Finite Group Databasebeta

← #14538087

Subgroups — #14538087

Order 2176.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587691C211yesmaximal, characteristic, complemented
2C2×#10587636C211yesmaximal, characteristic, complemented
3#11763387C211yesmaximal, characteristic, complemented
4#11762968C212yesmaximal, complemented
5#11762958C212yesmaximal, complemented
6#11762951C212yesmaximal, complemented
7#11762941C212yesmaximal, complemented
8#11762811C211yesmaximal, characteristic, complemented
9C2×#10587503C211yesmaximal, characteristic, complemented
10#11762777C211yesmaximal, characteristic, complemented
11C2×C17×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic, complemented
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17—22no—
15#10587691—24no—
16#10587691—24no—
17C2×#63254—21no—
18C2×#63254C2212yes—
19C2×D4×D17—21no—
20C2×D4×D17C2212yes—
21C2×#63252C2211yescharacteristic
22C2×#63252C2211yescharacteristic
23C17×AGL(1,ℤ/8)—24no—
24C2×SD16×C17C2212yescomplemented
25C2×D8×C17C2212yescomplemented
26C2×M16×C17C2211yescharacteristic, complemented
27#10587636—24no—
28#10587632C2212yescomplemented
29C2×#63231C2211yescharacteristic, complemented
30C2×#63230C2211yescharacteristic, complemented
31#10587625—22no—
32#10587624C2212yescomplemented
33C2×#63229C2211yescharacteristic, complemented
34C2×#63228C2211yescharacteristic, complemented
35#10587616—21no—
36C2×C4⋊Dic17C2211yescharacteristic, complemented
37#10587610—21no—
38C22×#3622C2211yescharacteristic, complemented
39#10587503C2212yescomplemented
40#10587503C2212yescomplemented
41#10587492C2212yescomplemented
42#10587491C2212yescomplemented
43#10587491C2212yescomplemented
44#10587487C2212yescomplemented
45C24×C17—21no—
46C23×D17—22no—
47#41×C17—22no—
48#41×C17—22no—
49C2×Q8×C17C2311yescharacteristic
50C2×D4×C17—21no—
51C2×D4×C17—21no—
52C2×D4×C17—22no—
53C2×D4×C17D414yescomplemented
54C2×D4×C17C2311yescharacteristic
55C2×D4×C17C2312yes—
56C22×C4×C17—21no—
57C22×C4×C17C2311yescharacteristic
58C2×C22⋊D17—216no—
59C2×C22⋊D17—21no—
60C2×C22⋊D17—22no—
61C2×C22⋊D17—22no—
62C2×C22⋊D17—22no—
63C2×C22⋊D17D412yes—
64C22×Dic17—22no—
65C22×Dic17—21no—
66#63254—28no—
67#63254—24no—
68#63254—24no—
69D4×D17—28no—
70D4×D17—24no—
71D4×D17—24no—
72#63252—22no—
73#63252—22no—
74#63252—22no—
75#63252D414yescomplemented
76C2×D68D411yescharacteristic
77C2×D68C2311yescharacteristic
78C2×C4×D17—21no—
79C2×C4×D17C2312yes—
80C2×Dic34D411yescharacteristic
81C2×Dic34C2311yescharacteristic
82SD16×C17—24no—
83D8×C17—24no—
84M16×C17—22no—
85C2×C8×C17C2312yes—
86C22⋊Dic17—22no—
87#63231—22no—
88#63230—22no—
89#63229—22no—
90#63228—22no—
91#63227—22no—
92C4⋊Dic17—21no—
93C4⋊Dic17C2312yes—
94#63225—22no—
95C2×#3622—22no—
96C2×#3622C2311yescharacteristic
97C2×#3622C2311yescharacteristic
98C23×C17—22no—
99C23×C17—24no—
100C23×C17—21no—
101C23×C17—81no—
102C23×C17—81no—
103C23×C17C2×D411yescharacteristic
104C22×D17—21no—
105C22×D17—28no—
106C22×D17—82no—
107C22×D17—82no—
108C22×D17C2×D412yes—
109Q8×C17—21no—
110Q8×C17—21no—
111D4×C17—24no—
112D4×C17—22no—
113D4×C17—21no—
114D4×C17—21no—
115D4×C17—42no—
116D4×C17C2×D414yes—
117C2×C4×C17—21no—
118C2×C4×C17—42no—
119C2×C4×C17C2×D411yescharacteristic
120C2×C4×C17C2×D412yes—
121C2×C4×C17C2×D412yes—
122C2×C4×C17C2411yescharacteristic
123C22⋊D17—28no—
124C22⋊D17—28no—
125C22⋊D17—24no—
126C22⋊D17—24no—
127C22⋊D17—82no—
128C2×Dic17—28no—
129C2×Dic17—21no—
130C2×Dic17—22no—
131C2×Dic17—81no—
132C2×Dic17C2×D412yes—
133D68—22no—
134D68—21no—
135D68C2×D412yes—
136C4×D17—24no—
137C4×D17—22no—
138C4×D17—22no—
139Dic34—22no—
140Dic34—21no—
141Dic34C2×D412yes—
142C8×C17—22no—
143#3622—21no—
144#3622—21no—
145#3012—171nomaximal, Sylow
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—24no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—84no—
155C22×C17#12012yes—
156C22×C17C22×D411yescharacteristic
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—22no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—88no—
164C4×C17—41no—
165C4×C17#12012yes—
166C4×C17C22×D411yescharacteristic
167C4×C17C22×D411yescharacteristic
168Dic17—22no—
169Dic17—22no—
170Dic17—42no—
171C2×H(5,𝔽2)—171no—
172C2×AGL(1,ℤ/8)D1711yescharacteristic, complemented
173C2×#135—171no—
174#537—171no—
175#452—172no—
176#450—172no—
177#449—172no—
178#447—172no—
179#417—171no—
180C2×#102—171no—
181#408—171no—
182C2×C17—22no—
183C2×C17—44no—
184C2×C17—81no—
185C2×C17C2×#12011yescharacteristic
186C2×C17#57611yescharacteristic
187C2×C17#57711yescharacteristic
188D17—42no—
189D17—42no—
190H(5,𝔽2)—344no—
191H(5,𝔽2)—344no—
192C2×#41—171no—
193C2×#41—172no—
194C2×#41—171no—
195C2×#41—341no—
196C2×#41C2×D1711yescharacteristic, complemented
197C22×D4—172no—
198C22×D4—341no—
199C22×D4—342no—
200C22×D4C2×D1711yescharacteristic
201AGL(1,ℤ/8)—24no—
202#135—344no—
203C2×Q16—171no—
204C2×SD16—171no—
205C2×SD16—171no—
206C2×SD16C2×D1712yescomplemented
207C2×D8—171no—
208C2×D8C2×D1712yescomplemented
209C2×M16C2×D1711yescharacteristic, complemented
210C22×C8—171no—
211C22⋊Q8—172no—
212C22⋊Q8—341no—
213#121—172no—
214#121—341no—
215#121—342no—
216C2×C4⋊C4—171no—
217#103—172no—
218#102—172no—
219#102—172no—
220#102—172no—
221C22⋊C8—172no—
222C22⋊C8—172no—
223C17#301211yescharacteristic, complemented, Sylow
224C24—21no—
225C24—342no—
226#41—22no—
227#41—22no—
228#41—174no—
229#41—344no—
230#41—348no—
231#41—344no—
232#41—342no—
233#41—342no—
234#41—342no—
235C2×Q8—171no—
236C2×Q8—171no—
237C2×Q8C22×D1711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—172no—
242C2×D4—171no—
243C2×D4—171no—
244C2×D4—342no—
245C2×D4—344no—
246C2×D4—348no—
247C2×D4—342no—
248C2×D4—3416no—
249C2×D4—341no—
250C2×D4—344no—
251C2×D4—342no—
252C2×D4C22⋊D1714yescomplemented
253C2×D4C22×D1711yescharacteristic
254C2×D4C22×D1712yes—
255C22×C4—21no—
256C22×C4—172no—
257C22×C4—342no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4C22×D1711yescharacteristic
261Q16—342no—
262SD16—24no—
263SD16—342no—
264SD16—342no—
265D8—24no—
266D8—342no—
267M16—22no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8—342no—
271C2×C8C22×D1712yes—
272C4⋊C4—172no—
273C4⋊C4—342no—
274C4⋊C4—341no—
275C22⋊C4—342no—
276C22⋊C4—342no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—172no—
283C23—348no—
284C23—341no—
285C23—1362no—
286C23—1362no—
287C23D4×D1711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—172no—
291Q8—341no—
292Q8—342no—
293D4—22no—
294D4—21no—
295D4—21no—
296D4—24no—
297D4—42no—
298D4—172no—
299D4—348no—
300D4—348no—
301D4—344no—
302D4—344no—
303D4—341no—
304D4—342no—
305D4—1362no—
306D4C2×C22⋊D1714yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—172no—
310C2×C4—348no—
311C2×C4—341no—
312C2×C4—342no—
313C2×C4—342no—
314C2×C4—344no—
315C2×C4—342no—
316C2×C4—1361no—
317C2×C4D4×D1712yes—
318C2×C4D4×D1711yescharacteristic
319C2×C4C2×C22⋊D1712yes—
320C2×C4C23×D1711yescharacteristic
321C8—22no—
322C8—341no—
323C8—341no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—342no—
334C22—342no—
335C22—342no—
336C22—342no—
337C22—681no—
338C22—681no—
339C22—1368no—
340C22#1058762312yes—
341C22C2×D4×D1711yescharacteristic
342C4—41no—
343C4—342no—
344C4—342no—
345C4—682no—
346C4#1058762312yes—
347C4C2×D4×D1711yescharacteristic
348C4C22×C22⋊D1711yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—682no—
353C2—682no—
354C2C2×#1058762311yescharacteristic
355C2#1176361311yescharacteristic
356C2#1176361911yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×AGL(1,ℤ/8)C2
socleC22×C17C22×D4
hypercenterC2×AGL(1,ℤ/8)D17
hypocenterC17#3012
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#10587691C2semidirect
2. C2×#10587636C2semidirect
3. #11763387C2semidirect
4. #11762968C2semidirect
5. #11762958C2semidirect
6. #11762951C2semidirect
7. #11762941C2semidirect
8. #11762811C2semidirect
9. C2×#10587503C2semidirect
10. #11762777C2semidirect
11. C2×C17×AGL(1,ℤ/8)C2semidirect
12. C2×#41×C17C22semidirect
24. C2×SD16×C17C22semidirect
25. C2×D8×C17C22semidirect
26. C2×M16×C17C22semidirect
28. #10587632C22semidirect
29. C2×#63231C22semidirect
30. C2×#63230C22semidirect
32. #10587624C22semidirect
33. C2×#63229C22semidirect
34. C2×#63228C22semidirect
36. C2×C4⋊Dic17C22semidirect
38. C22×#3622C22semidirect
39. #10587503C22semidirect
40. #10587503C22semidirect
41. #10587492C22semidirect
42. #10587491C22semidirect
43. #10587491C22semidirect
44. #10587487C22semidirect
53. C2×D4×C17D4semidirect
75. #63252D4semidirect
172. C2×AGL(1,ℤ/8)D17semidirect
196. C2×#41C2×D17semidirect
206. C2×SD16C2×D17semidirect
208. C2×D8C2×D17semidirect
209. C2×M16C2×D17semidirect
223. C17#3012semidirect
252. C2×D4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876912
C2×#105876362
#117633872
#117629682
#117629582
#117629512
#117629412
#117628112
C2×#105875032
#117627772
C2×C17×AGL(1,ℤ/8)2
#301217