Finite Group Databasebeta

← #14538012

Subgroups — #14538012

Order 2176.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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×D4×D17C211yesmaximal, characteristic, complemented
2C2×#10587600C212yesmaximal, complemented
3#11763322C211yesmaximal, characteristic, complemented
4#11762959C211yesmaximal, characteristic, complemented
5#11762958C212yesmaximal, complemented
6#11762945C211yesmaximal, characteristic, complemented
7#11762944C212yesmaximal, complemented
8#11762943C211yesmaximal, characteristic, complemented
9#11762812C212yesmaximal, complemented
10#11762771C211yesmaximal, characteristic, complemented
11C17×#450C211yesmaximal, characteristic, complemented
12C24×D17—21no—
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17—21no—
15C2×#63256C2211yescharacteristic, complemented
16C2×D4×D17—28no—
17C2×D4×D17—21no—
18C2×D4×D17—21no—
19C2×D4×D17C2212yes—
20C2×D4×D17C2212yes—
21C22×D68C2211yescharacteristic
22C22×C4×D17C2211yescharacteristic
23C2×SD16×C17C2212yescomplemented
24C17×C22⋊Q8C2211yescharacteristic, complemented
25#10587632C2211yescharacteristic, complemented
26C2×#63230C2212yescomplemented
27C2×#63228C2212yescomplemented
28#10587616C2211yescharacteristic, complemented
29#10587600—28no—
30C2×D136C2212yescomplemented
31C2×#63218C2212yescomplemented
32#10587581C2211yescharacteristic, complemented
33#10587580—21no—
34#10587579—21no—
35#10587578C2211yescharacteristic, complemented
36#10587568—21no—
37#10587567—21no—
38C17×#102C2212yescomplemented
39C17×C22⋊C8C2211yescharacteristic, complemented
40#10587503C2212yescomplemented
41C22⋊#3622C2211yescharacteristic, complemented
42#10587492C2212yescomplemented
43#10587491C2212yescomplemented
44#10587480C2212yescomplemented
45C24×C17—21no—
46C23×D17—28no—
47C23×D17—21no—
48C23×D17—22no—
49C23×D17—22no—
50C23×D17—44no—
51C23×D17—44no—
52C23×D17D411yescharacteristic
53C2×Q8×C17C2311yescharacteristic, complemented
54C2×D4×C17—24no—
55C2×D4×C17—21no—
56C2×D4×C17C2312yes—
57C22×C4×C17C2311yescharacteristic
58C2×C22⋊D17—28no—
59C2×C22⋊D17—21no—
60C2×C22⋊D17—22no—
61C2×C22⋊D17—22no—
62C22×Dic17D411yescharacteristic
63#63256—24no—
64D4×D17—28no—
65D4×D17—28no—
66D4×D17—24no—
67D4×D17—24no—
68D4×D17—24no—
69D4×D17D418yescomplemented
70C2×D68—21no—
71C2×D68—24no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68C2312yes—
75C2×D68C2311yescharacteristic, complemented
76C2×C4×D17—24no—
77C2×C4×D17—21no—
78C2×C4×D17D411yescharacteristic
79C2×C4×D17D411yescharacteristic
80C2×C4×D17C2311yescharacteristic
81C2×C4×D17C2311yescharacteristic
82SD16×C17—24no—
83C2×C8×C17C2312yescomplemented
84C4⋊C4×C17—21no—
85C4⋊C4×C17C2311yescharacteristic, complemented
86C22⋊C4×C17—21no—
87#63230—24no—
88#63228—24no—
89#63227—21no—
90#63227—21no—
91#63227—21no—
92#63227—21no—
93C4⋊Dic17C2311yescharacteristic, complemented
94#63225—21no—
95C4×Dic17—21no—
96C2×#3622C2312yescomplemented
97D136—24no—
98#63218—24no—
99C23×C17—24no—
100C23×C17—22no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17C2×D411yescharacteristic
104C22×D17—22no—
105C22×D17—22no—
106C22×D17—22no—
107C22×D17—24no—
108C22×D17—28no—
109C22×D17—22no—
110C22×D17—28no—
111C22×D17—21no—
112C22×D17—432no—
113C22×D17—44no—
114C22×D17—44no—
115C22×D17—44no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—44no—
119C22×D17C2×D411yescharacteristic
120C22×D17C2×D411yescharacteristic
121Q8×C17—22no—
122D4×C17—24no—
123D4×C17—22no—
124D4×C17C2×D414yes—
125C2×C4×C17—22no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C22⋊D17—28no—
131C22⋊D17—28no—
132C22⋊D17—42no—
133C2×Dic17—22no—
134C2×Dic17—21no—
135C2×Dic17C2×D411yescharacteristic
136C2×Dic17C2×D411yescharacteristic
137D68—24no—
138D68—22no—
139D68—22no—
140D68—42no—
141D68—42no—
142D68C2×D414yes—
143C4×D17—22no—
144C4×D17—22no—
145C4×D17—22no—
146C4×D17—42no—
147C4×D17C2×D414yes—
148C8×C17—22no—
149#3622—22no—
150#3003—171nomaximal, Sylow
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—24no—
154C22×C17—48no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17AGL(1,ℤ/8)12yescomplemented
158C22×C17C22×D411yescharacteristic
159C2×D17—28no—
160C2×D17—21no—
161C2×D17—21no—
162C2×D17—21no—
163C2×D17—21no—
164C2×D17—24no—
165C2×D17—416no—
166C2×D17—416no—
167C2×D17—48no—
168C2×D17—42no—
169C2×D17—42no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17#12012yes—
174C2×D17AGL(1,ℤ/8)12yescomplemented
175C4×C17—21no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17C22×D412yes—
179Dic17—21no—
180Dic17—41no—
181Dic17#12012yes—
182C23×D4—171no—
183C2×AGL(1,ℤ/8)—172no—
184#525—171no—
185#450—172no—
186#450—171no—
187#450D1711yescharacteristic, complemented
188#447—171no—
189#447—172no—
190#447—171no—
191#418—172no—
192#407—171no—
193C2×C17—22no—
194C2×C17—44no—
195C2×C17C2×#12011yescharacteristic
196C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
198D17—22no—
199D17—22no—
200D17—44no—
201D17—81no—
202C25—341no—
203C2×#41—171no—
204C22×D4—172no—
205C22×D4—172no—
206C22×D4—171no—
207C22×D4—341no—
208C22×D4—341no—
209C22×D4—341no—
210C22×D4—348no—
211C22×D4C2×D1711yescharacteristic
212C23×C4—171no—
213AGL(1,ℤ/8)—348no—
214C2×SD16—172no—
215C2×SD16C2×D1712yescomplemented
216C2×D8—172no—
217C2×D8—172no—
218C2×M16—172no—
219#123—341no—
220C22⋊Q8—171no—
221C22⋊Q8C2×D1711yescharacteristic, complemented
222#121—171no—
223#121—171no—
224#120—341no—
225C4×D4—341no—
226C4×D4—341no—
227#117—171no—
228#102—172no—
229#102—172no—
230#102—172no—
231#102C2×D1712yescomplemented
232C22⋊C8—171no—
233C22⋊C8—172no—
234C22⋊C8C2×D1711yescharacteristic, complemented
235C17#300311yescharacteristic, complemented, Sylow
236C24—21no—
237C24—171no—
238C24—342no—
239C24—342no—
240C24—348no—
241C24—341no—
242C24—684no—
243C24—684no—
244#41—344no—
245C2×Q8C22×D1711yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—178no—
249C2×D4—172no—
250C2×D4—171no—
251C2×D4—344no—
252C2×D4—341no—
253C2×D4—341no—
254C2×D4—344no—
255C2×D4—342no—
256C2×D4—342no—
257C2×D4—344no—
258C2×D4—348no—
259C2×D4—348no—
260C2×D4—348no—
261C2×D4—344no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4C22×D1712yes—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—344no—
271C22×C4—341no—
272C22×C4C22×D1711yescharacteristic
273SD16—24no—
274SD16—344no—
275D8—344no—
276D8—344no—
277M16—344no—
278C2×C8—172no—
279C2×C8C22×D1712yescomplemented
280C4⋊C4—21no—
281C4⋊C4—171no—
282C4⋊C4—341no—
283C4⋊C4C22×D1711yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—341no—
286C22⋊C4—341no—
287C22⋊C4—341no—
288C22⋊C4—341no—
289C42—341no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—171no—
295C23—171no—
296C23—342no—
297C23—342no—
298C23—348no—
299C23—344no—
300C23—348no—
301C23—342no—
302C23—342no—
303C23—341no—
304C23—682no—
305C23—684no—
306C23—6832no—
307C23—684no—
308C23—684no—
309C23—684no—
310C23—682no—
311C23D4×D1711yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—174no—
316D4—344no—
317D4—348no—
318D4—348no—
319D4—342no—
320D4—342no—
321D4—682no—
322D4—682no—
323D4—682no—
324D4D4×D1714yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—171no—
329C2×C4—174no—
330C2×C4—171no—
331C2×C4—342no—
332C2×C4—342no—
333C2×C4—342no—
334C2×C4—342no—
335C2×C4—341no—
336C2×C4—682no—
337C2×C4D4×D1711yescharacteristic
338C2×C4C23×D1711yescharacteristic
339C8—22no—
340C8—342no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—172no—
348C22—172no—
349C22—348no—
350C22—341no—
351C22—341no—
352C22—341no—
353C22—341no—
354C22—344no—
355C22—688no—
356C22—6816no—
357C22—6816no—
358C22—682no—
359C22—682no—
360C22—681no—
361C22—681no—
362C22—681no—
363C22#1058760012yescomplemented
364C22C2×D4×D1711yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—172no—
369C4—341no—
370C4—681no—
371C4C2×D4×D1712yes—
372C2—22no—
373C2—44no—
374C2—342no—
375C2—342no—
376C2—684no—
377C2—1361no—
378C2#120×D1711yescharacteristic
379C2AGL(1,ℤ/8)×D1711yescharacteristic
380C2C2×#1058760011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#450C2
socleC22×C17C22×D4
hypercenter#450D17
hypocenterC17#3003
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D17C2semidirect
2. C2×#10587600C2semidirect
3. #11763322C2semidirect
4. #11762959C2semidirect
5. #11762958C2semidirect
6. #11762945C2semidirect
7. #11762944C2semidirect
8. #11762943C2semidirect
9. #11762812C2semidirect
10. #11762771C2semidirect
11. C17×#450C2semidirect
15. C2×#63256C22semidirect
23. C2×SD16×C17C22semidirect
24. C17×C22⋊Q8C22semidirect
25. #10587632C22semidirect
26. C2×#63230C22semidirect
27. C2×#63228C22semidirect
28. #10587616C22semidirect
30. C2×D136C22semidirect
31. C2×#63218C22semidirect
32. #10587581C22semidirect
35. #10587578C22semidirect
38. C17×#102C22semidirect
39. C17×C22⋊C8C22semidirect
40. #10587503C22semidirect
41. C22⋊#3622C22semidirect
42. #10587492C22semidirect
43. #10587491C22semidirect
44. #10587480C22semidirect
53. C2×Q8×C17C23semidirect
69. D4×D17D4semidirect
75. C2×D68C23semidirect
83. C2×C8×C17C23semidirect
85. C4⋊C4×C17C23semidirect
93. C4⋊Dic17C23semidirect
96. C2×#3622C23semidirect
157. C22×C17AGL(1,ℤ/8)semidirect
174. C2×D17AGL(1,ℤ/8)semidirect
187. #450D17semidirect
215. C2×SD16C2×D17semidirect
221. C22⋊Q8C2×D17semidirect
231. #102C2×D17semidirect
234. C22⋊C8C2×D17semidirect
235. C17#3003semidirect
245. C2×Q8C22×D17semidirect
279. C2×C8C22×D17semidirect
283. C4⋊C4C22×D17semidirect
363. C22#10587600semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D172
C2×#105876002
#117633222
#117629592
#117629582
#117629452
#117629442
#117629432
#117628122
#117627712
C17×#4502
#300317