Finite Group Databasebeta

← #14538261

Subgroups — #14538261

Order 2176.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587600C212yesmaximal, complemented
2#127×D17C211yesmaximal, characteristic, complemented
3C4×#63256C211yesmaximal, characteristic, complemented
4#11763009C211yesmaximal, characteristic, complemented
5#11763007C212yesmaximal, complemented
6#11763005C211yesmaximal, characteristic, complemented
7#11763003C212yesmaximal, complemented
8#11763001C211yesmaximal, characteristic, complemented
9#11762831C211yesmaximal, characteristic, complemented
10#11762812C212yesmaximal, complemented
11C17×#460C211yesmaximal, characteristic, complemented
12C2×#63256C2211yescharacteristic
13C2×D4×D17—21no—
14C2×D4×D17—21no—
15C2×D4×D17C2212yes—
16C2×SD16×C17C2212yescomplemented
17C17×#127C2211yescharacteristic
18C4×Q8×C17C2211yescharacteristic, complemented
19Q8×Dic17C2211yescharacteristic, complemented
20C2×#63230C2212yescomplemented
21#10587626—21no—
22#10587626—21no—
23#10587626C2212yes—
24C2×#63228C2212yescomplemented
25#10587600—28no—
26C2×D136C2212yescomplemented
27C2×#63218C2212yes—
28#10587579—21no—
29#10587579C2211yescharacteristic, complemented
30#10587578—21no—
31#10587578C2211yescharacteristic
32#10587560C2211yescharacteristic
33C4×D68—21no—
34C4×D68C2211yescharacteristic, complemented
35C42×D17—21no—
36C42×D17C2211yescharacteristic
37C17×C4⋊C8C2211yescharacteristic, complemented
38C17×#102C2212yescomplemented
39#10587503C2212yescomplemented
40#10587492C2212yescomplemented
41#10587485C2212yescomplemented
42#10587480C2212yescomplemented
43C4⋊#3622C2211yescharacteristic, complemented
44C23×D17—22no—
45C2×Q8×C17C2311yescharacteristic
46C2×D4×C17—21no—
47C2×D4×C17—21no—
48C2×D4×C17C2312yes—
49C2×C22⋊D17—22no—
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52#63256—22no—
53#63256D414yescomplemented
54D4×D17—24no—
55D4×D17—24no—
56D4×D17—44no—
57D4×D17—44no—
58C2×D68—21no—
59C2×D68—21no—
60C2×D68—21no—
61C2×D68C2312yes—
62C2×D68C2311yescharacteristic
63C2×C4×D17—21no—
64C2×C4×D17—21no—
65C2×C4×D17—21no—
66C2×C4×D17D411yescharacteristic
67C2×C4×D17D411yescharacteristic
68C2×C4×D17C2311yescharacteristic
69SD16×C17—24no—
70C2×C8×C17C2312yes—
71C4⋊C4×C17—21no—
72C4⋊C4×C17C2311yescharacteristic
73C42×C17—21no—
74C42×C17C2311yescharacteristic
75#63230—24no—
76#63228—24no—
77#63227—21no—
78#63227—21no—
79#63227—21no—
80C4⋊Dic17—21no—
81C4⋊Dic17C2311yescharacteristic
82C4×Dic17—21no—
83C4×Dic17—21no—
84C4×Dic17D411yescharacteristic
85C4×Dic17D411yescharacteristic
86C4×Dic17C2311yescharacteristic
87C2×#3622C2312yes—
88D136—24no—
89#63218—24no—
90C23×C17—22no—
91C22×D17—21no—
92C22×D17—22no—
93C22×D17—44no—
94C22×D17—44no—
95C22×D17—44no—
96C22×D17C2×D411yescharacteristic
97Q8×C17—21no—
98Q8×C17C2×D412yescomplemented
99D4×C17—22no—
100D4×C17—22no—
101D4×C17—42no—
102D4×C17—42no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17C2×D411yescharacteristic
106C2×C4×C17C2×D411yescharacteristic
107C2×C4×C17C2411yescharacteristic
108C22⋊D17—44no—
109C22⋊D17—44no—
110C22⋊D17—44no—
111C2×Dic17—21no—
112C2×Dic17C2×D411yescharacteristic
113C2×Dic17C2×D411yescharacteristic
114C2×Dic17C2×D411yescharacteristic
115D68—24no—
116D68—22no—
117D68—21no—
118D68—22no—
119D68—42no—
120D68—42no—
121D68C2×D412yescomplemented
122C4×D17—22no—
123C4×D17—24no—
124C4×D17—22no—
125C4×D17—42no—
126C4×D17—42no—
127C4×D17C2×D412yes—
128C4×D17#4112yes—
129C8×C17—22no—
130#3622—22no—
131#3038—171nomaximal, Sylow
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17C22×D411yescharacteristic
136C2×D17—21no—
137C2×D17—21no—
138C2×D17—22no—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17—42no—
142C2×D17—42no—
143C2×D17—42no—
144C2×D17—88no—
145C2×D17#12112yes—
146C4×C17—22no—
147C4×C17—21no—
148C4×C17—41no—
149C4×C17AGL(1,ℤ/8)12yescomplemented
150C4×C17C22×D411yescharacteristic
151C4×C17C2×#4111yescharacteristic
152Dic17—21no—
153Dic17—41no—
154Dic17#12112yes—
155Dic17AGL(1,ℤ/8)12yescomplemented
156C2×AGL(1,ℤ/8)—172no—
157C2×#127—171no—
158C4×#41—171no—
159#460—172no—
160#460—171no—
161#460D1711yescharacteristic, complemented
162#459—171no—
163#459—171no—
164#459—172no—
165#423—171no—
166#418—172no—
167C2×C17—82no—
168C2×C17C2×#12111yescharacteristic
169C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
171D17—22no—
172D17—42no—
173D17—82no—
174C2×#41—171no—
175C22×D4—172no—
176C22×D4—341no—
177C22×D4—341no—
178AGL(1,ℤ/8)—348no—
179C2×SD16—172no—
180C2×SD16C2×D1712yescomplemented
181C2×D8—172no—
182C2×D8—172no—
183C2×M16—172no—
184#127—172no—
185#127—171no—
186#127—341no—
187#127—341no—
188#127C2×D1711yescharacteristic
189C4×Q8—171no—
190C4×Q8C2×D1711yescharacteristic, complemented
191C4×D4—171no—
192C4×D4—171no—
193C4×D4—341no—
194C4×D4—341no—
195#117—171no—
196#117—341no—
197C2×C42—171no—
198C2×C42—341no—
199C4⋊C8—172no—
200C4⋊C8—171no—
201C4⋊C8C2×D1711yescharacteristic, complemented
202#102—172no—
203#102—172no—
204#102—172no—
205#102C2×D1712yescomplemented
206C17#303811yescharacteristic, complemented, Sylow
207C24—342no—
208#41—174no—
209#41—342no—
210C2×Q8C22×D1711yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—171no—
214C2×D4—172no—
215C2×D4—341no—
216C2×D4—344no—
217C2×D4—344no—
218C2×D4—342no—
219C2×D4—342no—
220C2×D4—342no—
221C2×D4—341no—
222C2×D4—341no—
223C2×D4—684no—
224C2×D4—684no—
225C2×D4C22×D1712yes—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—341no—
232SD16—24no—
233SD16—344no—
234D8—344no—
235D8—344no—
236M16—344no—
237C2×C8—172no—
238C2×C8C22×D1712yes—
239C4⋊C4—21no—
240C4⋊C4—171no—
241C4⋊C4—341no—
242C4⋊C4C22×D1711yescharacteristic
243C22⋊C4—341no—
244C22⋊C4—341no—
245C22⋊C4—341no—
246C42—21no—
247C42—171no—
248C42—171no—
249C42—171no—
250C42—341no—
251C42—341no—
252C42C22×D1711yescharacteristic
253C23—22no—
254C23—171no—
255C23—341no—
256C23—342no—
257C23—684no—
258C23—684no—
259C23—684no—
260Q8—21no—
261Q8D4×D1712yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—172no—
267D4—344no—
268D4—342no—
269D4—341no—
270D4—342no—
271D4—682no—
272D4—684no—
273D4—684no—
274D4—684no—
275D4—682no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—171no—
279C2×C4—171no—
280C2×C4—172no—
281C2×C4—172no—
282C2×C4—171no—
283C2×C4—342no—
284C2×C4—344no—
285C2×C4—342no—
286C2×C4—341no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4D4×D1711yescharacteristic
290C2×C4D4×D1711yescharacteristic
291C2×C4C23×D1711yescharacteristic
292C8—22no—
293C8—342no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—172no—
298C22—341no—
299C22—341no—
300C22—342no—
301C22—681no—
302C22—681no—
303C22—682no—
304C22—682no—
305C22—682no—
306C22—1368no—
307C22C2×D4×D1711yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—172no—
312C4—172no—
313C4—341no—
314C4—681no—
315C4#1058760012yescomplemented
316C4C2×D4×D1711yescharacteristic
317C4#41×D1711yescharacteristic
318C2—82no—
319C2—342no—
320C2—682no—
321C2—1362no—
322C2#121×D1711yescharacteristic
323C2AGL(1,ℤ/8)×D1711yescharacteristic
324C2C2×#1058760011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#460C2
socleC22×C17C22×D4
hypercenter#460D17
hypocenterC17#3038
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#10587600C2semidirect
2. #127×D17C2semidirect
3. C4×#63256C2semidirect
4. #11763009C2semidirect
5. #11763007C2semidirect
6. #11763005C2semidirect
7. #11763003C2semidirect
8. #11763001C2semidirect
9. #11762831C2semidirect
10. #11762812C2semidirect
11. C17×#460C2semidirect
16. C2×SD16×C17C22semidirect
18. C4×Q8×C17C22semidirect
19. Q8×Dic17C22semidirect
20. C2×#63230C22semidirect
24. C2×#63228C22semidirect
26. C2×D136C22semidirect
29. #10587579C22semidirect
34. C4×D68C22semidirect
37. C17×C4⋊C8C22semidirect
38. C17×#102C22semidirect
39. #10587503C22semidirect
40. #10587492C22semidirect
41. #10587485C22semidirect
42. #10587480C22semidirect
43. C4⋊#3622C22semidirect
53. #63256D4semidirect
98. Q8×C17C2×D4semidirect
121. D68C2×D4semidirect
149. C4×C17AGL(1,ℤ/8)semidirect
155. Dic17AGL(1,ℤ/8)semidirect
161. #460D17semidirect
180. C2×SD16C2×D17semidirect
190. C4×Q8C2×D17semidirect
201. C4⋊C8C2×D17semidirect
205. #102C2×D17semidirect
206. C17#3038semidirect
261. Q8D4×D17semidirect
315. C4#10587600semidirect

Maximal subgroups

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

subgroupindex
C2×#105876002
#127×D172
C4×#632562
#117630092
#117630072
#117630052
#117630032
#117630012
#117628312
#117628122
C17×#4602
#303817