semidirect
42. #10587479C22semidirect
141. #450D17semidirect
183. C17#3114semidirect

Maximal subgroups

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

subgroupindex
#117633452
C2×D4×Dic172
#117631372
#117631232
#117631072
#117630262
#117629432
#117629182
#117629102
#117628632
C17×#4502
#311417