Finite Group Databasebeta

← #14540463

Subgroups — #14540463

Order 2176.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×D8×D17C211yesmaximal, characteristic, complemented
2#11763447C212yesmaximal, complemented
3#11763194C211yesmaximal, characteristic, complemented
4#11763047C211yesmaximal, characteristic, complemented
5#11763045C211yesmaximal, characteristic, complemented
6#11763026C212yesmaximal, complemented
7#11762958C212yesmaximal, complemented
8#11762954C212yesmaximal, complemented
9D8×Dic17C211yesmaximal, characteristic, complemented
10C8×C22⋊D17C211yesmaximal, characteristic, complemented
11C2×C17×#135C211yesmaximal, characteristic, complemented
12C2×#41×C17C2212yescomplemented
13C2×D4×D17C2212yes—
14C17×#135—24no—
15C2×Q16×C17C2211yescharacteristic, complemented
16C2×SD16×C17C2212yescomplemented
17C2×D8×C17C2211yescharacteristic
18C22×C8×C17C2211yescharacteristic, complemented
19#10587633C2212yescomplemented
20#10587632C2212yes—
21C2×#63230C2212yescomplemented
22#10587625—22no—
23#10587623—22no—
24D4×Dic17C2212yescomplemented
25C2×#63228C2212yes—
26#10587616C2212yescomplemented
27#10587615—22no—
28C4×C22⋊D17C2211yescharacteristic
29C2×#63227—22no—
30D8×D17—24no—
31C2×D136C2211yescharacteristic
32C2×C8×D17C2211yescharacteristic
33#10587506C2212yescomplemented
34#10587503C2212yescomplemented
35C22⋊#3622C2211yescharacteristic, complemented
36#10587492C2212yescomplemented
37#10587491C2211yescharacteristic
38#10587489C2211yescharacteristic, complemented
39#10587485C2211yescharacteristic, complemented
40C8×Dic17C2211yescharacteristic, complemented
41C23×D17—22no—
42#41×C17—24no—
43#41×C17—42no—
44C2×Q8×C17C2312yes—
45C2×D4×C17—22no—
46C2×D4×C17C2312yes—
47C22×C4×C17—22no—
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—22no—
50C2×C22⋊D17—22no—
51C2×C22⋊D17D411yescharacteristic
52C22×Dic17—22no—
53D4×D17—24no—
54D4×D17—42no—
55C2×D68C2312yes—
56C2×C4×D17C2311yescharacteristic
57Q16×C17—22no—
58SD16×C17—24no—
59D8×C17D414yescomplemented
60C2×C8×C17—22no—
61C2×C8×C17C2311yescharacteristic
62C2×C8×C17C2311yescharacteristic
63C22⋊Dic17—22no—
64C22⋊Dic17D411yescharacteristic
65#63230—42no—
66#63228—24no—
67#63227—22no—
68#63227—22no—
69#63227—22no—
70#63227—22no—
71#63227D411yescharacteristic
72C4⋊Dic17C2312yes—
73#63225—22no—
74#63225D411yescharacteristic
75C4×Dic17C2311yescharacteristic
76C2×#3622C2311yescharacteristic
77C2×#3622C2311yescharacteristic
78D136—22no—
79C8×D17—22no—
80C23×C17—22no—
81C23×C17C2×D411yescharacteristic
82C22×D17—22no—
83C22×D17—44no—
84C22×D17—44no—
85C22×D17—82no—
86C22×D17C2×D411yescharacteristic
87Q8×C17—22no—
88Q8×C17—22no—
89D4×C17—22no—
90D4×C17—44no—
91D4×C17C2×D414yes—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—44no—
96C2×C4×C17C2×D411yescharacteristic
97C2×C4×C17C2411yescharacteristic
98C22⋊D17—22no—
99C22⋊D17—44no—
100C22⋊D17—82no—
101C2×Dic17—22no—
102C2×Dic17—82no—
103C2×Dic17C2×D411yescharacteristic
104C2×Dic17C2×D411yescharacteristic
105C2×Dic17C2×D411yescharacteristic
106D68—22no—
107D68—22no—
108C4×D17—21no—
109C4×D17—21no—
110C8×C17—21no—
111C8×C17C2×D412yes—
112#3622—21no—
113#3622—21no—
114#3280—171nomaximal, Sylow
115C22×C17—24no—
116C22×C17—21no—
117C22×C17—21no—
118C22×C17—21no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17C22×D411yescharacteristic
122C2×D17—21no—
123C2×D17—21no—
124C2×D17—42no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—88no—
128C2×D17#13512yescomplemented
129C4×C17—21no—
130C4×C17—42no—
131C4×C17C22×D411yescharacteristic
132C4×C17H(5,𝔽2)11yescharacteristic
133Dic17—21no—
134Dic17—21no—
135Dic17—21no—
136Dic17—42no—
137C2×#135D1711yescharacteristic, complemented
138C22×D8—171no—
139#546—172no—
140#495—171no—
141#467—171no—
142#466—171no—
143#463—172no—
144#450—172no—
145#449—172no—
146C4×D8—171no—
147C8×D4—171no—
148C2×C17—44no—
149C2×C17—41no—
150C2×C17D4211yescharacteristic
151C2×C17C2×#13511yescharacteristic
152C2×C17#57611yescharacteristic
153D17—22no—
154D17—82no—
155C2×#41C2×D1712yescomplemented
156C22×D4—172no—
157#135—24no—
158C2×Q16C2×D1711yescharacteristic, complemented
159C2×SD16—172no—
160C2×SD16C2×D1712yescomplemented
161C2×D8—172no—
162C2×D8—171no—
163C2×D8—344no—
164C2×D8C2×D1711yescharacteristic
165C22×C8—171no—
166C22×C8C2×D1711yescharacteristic, complemented
167#124—172no—
168#123—342no—
169C22⋊Q8—172no—
170#121—172no—
171#121—342no—
172#120—342no—
173C4×D4—171no—
174C4×D4—172no—
175C2×C22⋊C4—342no—
176#107—171no—
177C4⋊C8—171no—
178#103—172no—
179#102—172no—
180#102—172no—
181C22⋊C8—171no—
182C22⋊C8—171no—
183C4×C8—171no—
184C17#328011yescharacteristic, complemented, Sylow
185C24—342no—
186#41—24no—
187#41—42no—
188C2×Q8C22×D1712yes—
189C2×D4—22no—
190C2×D4—171no—
191C2×D4—172no—
192C2×D4—342no—
193C2×D4—344no—
194C2×D4—342no—
195C2×D4—682no—
196C2×D4C22×D1712yes—
197C22×C4—22no—
198C22×C4—171no—
199C22×C4—342no—
200C22×C4C22×D1711yescharacteristic
201Q16—22no—
202SD16—24no—
203SD16—682no—
204D8—344no—
205D8—342no—
206D8C22⋊D1714yescomplemented
207C2×C8—22no—
208C2×C8—171no—
209C2×C8—171no—
210C2×C8—342no—
211C2×C8C22×D1711yescharacteristic
212C2×C8C22×D1711yescharacteristic
213C4⋊C4—171no—
214C4⋊C4—172no—
215C4⋊C4—342no—
216C22⋊C4—171no—
217C22⋊C4—171no—
218C22⋊C4—342no—
219C22⋊C4—342no—
220C22⋊C4—342no—
221C22⋊C4—342no—
222C22⋊C4—342no—
223C42—171no—
224C23—22no—
225C23—171no—
226C23—342no—
227C23—684no—
228C23—684no—
229C23—1362no—
230C23D4×D1711yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—44no—
235D4—342no—
236D4—342no—
237D4—342no—
238D4—684no—
239D4—1362no—
240D4C2×C22⋊D1714yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—171no—
246C2×C4—171no—
247C2×C4—171no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—342no—
251C2×C4—1362no—
252C2×C4D4×D1711yescharacteristic
253C2×C4C23×D1711yescharacteristic
254C8—21no—
255C8—341no—
256C8—341no—
257C8C2×C22⋊D1712yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22—172no—
265C22—341no—
266C22—341no—
267C22—682no—
268C22—682no—
269C22—682no—
270C22—1368no—
271C22C2×D4×D1711yescharacteristic
272C4—21no—
273C4—42no—
274C4—341no—
275C4—341no—
276C4—341no—
277C4—682no—
278C4#1058769111yescharacteristic
279C4C22×C22⋊D1711yescharacteristic
280C2—41no—
281C2—44no—
282C2—342no—
283C2—1362no—
284C2D4×C22⋊D1711yescharacteristic
285C2#135×D1711yescharacteristic
286C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#135C2
socleC22×C17C22×D4
hypercenterC2×#135D17
hypocenterC17#3280
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×D8×D17C2semidirect
2. #11763447C2semidirect
3. #11763194C2semidirect
4. #11763047C2semidirect
5. #11763045C2semidirect
6. #11763026C2semidirect
7. #11762958C2semidirect
8. #11762954C2semidirect
9. D8×Dic17C2semidirect
10. C8×C22⋊D17C2semidirect
11. C2×C17×#135C2semidirect
12. C2×#41×C17C22semidirect
15. C2×Q16×C17C22semidirect
16. C2×SD16×C17C22semidirect
18. C22×C8×C17C22semidirect
19. #10587633C22semidirect
21. C2×#63230C22semidirect
24. D4×Dic17C22semidirect
26. #10587616C22semidirect
33. #10587506C22semidirect
34. #10587503C22semidirect
35. C22⋊#3622C22semidirect
36. #10587492C22semidirect
38. #10587489C22semidirect
39. #10587485C22semidirect
40. C8×Dic17C22semidirect
59. D8×C17D4semidirect
128. C2×D17#135semidirect
137. C2×#135D17semidirect
155. C2×#41C2×D17semidirect
158. C2×Q16C2×D17semidirect
160. C2×SD16C2×D17semidirect
166. C22×C8C2×D17semidirect
184. C17#3280semidirect
206. D8C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D172
#117634472
#117631942
#117630472
#117630452
#117630262
#117629582
#117629542
D8×Dic172
C8×C22⋊D172
C2×C17×#1352
#328017