Finite Group Databasebeta

← #14540403

Subgroups — #14540403

Order 2176.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587597C212yesmaximal, complemented
2#11763479C211yesmaximal, characteristic, complemented
3#11763444C211yesmaximal, characteristic, complemented
4#11763059C212yesmaximal, complemented
5#11763006C211yesmaximal, characteristic, complemented
6#11763001C211yesmaximal, characteristic, complemented
7#11762951C212yesmaximal, complemented
8#11762941C212yesmaximal, complemented
9C4×#63219C211yesmaximal, characteristic, complemented
10#11762888C211yesmaximal, characteristic, complemented
11C17×#493C211yesmaximal, characteristic, complemented
12C2×#63254C2212yescomplemented
13C2×D4×D17—21no—
14C2×D4×D17C2212yes—
15C2×D8×C17C2212yescomplemented
16C2×D8×C17C2212yescomplemented
17C17×#127C2211yescharacteristic
18C17×#127C2211yescharacteristic, complemented
19#10587626—21no—
20#10587625—22no—
21#10587624—21no—
22#10587624C2212yes—
23#10587624C2212yescomplemented
24#10587623—22no—
25D4×Dic17—21no—
26C2×#63229C2212yescomplemented
27C2×#63228C2212yes—
28#10587597—28no—
29C2×#63219C2211yescharacteristic
30C2×#63218C2212yescomplemented
31C4×D68C2211yescharacteristic
32C4×D68C2211yescharacteristic
33C4×Dic34C2211yescharacteristic, complemented
34C4×C8×C17C2211yescharacteristic, complemented
35#10587503C2212yescomplemented
36#10587503C2212yescomplemented
37#10587492C2211yescharacteristic
38#10587491C2212yescomplemented
39#10587488C2211yescharacteristic, complemented
40#10587487C2211yescharacteristic, complemented
41C4⋊#3622C2211yescharacteristic, complemented
42C23×D17—22no—
43C2×D4×C17—21no—
44C2×D4×C17—21no—
45C2×D4×C17—21no—
46C2×D4×C17—21no—
47C2×D4×C17C2312yes—
48C2×D4×C17C2312yes—
49C2×C22⋊D17—22no—
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C22×Dic17—22no—
54#63254—24no—
55#63254—42no—
56D4×D17—24no—
57D4×D17—42no—
58D4×D17—44no—
59C2×D68D411yescharacteristic
60C2×D68C2311yescharacteristic
61C2×C4×D17—21no—
62C2×C4×D17C2312yes—
63C2×Dic34C2311yescharacteristic
64D8×C17—24no—
65D8×C17—42no—
66C2×C8×C17C2311yescharacteristic
67C2×C8×C17C2311yescharacteristic
68C42×C17C2311yescharacteristic
69C22⋊Dic17—22no—
70C22⋊Dic17—22no—
71#63229—24no—
72#63228—24no—
73#63227—21no—
74#63227D412yes—
75C4⋊Dic17D411yescharacteristic
76C4⋊Dic17C2311yescharacteristic
77C4⋊Dic17C2311yescharacteristic
78#63225—21no—
79C4×Dic17—21no—
80C2×#3622C2312yes—
81#63219D414yescomplemented
82#63218—24no—
83C23×C17—22no—
84C23×C17—22no—
85C22×D17—21no—
86C22×D17—44no—
87C22×D17—44no—
88C22×D17—82no—
89C22×D17C2×D412yes—
90D4×C17—22no—
91D4×C17—22no—
92D4×C17—22no—
93D4×C17—22no—
94D4×C17—42no—
95D4×C17—42no—
96D4×C17—81no—
97D4×C17—81no—
98C2×C4×C17C2×D411yescharacteristic
99C2×C4×C17C2×D411yescharacteristic
100C2×C4×C17C2411yescharacteristic
101C22⋊D17—44no—
102C22⋊D17—44no—
103C22⋊D17—44no—
104C22⋊D17—82no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—44no—
109C2×Dic17C2×D412yes—
110D68—22no—
111D68—21no—
112D68C2×D412yes—
113C4×D17—22no—
114C4×D17—22no—
115C4×D17—42no—
116Dic34—21no—
117Dic34C2×D412yescomplemented
118C8×C17—21no—
119C8×C17C2×D412yescomplemented
120#3622—22no—
121#3274—171nomaximal, Sylow
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17C22×D411yescharacteristic
129C2×D17—22no—
130C2×D17—22no—
131C2×D17—22no—
132C2×D17—22no—
133C2×D17—41no—
134C2×D17—41no—
135C2×D17—88no—
136C4×C17—21no—
137C4×C17AGL(1,ℤ/8)12yescomplemented
138C4×C17C22×D411yescharacteristic
139C4×C17H(5,𝔽2)11yescharacteristic
140Dic17—22no—
141Dic17—22no—
142Dic17—41no—
143Dic17—41no—
144C2×AGL(1,ℤ/8)—172no—
145#550—171no—
146D42—171no—
147#493D1711yescharacteristic, complemented
148#469—172no—
149#460—171no—
150#459—171no—
151#449—172no—
152#447—172no—
153C4×SD16—171no—
154#436—171no—
155C2×C17—82no—
156C2×C17—82no—
157C2×C17D4211yescharacteristic
158C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C17#57611yescharacteristic
160D17—42no—
161D17—42no—
162C2×#41—172no—
163C22×D4—172no—
164C22×D4—341no—
165AGL(1,ℤ/8)—348no—
166C2×SD16—172no—
167C2×SD16—171no—
168C2×D8—172no—
169C2×D8C2×D1712yescomplemented
170C2×D8C2×D1712yescomplemented
171C2×M16—172no—
172#127—341no—
173#127C2×D1711yescharacteristic
174#127C2×D1711yescharacteristic, complemented
175#121—172no—
176#121—172no—
177#121—342no—
178#121—341no—
179#120—342no—
180C4×Q8—171no—
181C4×D4—171no—
182C4×D4—171no—
183C4×D4—341no—
184#106—171no—
185C4⋊C8—171no—
186#103—171no—
187#102—172no—
188#102—172no—
189#102—171no—
190C22⋊C8—172no—
191C4×C8C2×D1711yescharacteristic, complemented
192C17#327411yescharacteristic, complemented, Sylow
193C24—342no—
194#41—344no—
195#41—682no—
196C2×Q8—171no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—171no—
202C2×D4—171no—
203C2×D4—342no—
204C2×D4—344no—
205C2×D4—342no—
206C2×D4—342no—
207C2×D4—342no—
208C2×D4—682no—
209C2×D4—684no—
210C2×D4C22×D1712yes—
211C2×D4C22×D1712yes—
212C22×C4—172no—
213C22×C4—342no—
214C22×C4—341no—
215SD16—174no—
216SD16—344no—
217D8—24no—
218D8—42no—
219D8—344no—
220M16—344no—
221C2×C8—172no—
222C2×C8C22×D1711yescharacteristic
223C2×C8C22×D1711yescharacteristic
224C4⋊C4—171no—
225C4⋊C4—171no—
226C4⋊C4—171no—
227C4⋊C4—341no—
228C22⋊C4—172no—
229C22⋊C4—342no—
230C22⋊C4—342no—
231C22⋊C4—341no—
232C42—341no—
233C42C22×D1711yescharacteristic
234C23—22no—
235C23—22no—
236C23—172no—
237C23—341no—
238C23—684no—
239C23—684no—
240C23—1362no—
241Q8—172no—
242Q8—341no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—172no—
252D4—341no—
253D4—342no—
254D4—684no—
255D4—684no—
256D4—684no—
257D4—1362no—
258C2×C4—172no—
259C2×C4—341no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—341no—
263C2×C4—341no—
264C2×C4—684no—
265C2×C4—682no—
266C2×C4D4×D1711yescharacteristic
267C2×C4D4×D1711yescharacteristic
268C2×C4C23×D1711yescharacteristic
269C8—21no—
270C8—342no—
271C8D4×D1712yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—342no—
279C22—342no—
280C22—342no—
281C22—342no—
282C22—681no—
283C22—681no—
284C22—1368no—
285C22C2×D4×D1711yescharacteristic
286C4—21no—
287C4—342no—
288C4—342no—
289C4—681no—
290C4—681no—
291C4#1058759712yescomplemented
292C4C2×D4×D1711yescharacteristic
293C4#1058769111yescharacteristic
294C2—82no—
295C2—82no—
296C2—682no—
297C2—682no—
298C2#1176344411yescharacteristic
299C2C2×#1058759711yescharacteristic
300C2#1176361011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#493C2
socleC22×C17C22×D4
hypercenter#493D17
hypocenterC17#3274
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#10587597C2semidirect
2. #11763479C2semidirect
3. #11763444C2semidirect
4. #11763059C2semidirect
5. #11763006C2semidirect
6. #11763001C2semidirect
7. #11762951C2semidirect
8. #11762941C2semidirect
9. C4×#63219C2semidirect
10. #11762888C2semidirect
11. C17×#493C2semidirect
12. C2×#63254C22semidirect
15. C2×D8×C17C22semidirect
16. C2×D8×C17C22semidirect
18. C17×#127C22semidirect
23. #10587624C22semidirect
26. C2×#63229C22semidirect
30. C2×#63218C22semidirect
33. C4×Dic34C22semidirect
34. C4×C8×C17C22semidirect
35. #10587503C22semidirect
36. #10587503C22semidirect
38. #10587491C22semidirect
39. #10587488C22semidirect
40. #10587487C22semidirect
41. C4⋊#3622C22semidirect
81. #63219D4semidirect
117. Dic34C2×D4semidirect
119. C8×C17C2×D4semidirect
137. C4×C17AGL(1,ℤ/8)semidirect
147. #493D17semidirect
169. C2×D8C2×D17semidirect
170. C2×D8C2×D17semidirect
174. #127C2×D17semidirect
191. C4×C8C2×D17semidirect
192. C17#3274semidirect
271. C8D4×D17semidirect
291. C4#10587597semidirect

Maximal subgroups

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

subgroupindex
C2×#105875972
#117634792
#117634442
#117630592
#117630062
#117630012
#117629512
#117629412
C4×#632192
#117628882
C17×#4932
#327417