Finite Group Databasebeta

← D17×#484 #14538686

Subgroups — D17×#484 #14538686

Order 2176.

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

normal subgroups only

This group has 460 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
1C22⋊Q8×D17C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic
3#11763146C212yesmaximal, complemented
4#11763145C211yesmaximal, characteristic, complemented
5#11763144C211yesmaximal, characteristic, complemented
6#11763143C212yesmaximal, complemented
7#11763142C211yesmaximal, characteristic, complemented
8#107×D17C212yesmaximal, complemented
9#103×D17C212yesmaximal, complemented
10C22⋊C8×D17C211yesmaximal, characteristic
11C17×#484C211yesmaximal, characteristic, complemented
12C2×Q8×D17C2211yescharacteristic
13C22×C4×D17—21no—
14C22×C4×D17C2211yescharacteristic
15C17×C22⋊Q8C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17#10587632C2211yescharacteristic
18C2×C4⋊Dic17C2211yescharacteristic
19#10587610C2211yescharacteristic
20C2×#63225—21no—
21C2×C8×D17C2212yes—
22#10587583C2211yescharacteristic
23#10587582—21no—
24C4⋊C4×D17—21no—
25C4⋊C4×D17—21no—
26C4⋊C4×D17—21no—
27C4⋊C4×D17C2212yes—
28C4⋊C4×D17C2212yes—
29C4⋊C4×D17C2211yescharacteristic
30C22⋊C4×D17—21no—
31#10587564—21no—
32C17×#107C2212yescomplemented
33C17×#103C2212yescomplemented
34C17×C22⋊C8C2211yescharacteristic
35#10587506C2212yescomplemented
36C22⋊#3622C2211yescharacteristic
37#10587491C2212yes—
38#10587489C2212yescomplemented
39#10587487C2212yescomplemented
40#10587481C2212yescomplemented
41#10587478C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17D411yescharacteristic
44C2×Q8×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C22×C4×C17C2311yescharacteristic
47C22×Dic17—21no—
48C22×Dic17D411yescharacteristic
49Q8×D17—24no—
50C2×C4×D17—24no—
51C2×C4×D17—22no—
52C2×C4×D17—22no—
53C2×C4×D17—21no—
54C2×C4×D17—21no—
55C2×C4×D17—44no—
56C2×C4×D17D411yescharacteristic
57C2×C4×D17D411yescharacteristic
58C2×C4×D17C2311yescharacteristic
59C2×C4×D17C2311yescharacteristic
60C2×Dic34—21no—
61C2×Dic34C2311yescharacteristic
62C2×C8×C17C2312yes—
63C4⋊C4×C17—21no—
64C4⋊C4×C17—21no—
65C4⋊C4×C17C2311yescharacteristic
66C4⋊C4×C17C2312yes—
67C22⋊C4×C17—21no—
68C22⋊Dic17—21no—
69#63227—21no—
70#63227—21no—
71C4⋊Dic17—21no—
72C4⋊Dic17—21no—
73C4⋊Dic17C2311yescharacteristic
74C4⋊Dic17C2312yes—
75#63225—21no—
76#63225—21no—
77#63225—21no—
78#63225—22no—
79#63225—22no—
80C2×#3622C2312yes—
81C8×D17—24no—
82C23×C17C2×D411yescharacteristic
83C22×D17—22no—
84C22×D17—22no—
85C22×D17Q1614yes—
86C22×D17C2×D411yescharacteristic
87C22×D17C2×D411yescharacteristic
88Q8×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—42no—
94C2×C4×C17C2×D411yescharacteristic
95C2×C4×C17C2411yescharacteristic
96C2×Dic17—22no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—22no—
100C2×Dic17—42no—
101C2×Dic17C2×D411yescharacteristic
102C2×Dic17C2×D411yescharacteristic
103C4×D17—22no—
104C4×D17—22no—
105C4×D17—22no—
106C4×D17—44no—
107C4×D17—42no—
108C4×D17—42no—
109C4×D17—44no—
110C4×D17#4114yes—
111Dic34—22no—
112Dic34—42no—
113C8×C17—22no—
114#3622—22no—
115C2×#484—171nomaximal, Sylow
116C22×C17—21no—
117C22×C17—21no—
118C22×C17C2×Q1612yes—
119C22×C17C22×D411yescharacteristic
120C2×D17—21no—
121C2×D17—21no—
122C2×D17—28no—
123C2×D17#12312yes—
124C2×D17C2×Q1612yes—
125C2×D17C2×Q1612yes—
126C2×D17AGL(1,ℤ/8)12yes—
127C4×C17—21no—
128C4×C17—41no—
129C4×C17—42no—
130C4×C17—41no—
131C4×C17C2×#4112yes—
132Dic17—21no—
133Dic17—41no—
134Dic17—42no—
135Dic17—41no—
136Dic17#12312yes—
137C2×C22⋊Q8—171no—
138C22×C4⋊C4—171no—
139#484—171no—
140#484—171no—
141#484—172no—
142#484—172no—
143#484—171no—
144#484D1711yescharacteristic, direct product factor
145C2×#107—172no—
146C2×#103—172no—
147C2×C22⋊C8—171no—
148C2×C17—22no—
149C2×C17C2×#12311yescharacteristic
150C2×C17C22×Q1611yescharacteristic
151C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
152D17—22no—
153D17#48414yesdirect product factor
154C22×Q8—171no—
155C23×C4—171no—
156C23×C4—341no—
157C22×C8—172no—
158C22⋊Q8—171no—
159C22⋊Q8—171no—
160C22⋊Q8—171no—
161C22⋊Q8—341no—
162C22⋊Q8—341no—
163C22⋊Q8C2×D1711yescharacteristic
164C2×C4⋊C4—171no—
165C2×C4⋊C4—172no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4—171no—
168C2×C4⋊C4—341no—
169C2×C4⋊C4—341no—
170C2×C4⋊C4—341no—
171C2×C4⋊C4—341no—
172C2×C4⋊C4C2×D1711yescharacteristic
173C2×C22⋊C4—341no—
174#107—172no—
175#107—172no—
176#107—172no—
177#107C2×D1712yescomplemented
178#103—172no—
179#103—172no—
180#103—172no—
181#103C2×D1712yescomplemented
182C22⋊C8—171no—
183C22⋊C8—172no—
184C22⋊C8C2×D1711yescharacteristic
185C17C2×#48411yescharacteristic, complemented, Sylow
186C24—171no—
187C2×Q8—171no—
188C2×Q8—341no—
189C2×Q8—344no—
190C2×Q8C22×D1711yescharacteristic
191C22×C4—21no—
192C22×C4—171no—
193C22×C4—171no—
194C22×C4—171no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—344no—
198C22×C4—341no—
199C22×C4—341no—
200C22×C4—342no—
201C22×C4—342no—
202C22×C4—341no—
203C22×C4—684no—
204C22×C4C22×D1711yescharacteristic
205C2×C8—172no—
206C2×C8—344no—
207C2×C8C22×D1712yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—172no—
211C4⋊C4—171no—
212C4⋊C4—342no—
213C4⋊C4—342no—
214C4⋊C4—341no—
215C4⋊C4—341no—
216C4⋊C4—341no—
217C4⋊C4—341no—
218C4⋊C4—341no—
219C4⋊C4C22×D1712yes—
220C4⋊C4C22×D1711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—341no—
223C22⋊C4—341no—
224C22⋊C4—341no—
225C23—171no—
226C23—171no—
227C23—174no—
228C23—342no—
229C23—342no—
230C23D4×D1711yescharacteristic
231Q8—22no—
232Q8—342no—
233Q8—682no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—171no—
240C2×C4—174no—
241C2×C4—171no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—341no—
247C2×C4—341no—
248C2×C4—342no—
249C2×C4—684no—
250C2×C4—682no—
251C2×C4—682no—
252C2×C4—684no—
253C2×C4—682no—
254C2×C4D4×D1711yescharacteristic
255C2×C4C23×D1711yescharacteristic
256C8—22no—
257C8—342no—
258C22—21no—
259C22—21no—
260C22—172no—
261C22—172no—
262C22—172no—
263C22—172no—
264C22—348no—
265C22—341no—
266C22—341no—
267C22Q16×D1712yes—
268C22C2×D4×D1711yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—172no—
274C4—341no—
275C4—681no—
276C4—682no—
277C4—681no—
278C4#41×D1712yes—
279C2—22no—
280C2—174no—
281C2—342no—
282C2#123×D1711yescharacteristic
283C2C2×Q16×D1711yescharacteristic
284C2AGL(1,ℤ/8)×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#484C2
socleC22×C17C22×D4
prime radicalC23×D17D4
hypercenter#484D17
hypocenterC17C2×#484
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #11763146C2semidirect
4. #11763145C2semidirect
5. #11763144C2semidirect
6. #11763143C2semidirect
7. #11763142C2semidirect
8. #107×D17C2semidirect
9. #103×D17C2semidirect
11. C17×#484C2semidirect
32. C17×#107C22semidirect
33. C17×#103C22semidirect
35. #10587506C22semidirect
38. #10587489C22semidirect
39. #10587487C22semidirect
40. #10587481C22semidirect
41. #10587478C22semidirect
42. #10587478C22semidirect
144. #484D17direct (N × H)
153. D17#484direct (N × H)
177. #107C2×D17semidirect
181. #103C2×D17semidirect
185. C17C2×#484semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D172
C2×C4⋊C4×D172
#117631462
#117631452
#117631442
#117631432
#117631422
#107×D172
#103×D172
C22⋊C8×D172
C17×#4842
C2×#48417