Finite Group Databasebeta

← D17×#481 #14538680

Subgroups — D17×#481 #14538680

Order 2176.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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#121×D17C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3#11763129C212yesmaximal, complemented
4#11763128C211yesmaximal, characteristic, complemented
5#11763127C211yesmaximal, characteristic, complemented
6#11763126C212yesmaximal, complemented
7#11763125C211yesmaximal, characteristic, complemented
8#106×D17C212yesmaximal, complemented
9#102×D17C212yesmaximal, complemented
10C22⋊C8×D17C211yesmaximal, characteristic, complemented
11C17×#481C211yesmaximal, characteristic, complemented
12C2×D4×D17—21no—
13C2×D4×D17C2211yescharacteristic
14C22×C4×D17—21no—
15C22×C4×D17C2211yescharacteristic
16C17×#121C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic, complemented
18#10587625—21no—
19#10587624C2211yescharacteristic
20#10587616C2211yescharacteristic
21C2×C4⋊Dic17C2211yescharacteristic, complemented
22C2×#63225—21no—
23C2×C8×D17C2212yes—
24#10587581C2211yescharacteristic
25C4⋊C4×D17—21no—
26C4⋊C4×D17—21no—
27C4⋊C4×D17C2212yes—
28C4⋊C4×D17C2211yescharacteristic
29C4⋊C4×D17C2212yescomplemented
30#10587570—21no—
31C22⋊C4×D17—21no—
32C17×#106C2212yescomplemented
33C17×#102C2212yescomplemented
34C17×C22⋊C8C2211yescharacteristic, complemented
35#10587503C2212yescomplemented
36C22⋊#3622C2211yescharacteristic, complemented
37#10587492C2212yescomplemented
38#10587491C2212yescomplemented
39#10587488C2212yescomplemented
40#10587480C2212yescomplemented
41#10587479C2212yescomplemented
42#10587479C2212yescomplemented
43C23×D17—21no—
44C23×D17D411yescharacteristic
45C2×D4×C17—21no—
46C2×D4×C17C2311yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—21no—
50C2×C22⋊D17—21no—
51C2×C22⋊D17—21no—
52C22×Dic17—21no—
53C22×Dic17D411yescharacteristic
54D4×D17—24no—
55D4×D17—44no—
56C2×D68—21no—
57C2×D68C2311yescharacteristic
58C2×C4×D17—24no—
59C2×C4×D17—22no—
60C2×C4×D17—22no—
61C2×C4×D17—21no—
62C2×C4×D17—44no—
63C2×C4×D17D411yescharacteristic
64C2×C4×D17D411yescharacteristic
65C2×C4×D17C2311yescharacteristic
66C2×C4×D17C2311yescharacteristic
67C2×C8×C17C2312yes—
68C4⋊C4×C17—21no—
69C4⋊C4×C17C2311yescharacteristic
70C4⋊C4×C17C2312yes—
71C22⋊C4×C17—21no—
72C22⋊Dic17—21no—
73#63227—21no—
74#63227—21no—
75C4⋊Dic17—21no—
76C4⋊Dic17C2311yescharacteristic
77C4⋊Dic17C2312yes—
78#63225—21no—
79#63225—22no—
80#63225—22no—
81C2×#3622C2312yes—
82C8×D17—24no—
83C23×C17—21no—
84C23×C17C2×D411yescharacteristic
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—21no—
88C22×D17—42no—
89C22×D17—42no—
90C22×D17—42no—
91C22×D17SD1614yes—
92C22×D17C2×D411yescharacteristic
93C22×D17C2×D411yescharacteristic
94D4×C17—22no—
95D4×C17—42no—
96C2×C4×C17—22no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—42no—
100C2×C4×C17C2×D411yescharacteristic
101C2×C4×C17C2411yescharacteristic
102C22⋊D17—42no—
103C22⋊D17—42no—
104C22⋊D17—42no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—42no—
109C2×Dic17C2×D411yescharacteristic
110C2×Dic17C2×D411yescharacteristic
111D68—22no—
112D68—42no—
113C4×D17—22no—
114C4×D17—22no—
115C4×D17—22no—
116C4×D17—44no—
117C4×D17—42no—
118C4×D17—44no—
119C4×D17#4114yes—
120C8×C17—22no—
121#3622—22no—
122C2×#481—171nomaximal, Sylow
123C22×C17—21no—
124C22×C17—21no—
125C22×C17—41no—
126C22×C17—41no—
127C22×C17—41no—
128C22×C17C2×SD1612yes—
129C22×C17C22×D411yescharacteristic
130C2×D17—21no—
131C2×D17—21no—
132C2×D17—28no—
133C2×D17—41no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—84no—
137C2×D17#12312yes—
138C2×D17C2×SD1612yes—
139C2×D17C2×SD1612yes—
140C2×D17AGL(1,ℤ/8)12yes—
141C4×C17—21no—
142C4×C17—41no—
143C4×C17—42no—
144C4×C17C2×#4112yes—
145Dic17—21no—
146Dic17—41no—
147Dic17—42no—
148Dic17#12312yes—
149C2×#121—171no—
150C22×C4⋊C4—171no—
151#481—171no—
152#481—171no—
153#481—172no—
154#481—172no—
155#481—171no—
156#481D1711yescharacteristic, direct product factor
157C2×#106—172no—
158C2×#102—172no—
159C2×C22⋊C8—171no—
160C2×C17—22no—
161C2×C17—81no—
162C2×C17C2×#12311yescharacteristic
163C2×C17C22×SD1611yescharacteristic
164C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
165D17—22no—
166D17—81no—
167D17#48114yesdirect product factor
168C22×D4—171no—
169C22×D4—341no—
170C23×C4—171no—
171C23×C4—341no—
172C22×C8—172no—
173#121—171no—
174#121—171no—
175#121—171no—
176#121—341no—
177#121—341no—
178#121C2×D1711yescharacteristic
179C2×C4⋊C4—171no—
180C2×C4⋊C4—172no—
181C2×C4⋊C4—172no—
182C2×C4⋊C4—171no—
183C2×C4⋊C4—341no—
184C2×C4⋊C4—341no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4C2×D1711yescharacteristic, complemented
187C2×C22⋊C4—341no—
188#106—172no—
189#106—172no—
190#106—172no—
191#106C2×D1712yescomplemented
192#102—172no—
193#102—172no—
194#102—172no—
195#102C2×D1712yescomplemented
196C22⋊C8—171no—
197C22⋊C8—172no—
198C22⋊C8C2×D1711yescharacteristic, complemented
199C17C2×#48111yescharacteristic, complemented, Sylow
200C24—171no—
201C24—341no—
202C2×D4—21no—
203C2×D4—171no—
204C2×D4—341no—
205C2×D4—344no—
206C2×D4—341no—
207C2×D4—341no—
208C2×D4—341no—
209C2×D4—684no—
210C2×D4C22×D1711yescharacteristic
211C22×C4—21no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—171no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—344no—
218C22×C4—341no—
219C22×C4—342no—
220C22×C4—342no—
221C22×C4—341no—
222C22×C4—684no—
223C22×C4C22×D1711yescharacteristic
224C2×C8—172no—
225C2×C8—344no—
226C2×C8C22×D1712yes—
227C4⋊C4—21no—
228C4⋊C4—172no—
229C4⋊C4—171no—
230C4⋊C4—342no—
231C4⋊C4—342no—
232C4⋊C4—341no—
233C4⋊C4—341no—
234C4⋊C4C22×D1712yes—
235C4⋊C4C22×D1711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—341no—
238C22⋊C4—341no—
239C22⋊C4—341no—
240C23—21no—
241C23—171no—
242C23—171no—
243C23—174no—
244C23—342no—
245C23—342no—
246C23—341no—
247C23—682no—
248C23—682no—
249C23—682no—
250C23D4×D1711yescharacteristic
251D4—22no—
252D4—42no—
253D4—342no—
254D4—682no—
255D4—682no—
256D4—682no—
257D4—682no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—171no—
263C2×C4—174no—
264C2×C4—171no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—341no—
270C2×C4—342no—
271C2×C4—684no—
272C2×C4—682no—
273C2×C4—684no—
274C2×C4—682no—
275C2×C4D4×D1711yescharacteristic
276C2×C4C23×D1711yescharacteristic
277C8—22no—
278C8—342no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—172no—
285C22—172no—
286C22—172no—
287C22—172no—
288C22—348no—
289C22—341no—
290C22—341no—
291C22—681no—
292C22—681no—
293C22—681no—
294C22—1364no—
295C22SD16×D1712yes—
296C22C2×D4×D1711yescharacteristic
297C4—21no—
298C4—41no—
299C4—42no—
300C4—172no—
301C4—341no—
302C4—681no—
303C4—682no—
304C4#41×D1712yes—
305C2—22no—
306C2—81no—
307C2—174no—
308C2—342no—
309C2—1361no—
310C2#123×D1711yescharacteristic
311C2C2×SD16×D1711yescharacteristic
312C2AGL(1,ℤ/8)×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#481C2
socleC22×C17C22×D4
prime radical#121×D17C2
hypercenter#481D17
hypocenterC17C2×#481
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D17C2semidirect
3. #11763129C2semidirect
4. #11763128C2semidirect
5. #11763127C2semidirect
6. #11763126C2semidirect
7. #11763125C2semidirect
8. #106×D17C2semidirect
9. #102×D17C2semidirect
10. C22⋊C8×D17C2semidirect
11. C17×#481C2semidirect
17. C2×C4⋊C4×C17C22semidirect
21. C2×C4⋊Dic17C22semidirect
29. C4⋊C4×D17C22semidirect
32. C17×#106C22semidirect
33. C17×#102C22semidirect
34. C17×C22⋊C8C22semidirect
35. #10587503C22semidirect
36. C22⋊#3622C22semidirect
37. #10587492C22semidirect
38. #10587491C22semidirect
39. #10587488C22semidirect
40. #10587480C22semidirect
41. #10587479C22semidirect
42. #10587479C22semidirect
156. #481D17direct (N × H)
167. D17#481direct (N × H)
186. C2×C4⋊C4C2×D17semidirect
191. #106C2×D17semidirect
195. #102C2×D17semidirect
198. C22⋊C8C2×D17semidirect
199. C17C2×#481semidirect

Maximal subgroups

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

subgroupindex
#121×D172
C2×C4⋊C4×D172
#117631292
#117631282
#117631272
#117631262
#117631252
#106×D172
#102×D172
C22⋊C8×D172
C17×#4812
C2×#48117