Finite Group Databasebeta

← #14538885

Subgroups — #14538885

Order 2176.

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

normal subgroups only

This group has 444 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#11763364C211yesmaximal, characteristic, complemented
2C4×D4×D17C211yesmaximal, characteristic
3#11763146C212yesmaximal, complemented
4#11763086C211yesmaximal, characteristic, complemented
5#11763015C211yesmaximal, characteristic, complemented
6#11763014C211yesmaximal, characteristic, complemented
7#11762961C212yesmaximal
8#11762908C212yesmaximal, complemented
9#11762906C212yesmaximal, complemented
10#11762865C211yesmaximal, characteristic, complemented
11C17×#474C211yesmaximal, characteristic, complemented
12C2×D4×D17C2211yescharacteristic
13C22×C4×D17—21no—
14C17×C4⋊Q8C2211yescharacteristic, complemented
15C4×D4×C17C2211yescharacteristic
16#10587632—21no—
17Q8×Dic17—21no—
18D4×Dic17C2211yescharacteristic
19C2×#63229C2212yes—
20C4×C22⋊D17—21no—
21C2×#63219C2212yes—
22#10587583C2212yes—
23#10587579C2211yescharacteristic
24#10587578—21no—
25C4⋊C4×D17C2211yescharacteristic
26#10587576—21no—
27#10587573C2212yescomplemented
28#10587567—21no—
29C22⋊C4×D17—21no—
30C4×D68C2211yescharacteristic
31C42×D17C2211yescharacteristic
32#10587555C2211yescharacteristic, complemented
33C17×#107C2212yescomplemented
34C17×C4⋊C8C2211yescharacteristic, complemented
35C17×#102C2212yes—
36#10587491C2212yes—
37#10587487C2212yescomplemented
38#10587486C2212yescomplemented
39#10587481C2212yescomplemented
40#10587480C2212yes—
41#10587478C2212yescomplemented
42C4⋊#3622C2211yescharacteristic, complemented
43C23×D17—21no—
44C2×Q8×C17—21no—
45C2×D4×C17C2311yescharacteristic
46C22×C4×C17—21no—
47C2×C22⋊D17—21no—
48C22×Dic17—21no—
49D4×D17—22no—
50D4×D17—22no—
51C2×D68C2311yescharacteristic
52C2×C4×D17—21no—
53C2×C4×D17—21no—
54C2×C4×D17—21no—
55C2×C4×D17—44no—
56C2×C4×D17D411yescharacteristic
57C2×C4×D17D411yescharacteristic
58C2×C4×D17C2311yescharacteristic
59C2×Dic34C2312yes—
60C2×C8×C17C2312yes—
61C4⋊C4×C17—21no—
62C4⋊C4×C17C2312yes—
63C4⋊C4×C17C2311yescharacteristic
64C22⋊C4×C17—21no—
65C42×C17C2311yescharacteristic
66C22⋊Dic17—21no—
67#63229—24no—
68#63227—22no—
69#63227—21no—
70#63227—21no—
71C4⋊Dic17—21no—
72C4⋊Dic17—21no—
73C4⋊Dic17—22no—
74C4⋊Dic17C2311yescharacteristic
75#63225—21no—
76#63225—22no—
77C4×Dic17—22no—
78C4×Dic17D411yescharacteristic
79C4×Dic17D411yescharacteristic
80C4×Dic17C2311yescharacteristic
81C2×#3622C2312yes—
82#63219—24no—
83C23×C17—21no—
84C22×D17—24no—
85C22×D17—21no—
86C22×D17—42no—
87C22×D17—42no—
88C22×D17C2×D411yescharacteristic
89Q8×C17—42no—
90D4×C17—21no—
91D4×C17#4112yes—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—42no—
95C2×C4×C17C2×D411yescharacteristic
96C2×C4×C17C2×D411yescharacteristic
97C2×C4×C17C2411yescharacteristic
98C22⋊D17—24no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—42no—
102C2×Dic17C2×D411yescharacteristic
103C2×Dic17C2×D411yescharacteristic
104C2×Dic17C2×D411yescharacteristic
105D68—21no—
106D68#4112yes—
107C4×D17—22no—
108C4×D17—22no—
109C4×D17—21no—
110C4×D17—21no—
111C4×D17—42no—
112C4×D17—42no—
113C4×D17—42no—
114Dic34—22no—
115Dic34—22no—
116C8×C17—22no—
117#3622—22no—
118#3110—171nomaximal, Sylow
119C22×C17—22no—
120C22×C17—41no—
121C22×C17—41no—
122C22×C17C22×D411yescharacteristic
123C2×D17—21no—
124C2×D17—21no—
125C2×D17—22no—
126C2×D17—48no—
127C2×D17—41no—
128C2×D17—41no—
129C2×D17#13712yes—
130C4×C17—21no—
131C4×C17—41no—
132C4×C17—42no—
133C4×C17AGL(1,ℤ/8)12yes—
134C4×C17C2×#4111yescharacteristic
135C4×C17C2×#4111yescharacteristic
136Dic17—21no—
137Dic17—21no—
138Dic17—21no—
139Dic17—42no—
140Dic17—41no—
141#533—171no—
142C2×C4×D4—171no—
143#484—172no—
144#474—171no—
145#474D1711yescharacteristic, complemented
146#461—171no—
147#461—171no—
148#450—172no—
149#440—172no—
150#440—172no—
151#432—171no—
152C2×C17—42no—
153C2×C17#52511yescharacteristic
154C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C17C2×#13711yescharacteristic
156D17—22no—
157D17—42no—
158C22×D4—171no—
159C23×C4—341no—
160C2×SD16—172no—
161C2×SD16—172no—
162C4⋊Q8—171no—
163C4⋊Q8C2×D1711yescharacteristic, complemented
164#125—341no—
165C22⋊Q8—172no—
166C22⋊Q8—341no—
167C4×Q8—172no—
168C4×Q8—341no—
169C4×D4—171no—
170C4×D4—171no—
171C4×D4—171no—
172C4×D4—341no—
173C4×D4—341no—
174C4×D4C2×D1711yescharacteristic
175#117—341no—
176C2×C4⋊C4—171no—
177C2×C22⋊C4—341no—
178C2×C42—171no—
179#107—172no—
180#107C2×D1712yescomplemented
181C4⋊C8—171no—
182C4⋊C8C2×D1711yescharacteristic, complemented
183#103—172no—
184#103—172no—
185#102—172no—
186#102C2×D1712yes—
187C22⋊C8—172no—
188#97—172no—
189C17#311011yescharacteristic, complemented, Sylow
190C24—341no—
191C2×Q8—21no—
192C2×Q8—172no—
193C2×D4—171no—
194C2×D4—341no—
195C2×D4—342no—
196C2×D4—342no—
197C2×D4C22×D1711yescharacteristic
198C22×C4—21no—
199C22×C4—171no—
200C22×C4—171no—
201C22×C4—171no—
202C22×C4—341no—
203C22×C4—341no—
204C22×C4—341no—
205C22×C4—341no—
206C22×C4—684no—
207SD16—344no—
208SD16—344no—
209C2×C8—172no—
210C2×C8C22×D1712yes—
211C4⋊C4—21no—
212C4⋊C4—171no—
213C4⋊C4—341no—
214C4⋊C4—342no—
215C4⋊C4—342no—
216C4⋊C4—341no—
217C4⋊C4—341no—
218C4⋊C4C22×D1712yes—
219C4⋊C4C22×D1711yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—341no—
222C22⋊C4—342no—
223C22⋊C4—341no—
224C22⋊C4—341no—
225C42—171no—
226C42—171no—
227C42—171no—
228C42—342no—
229C42C22×D1711yescharacteristic
230C23—21no—
231C23—171no—
232C23—344no—
233C23—341no—
234C23—682no—
235C23—682no—
236Q8—42no—
237Q8—342no—
238Q8—342no—
239D4—21no—
240D4—172no—
241D4—344no—
242D4—341no—
243D4#6325612yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—171no—
248C2×C4—171no—
249C2×C4—171no—
250C2×C4—342no—
251C2×C4—341no—
252C2×C4—341no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—341no—
256C2×C4—682no—
257C2×C4—682no—
258C2×C4—682no—
259C2×C4—682no—
260C2×C4D4×D1711yescharacteristic
261C2×C4D4×D1711yescharacteristic
262C2×C4C23×D1711yescharacteristic
263C8—22no—
264C8—342no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—172no—
269C22—341no—
270C22—341no—
271C22—342no—
272C22—688no—
273C22—681no—
274C22—681no—
275C22C2×D4×D1711yescharacteristic
276C4—21no—
277C4—41no—
278C4—42no—
279C4—341no—
280C4—341no—
281C4—341no—
282C4—682no—
283C4—681no—
284C4#1058759712yes—
285C4C2×#6325611yescharacteristic
286C4#41×D1711yescharacteristic
287C2—42no—
288C2—342no—
289C2—682no—
290C2#1176332211yescharacteristic
291C2C2×#1058759711yescharacteristic
292C2#137×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#474C2
socleC22×C17C22×D4
prime radicalC2×D4×D17C22
hypercenter#474D17
hypocenterC17#3110
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #11763364C2semidirect
3. #11763146C2semidirect
4. #11763086C2semidirect
5. #11763015C2semidirect
6. #11763014C2semidirect
8. #11762908C2semidirect
9. #11762906C2semidirect
10. #11762865C2semidirect
11. C17×#474C2semidirect
14. C17×C4⋊Q8C22semidirect
27. #10587573C22semidirect
32. #10587555C22semidirect
33. C17×#107C22semidirect
34. C17×C4⋊C8C22semidirect
37. #10587487C22semidirect
38. #10587486C22semidirect
39. #10587481C22semidirect
41. #10587478C22semidirect
42. C4⋊#3622C22semidirect
145. #474D17semidirect
163. C4⋊Q8C2×D17semidirect
180. #107C2×D17semidirect
182. C4⋊C8C2×D17semidirect
189. C17#3110semidirect

Maximal subgroups

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

subgroupindex
#117633642
C4×D4×D172
#117631462
#117630862
#117630152
#117630142
#117629612
#117629082
#117629062
#117628652
C17×#4742
#311017