Finite Group Databasebeta

← #14542437

Subgroups — #14542437

Order 2176.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587697C211yesmaximal, characteristic, complemented
2C2×#41×D17C211yesmaximal, characteristic, complemented
3#11763479C213yesmaximal, complemented
4#11763478C211yesmaximal, characteristic, complemented
5#11763477C213yesmaximal, complemented
6#11763471C211yesmaximal, characteristic, complemented
7#11763468C213yesmaximal, complemented
8#11763466C211yesmaximal, characteristic, complemented
9#11763465C213yesmaximal, complemented
10#11763375C213yesmaximal, complemented
11#11763372C213yesmaximal, complemented
12#11763319C213yesmaximal, complemented
13#11763318C213yesmaximal, complemented
14#11763285C211yesmaximal, characteristic, complemented
15C17×#550C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#41×C17C2211yescharacteristic, complemented
18#10587697—28no—
19#41×D17—28no—
20C2×#63256C2211yescharacteristic
21C2×Q8×D17C2211yescharacteristic
22C2×Q8×D17C2211yescharacteristic, complemented
23C2×#63254C2213yes—
24C2×#63254C2213yes—
25C2×#63254C2213yescomplemented
26C2×D4×D17C2213yes—
27C2×#63252C2213yes—
28C2×#63252C2213yes—
29C22×C4×D17C2213yes—
30C22×Dic34C2213yescomplemented
31C17×#127C2213yescomplemented
32C17×#121C2213yescomplemented
33C17×#121C2213yescomplemented
34C4×Q8×C17C2211yescharacteristic, complemented
35C4×D4×C17C2213yescomplemented
36Q8×Dic17C2211yescharacteristic, complemented
37#10587626C2213yescomplemented
38#10587626C2216yescomplemented
39#10587625C2216yescomplemented
40#10587625C2216yescomplemented
41#10587625C2213yescomplemented
42#10587624C2213yescomplemented
43#10587624C2213yescomplemented
44#10587623C2213yes—
45#10587623C2216yes—
46#10587622C2213yescomplemented
47D4×Dic17C2216yescomplemented
48#10587616C2213yescomplemented
49C4×C22⋊D17C2213yescomplemented
50C4×C22⋊D17C2213yescomplemented
51#10587610C2213yescomplemented
52#10587583C2213yescomplemented
53#10587582C2213yescomplemented
54#10587581C2213yescomplemented
55#10587580C2211yescharacteristic
56#10587578C2213yes—
57#10587573C2213yescomplemented
58#10587571C2216yescomplemented
59#10587570C2213yescomplemented
60#10587569C2213yes—
61#10587568C2213yes—
62#10587567C2216yescomplemented
63#10587564C2213yescomplemented
64#10587561C2213yescomplemented
65C4×D68C2213yescomplemented
66#10587558C2213yes—
67C4×Dic34C2213yescomplemented
68C23×D17C2313yes—
69#41×C17—24no—
70#41×C17—24no—
71C2×Q8×C17C2311yescharacteristic
72C2×D4×C17C2313yes—
73C2×D4×C17C2313yes—
74C2×D4×C17C2313yes—
75C2×D4×C17C2316yes—
76C22×C4×C17C2313yes—
77C22×C4×C17C2313yes—
78C2×C22⋊D17C2316yes—
79C2×C22⋊D17C2313yes—
80C2×C22⋊D17C2313yes—
81C2×C22⋊D17C2316yes—
82C2×C22⋊D17C2313yes—
83C2×C22⋊D17C2313yes—
84C22×Dic17C2316yes—
85C22×Dic17C2313yes—
86#63256—24no—
87Q8×D17—24no—
88Q8×D17D418yescomplemented
89#63254—212no—
90#63254—212no—
91#63254—212no—
92D4×D17—212no—
93#63252—212no—
94#63252—212no—
95C2×D68C2311yescharacteristic
96C2×D68C2313yes—
97C2×D68C2313yes—
98C2×C4×D17—212no—
99C2×C4×D17C2311yescharacteristic
100C2×C4×D17C2313yes—
101C2×C4×D17C2313yes—
102C2×C4×D17C2313yes—
103C2×C4×D17C2313yes—
104C2×Dic34—212no—
105C2×Dic34C2313yes—
106C2×Dic34C2313yes—
107C2×Dic34C2313yes—
108C4⋊C4×C17C2313yes—
109C4⋊C4×C17C2311yescharacteristic
110C22⋊C4×C17C2313yes—
111C22⋊C4×C17C2313yes—
112C42×C17C2313yes—
113C22⋊Dic17C2313yes—
114C22⋊Dic17C2313yes—
115C22⋊Dic17C2316yes—
116#63227C2311yescharacteristic
117#63227C2313yes—
118#63227C2311yescharacteristic
119#63227C2313yes—
120#63227C2316yes—
121#63227C2313yes—
122#63227C2311yescharacteristic
123C4⋊Dic17C2313yes—
124C4⋊Dic17C2313yes—
125#63225C2313yes—
126#63225C2313yes—
127#63225C2311yescharacteristic
128#63225C2313yes—
129C4×Dic17C2316yes—
130C4×Dic17C2313yes—
131C23×C17C2413yes—
132C23×C17C2413yes—
133C22×D17—26no—
134C22×D17—43no—
135C22×D17—43no—
136C22×D17C2411yescharacteristic
137C22×D17C2413yes—
138C22×D17C2411yescharacteristic
139Q8×C17C2×D414yes—
140D4×C17—26no—
141D4×C17—26no—
142D4×C17—46no—
143D4×C17—43no—
144C2×C4×C17—26no—
145C2×C4×C17—26no—
146C2×C4×C17C2411yescharacteristic
147C2×C4×C17C2413yes—
148C2×C4×C17C2411yescharacteristic
149C2×C4×C17C2411yescharacteristic
150C2×C4×C17C2413yes—
151C22⋊D17—26no—
152C22⋊D17—212no—
153C22⋊D17—26no—
154C22⋊D17—43no—
155C22⋊D17—43no—
156C22⋊D17—46no—
157C2×Dic17—212no—
158C2×Dic17—26no—
159C2×Dic17C2413yes—
160C2×Dic17C2411yescharacteristic
161C2×Dic17C2411yescharacteristic
162C2×Dic17C2413yes—
163C2×Dic17C2413yes—
164D68—22no—
165D68—26no—
166D68—43no—
167C4×D17—22no—
168C4×D17—26no—
169C4×D17—26no—
170C4×D17—26no—
171C4×D17C2×D4112yes—
172Dic34—26no—
173Dic34—26no—
174Dic34C2×D4112yes—
175#3487—171nomaximal, Sylow
176C22×C17—23no—
177C22×C17—23no—
178C22×C17—23no—
179C22×C17—23no—
180C22×C17—23no—
181C22×C17—23no—
182C22×C17C2511yescharacteristic
183C2×D17—21no—
184C2×D17—21no—
185C2×D17—21no—
186C2×D17—23no—
187C2×D17—21no—
188C2×D17—23no—
189C2×D17—21no—
190C2×D17—23no—
191C2×D17—412no—
192C2×D17C22×D412yes—
193C4×C17—21no—
194C4×C17—21no—
195C4×C17—23no—
196C4×C17—21no—
197C4×C17C22×D416yes—
198Dic17—23no—
199Dic17—21no—
200Dic17—21no—
201Dic17—23no—
202Dic17C22×D416yes—
203C2×2−1+4—171no—
204C22×#41—171no—
205#550—171no—
206#550—173no—
207#550—173no—
208#550D1711yescharacteristic, complemented
209#548—171no—
210#548—173no—
211#548—171no—
212#548—173no—
213#535—173no—
214#535—173no—
215#525—173no—
216#525—173no—
217#519—171no—
218C2×C17—43no—
219C2×C17—43no—
220C2×C17C23×D411yescharacteristic
221C2×C17#58511yescharacteristic
222C2×C17#58511yescharacteristic
223D17—22no—
224D17—41no—
225D17—43no—
2262−1+4—348no—
227C2×#41—171no—
228C2×#41—173no—
229C2×#41—173no—
230C2×#41—173no—
231C2×#41—173no—
232C2×#41—173no—
233C2×#41—348no—
234C2×#41C2×D1711yescharacteristic
235C2×#41C2×D1711yescharacteristic, complemented
236C22×Q8—171no—
237C22×Q8—171no—
238C22×Q8—173no—
239C22×D4—173no—
240C23×C4—173no—
241#127—176no—
242#127—173no—
243#127C2×D1713yescomplemented
244#124—173no—
245#124—176no—
246#124—173no—
247#123—171no—
248#123—173no—
249C22⋊Q8—173no—
250C22⋊Q8—173no—
251C22⋊Q8—173no—
252C22⋊Q8—173no—
253#121—173no—
254#121—173no—
255#121—173no—
256#121—176no—
257#121—176no—
258#121—173no—
259#121—173no—
260#121—173no—
261#121C2×D1713yescomplemented
262#121C2×D1713yescomplemented
263#120—176no—
264#120—173no—
265#120—173no—
266C4×Q8—171no—
267C4×Q8—173no—
268C4×Q8—173no—
269C4×Q8C2×D1711yescharacteristic, complemented
270C4×D4—176no—
271C4×D4—173no—
272C4×D4—173no—
273C4×D4—176no—
274C4×D4—173no—
275C4×D4C2×D1713yescomplemented
276#117—173no—
277#117—173no—
278C17#348711yescharacteristic, complemented, Sylow
279C24—173no—
280#41—24no—
281#41—24no—
282#41—3412no—
283#41—3412no—
284#41—3412no—
285#41—3412no—
286#41—3412no—
287#41—344no—
288C2×Q8—178no—
289C2×Q8—173no—
290C2×Q8—173no—
291C2×Q8—173no—
292C2×Q8—3412no—
293C2×Q8—344no—
294C2×Q8C22×D1711yescharacteristic
295C2×D4—173no—
296C2×D4—176no—
297C2×D4—173no—
298C2×D4—176no—
299C2×D4—173no—
300C2×D4—173no—
301C2×D4—171no—
302C2×D4—173no—
303C2×D4—173no—
304C2×D4—3412no—
305C2×D4C22×D1713yes—
306C2×D4C22×D1713yes—
307C2×D4C22×D1716yes—
308C2×D4C22×D1713yes—
309C22×C4—173no—
310C22×C4—176no—
311C22×C4—173no—
312C22×C4—173no—
313C22×C4—173no—
314C22×C4—173no—
315C22×C4—171no—
316C22×C4—3412no—
317C22×C4C22×D1713yes—
318C22×C4C22×D1713yes—
319C4⋊C4—173no—
320C4⋊C4—173no—
321C4⋊C4—173no—
322C4⋊C4—173no—
323C4⋊C4—171no—
324C4⋊C4—173no—
325C4⋊C4C22×D1713yes—
326C4⋊C4C22×D1711yescharacteristic
327C22⋊C4—173no—
328C22⋊C4—173no—
329C22⋊C4—176no—
330C22⋊C4—173no—
331C22⋊C4—171no—
332C22⋊C4—171no—
333C22⋊C4—171no—
334C22⋊C4—173no—
335C22⋊C4—176no—
336C22⋊C4—173no—
337C22⋊C4C22×D1713yes—
338C22⋊C4C22×D1713yes—
339C42—176no—
340C42—173no—
341C42C22×D1713yes—
342C23—171no—
343C23—173no—
344C23—171no—
345C23—346no—
346C23—683no—
347C23—683no—
348C23C23×D1713yes—
349C23C23×D1713yes—
350Q8—1712no—
351Q8—346no—
352Q8—346no—
353Q8D4×D1714yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—342no—
359D4—3412no—
360D4—346no—
361D4—346no—
362D4—346no—
363D4—683no—
364D4—683no—
365D4—686no—
366D4—683no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—173no—
370C2×C4—1712no—
371C2×C4—171no—
372C2×C4—173no—
373C2×C4—173no—
374C2×C4—171no—
375C2×C4—346no—
376C2×C4—342no—
377C2×C4—346no—
378C2×C4—3412no—
379C2×C4—346no—
380C2×C4—346no—
381C2×C4C23×D1713yes—
382C2×C4C23×D1711yescharacteristic
383C2×C4C23×D1711yescharacteristic
384C2×C4C23×D1711yescharacteristic
385C2×C4C23×D1713yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—172no—
393C22—341no—
394C22—341no—
395C22—341no—
396C22—343no—
397C22—341no—
398C22—343no—
399C22—341no—
400C22—343no—
401C22—6812no—
402C22C24×D1711yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—176no—
408C4—341no—
409C4—343no—
410C4—341no—
411C4—343no—
412C4C2×D4×D1716yes—
413C2—43no—
414C2—43no—
415C2—342no—
416C2—683no—
417C2—681no—
418C2C22×D4×D1711yescharacteristic
419C2#1176364411yescharacteristic
420C2#1176364511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#550C2
hypercenter#550D17
hypocenterC17#3487
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#10587697C2semidirect
2. C2×#41×D17C2semidirect
3. #11763479C2semidirect
4. #11763478C2semidirect
5. #11763477C2semidirect
6. #11763471C2semidirect
7. #11763468C2semidirect
8. #11763466C2semidirect
9. #11763465C2semidirect
10. #11763375C2semidirect
11. #11763372C2semidirect
12. #11763319C2semidirect
13. #11763318C2semidirect
14. #11763285C2semidirect
15. C17×#550C2semidirect
17. C2×#41×C17C22semidirect
22. C2×Q8×D17C22semidirect
25. C2×#63254C22semidirect
30. C22×Dic34C22semidirect
31. C17×#127C22semidirect
32. C17×#121C22semidirect
33. C17×#121C22semidirect
34. C4×Q8×C17C22semidirect
35. C4×D4×C17C22semidirect
36. Q8×Dic17C22semidirect
37. #10587626C22semidirect
38. #10587626C22semidirect
39. #10587625C22semidirect
40. #10587625C22semidirect
41. #10587625C22semidirect
42. #10587624C22semidirect
43. #10587624C22semidirect
46. #10587622C22semidirect
47. D4×Dic17C22semidirect
48. #10587616C22semidirect
49. C4×C22⋊D17C22semidirect
50. C4×C22⋊D17C22semidirect
51. #10587610C22semidirect
52. #10587583C22semidirect
53. #10587582C22semidirect
54. #10587581C22semidirect
57. #10587573C22semidirect
58. #10587571C22semidirect
59. #10587570C22semidirect
62. #10587567C22semidirect
63. #10587564C22semidirect
64. #10587561C22semidirect
65. C4×D68C22semidirect
67. C4×Dic34C22semidirect
88. Q8×D17D4semidirect
208. #550D17semidirect
235. C2×#41C2×D17semidirect
243. #127C2×D17semidirect
261. #121C2×D17semidirect
262. #121C2×D17semidirect
269. C4×Q8C2×D17semidirect
275. C4×D4C2×D17semidirect
278. C17#3487semidirect

Maximal subgroups

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

subgroupindex
C2×#105876972
C2×#41×D172
#117634792
#117634782
#117634772
#117634712
#117634682
#117634662
#117634652
#117633752
#117633722
#117633192
#117633182
#117632852
C17×#5502
#348717