Finite Group Databasebeta

← #14542575

Subgroups — #14542575

Order 2176.

1540 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×#10587694C211yesmaximal, characteristic, complemented
2C2×#10587696C211yesmaximal, characteristic, complemented
3#11763471C211yesmaximal, characteristic, complemented
4#11763470C211yesmaximal, characteristic, complemented
5#11763466C211yesmaximal, characteristic, complemented
6#11763456C213yesmaximal, complemented
7#11763455C213yesmaximal, complemented
8#11763452C213yesmaximal, complemented
9#11763450C213yesmaximal, complemented
10#11763382C213yesmaximal, complemented
11#11763375C213yesmaximal, complemented
12#11763356C213yesmaximal, complemented
13#124×D17C213yesmaximal, complemented
14C4×#63256C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic, complemented
18#10587696—28no—
19#10587694—28no—
20C2×#63256C2211yescharacteristic
21C2×#63256C2211yescharacteristic
22C2×#63256C2213yescomplemented
23C2×Q8×D17C2211yescharacteristic, complemented
24C2×Q8×D17C2213yescomplemented
25C2×D4×D17C2213yes—
26C2×D4×D17C2213yes—
27C2×#63252C2213yes—
28C2×#63252C2213yescomplemented
29C2×#63252C2213yescomplemented
30C22×D68C2213yes—
31C17×#124C2213yescomplemented
32C17×C22⋊Q8C2213yescomplemented
33C17×#121C2213yescomplemented
34C4×Q8×C17C2211yescharacteristic, complemented
35C4×D4×C17C2213yescomplemented
36#10587633C2213yescomplemented
37#10587632C2213yescomplemented
38Q8×Dic17C2211yescharacteristic, complemented
39#10587626C2213yescomplemented
40#10587624C2213yescomplemented
41#10587624C2213yescomplemented
42#10587623C2216yescomplemented
43#10587622C2213yescomplemented
44#10587616C2213yescomplemented
45C4×C22⋊D17C2213yescomplemented
46C4×C22⋊D17C2213yescomplemented
47#10587610C2213yescomplemented
48#10587583C2213yescomplemented
49#10587581C2213yescomplemented
50#10587581C2213yescomplemented
51#10587580C2216yescomplemented
52#10587579C2213yescomplemented
53#10587578C2213yescomplemented
54#10587574C2213yescomplemented
55#10587571C2213yescomplemented
56#10587571C2216yescomplemented
57#10587571C2211yescharacteristic, complemented
58#10587570C2213yescomplemented
59#10587570C2213yescomplemented
60#10587569C2216yescomplemented
61#10587568C2216yescomplemented
62C22⋊C4×D17C2216yescomplemented
63C22⋊C4×D17C2216yescomplemented
64#10587561C2213yescomplemented
65C4×D68C2213yescomplemented
66C4×D68C2213yescomplemented
67C42×D17C2213yescomplemented
68C23×D17C2316yes—
69#41×C17—24no—
70C2×Q8×C17—24no—
71C2×Q8×C17C2313yes—
72C2×Q8×C17C2311yescharacteristic
73C2×D4×C17C2313yes—
74C2×D4×C17C2313yes—
75C22×C4×C17C2313yes—
76C22×C4×C17C2313yes—
77C2×C22⋊D17C2313yes—
78C2×C22⋊D17C2313yes—
79C2×C22⋊D17C2313yes—
80C2×C22⋊D17C2311yescharacteristic
81C2×C22⋊D17C2313yes—
82C2×C22⋊D17C2313yes—
83#63256—24no—
84#63256—212no—
85#63256D418yescomplemented
86Q8×D17—24no—
87Q8×D17—212no—
88D4×D17—212no—
89D4×D17—212no—
90#63252—212no—
91#63252—212no—
92#63252—212no—
93C2×D68—212no—
94C2×D68C2313yes—
95C2×D68C2313yes—
96C2×D68C2313yes—
97C2×D68C2313yes—
98C2×D68C2313yes—
99C2×C4×D17C2313yes—
100C2×C4×D17C2313yes—
101C2×C4×D17C2316yes—
102C2×C4×D17C2313yes—
103C2×C4×D17C2313yes—
104C2×C4×D17C2313yes—
105C2×Dic34C2311yescharacteristic
106C2×Dic34C2313yes—
107C2×Dic34C2313yes—
108C4⋊C4×C17C2313yes—
109C4⋊C4×C17C2313yes—
110C22⋊C4×C17C2316yes—
111C22⋊C4×C17C2313yes—
112C22⋊C4×C17C2311yescharacteristic
113C42×C17C2313yes—
114C22⋊Dic17C2313yes—
115C22⋊Dic17C2316yes—
116C22⋊Dic17C2311yescharacteristic
117#63227C2311yescharacteristic
118#63227C2313yes—
119#63227C2316yes—
120#63227C2316yes—
121#63227C2313yes—
122#63227C2316yes—
123#63227C2311yescharacteristic
124C4⋊Dic17C2313yes—
125C4⋊Dic17C2313yes—
126#63225C2313yes—
127#63225C2313yes—
128C4×Dic17C2311yescharacteristic
129C4×Dic17C2313yes—
130C4×Dic17C2313yes—
131C23×C17C2413yes—
132C23×C17C2411yescharacteristic
133C22×D17—212no—
134C22×D17—46no—
135C22×D17—46no—
136C22×D17C2411yescharacteristic
137C22×D17C2413yes—
138C22×D17C2413yes—
139Q8×C17—26no—
140Q8×C17C2×D414yes—
141D4×C17—26no—
142D4×C17—43no—
143C2×C4×C17—26no—
144C2×C4×C17—26no—
145C2×C4×C17C2411yescharacteristic
146C2×C4×C17C2413yes—
147C2×C4×C17C2413yes—
148C2×C4×C17C2411yescharacteristic
149C2×C4×C17C2413yes—
150C22⋊D17—26no—
151C22⋊D17—26no—
152C22⋊D17—22no—
153C22⋊D17—26no—
154C22⋊D17—43no—
155C22⋊D17—43no—
156C2×Dic17C2411yescharacteristic
157C2×Dic17C2413yes—
158C2×Dic17C2413yes—
159C2×Dic17C2411yescharacteristic
160C2×Dic17C2411yescharacteristic
161D68—26no—
162D68—26no—
163D68—26no—
164D68—43no—
165D68C2×D4112yes—
166C4×D17—212no—
167C4×D17—26no—
168C4×D17—26no—
169C4×D17—43no—
170C4×D17—43no—
171C4×D17C2×D4112yes—
172Dic34—26no—
173Dic34—22no—
174Dic34—26no—
175#3500—171nomaximal, Sylow
176C22×C17—21no—
177C22×C17—23no—
178C22×C17—21no—
179C22×C17—23no—
180C22×C17—21no—
181C22×C17—23no—
182C22×C17C2511yescharacteristic
183C2×D17—23no—
184C2×D17—23no—
185C2×D17—21no—
186C2×D17—23no—
187C2×D17—21no—
188C2×D17—23no—
189C2×D17—21no—
190C2×D17—23no—
191C2×D17—812no—
192C2×D17C22×D416yes—
193C4×C17—23no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—23no—
197C4×C17C22×D416yes—
198Dic17—23no—
199Dic17—21no—
200Dic17—21no—
201Dic17—23no—
202Dic17C22×D412yes—
203C2×2−1+4—171no—
204C2×H(5,𝔽2)—171no—
205#548—171no—
206#548—171no—
207#548—171no—
208#548D1711yescharacteristic, complemented
209#546—173no—
210#546—173no—
211#546—173no—
212#546—173no—
213#536—173no—
214#535—173no—
215#532—173no—
216C2×#124—173no—
217C4×#41—171no—
218C2×C17—43no—
219C2×C17—41no—
220C2×C17C23×D411yescharacteristic
221C2×C17#58511yescharacteristic
222C2×C17#58511yescharacteristic
223D17—26no—
224D17—41no—
225D17—43no—
2262−1+4—348no—
227H(5,𝔽2)—348no—
228C2×#41—171no—
229C2×#41—173no—
230C2×#41—171no—
231C2×#41—173no—
232C2×#41—173no—
233C2×#41—173no—
234C2×#41C2×D1711yescharacteristic
235C22×Q8—173no—
236C22×Q8—171no—
237C22×Q8C2×D1711yescharacteristic, complemented
238C22×D4—173no—
239C22×D4—173no—
240C22×D4—173no—
241C4⋊Q8—173no—
242#127—173no—
243#124—173no—
244#124—173no—
245#124—173no—
246#124—176no—
247#124—171no—
248#124—173no—
249#124C2×D1713yescomplemented
250#123—176no—
251#123—176no—
252C22⋊Q8—173no—
253C22⋊Q8—173no—
254C22⋊Q8—173no—
255C22⋊Q8C2×D1713yescomplemented
256#121—173no—
257#121—173no—
258#121—173no—
259#121—173no—
260#121—173no—
261#121—173no—
262#121—173no—
263#121C2×D1713yescomplemented
264#120—176no—
265#120—176no—
266C4×Q8—171no—
267C4×Q8C2×D1711yescharacteristic, complemented
268C4×D4—173no—
269C4×D4—173no—
270C4×D4—173no—
271C4×D4—173no—
272C4×D4—173no—
273C4×D4C2×D1713yescomplemented
274#117—173no—
275C2×C22⋊C4—176no—
276C2×C22⋊C4—176no—
277C2×C42—173no—
278C17#350011yescharacteristic, complemented, Sylow
279C24—176no—
280#41—24no—
281#41—178no—
282#41—3412no—
283#41—3412no—
284#41—3412no—
285#41—3412no—
286#41—344no—
287C2×Q8—24no—
288C2×Q8—173no—
289C2×Q8—173no—
290C2×Q8—171no—
291C2×Q8—3412no—
292C2×Q8—344no—
293C2×Q8C22×D1713yes—
294C2×Q8C22×D1711yescharacteristic
295C2×D4—173no—
296C2×D4—173no—
297C2×D4—173no—
298C2×D4—171no—
299C2×D4—173no—
300C2×D4—173no—
301C2×D4—173no—
302C2×D4—173no—
303C2×D4—173no—
304C2×D4—173no—
305C2×D4—173no—
306C2×D4—3412no—
307C2×D4—3412no—
308C2×D4—3412no—
309C2×D4C22×D1713yes—
310C2×D4C22×D1713yes—
311C22×C4—173no—
312C22×C4—176no—
313C22×C4—173no—
314C22×C4—173no—
315C22×C4—173no—
316C22×C4—173no—
317C22×C4C22×D1713yes—
318C22×C4C22×D1713yes—
319C4⋊C4—173no—
320C4⋊C4—173no—
321C4⋊C4—173no—
322C4⋊C4—173no—
323C4⋊C4C22×D1713yes—
324C4⋊C4C22×D1713yes—
325C22⋊C4—173no—
326C22⋊C4—176no—
327C22⋊C4—171no—
328C22⋊C4—173no—
329C22⋊C4—173no—
330C22⋊C4—176no—
331C22⋊C4—176no—
332C22⋊C4—176no—
333C22⋊C4—171no—
334C22⋊C4—171no—
335C22⋊C4C22×D1713yes—
336C22⋊C4C22×D1716yes—
337C22⋊C4C22×D1711yescharacteristic
338C42—171no—
339C42—173no—
340C42—173no—
341C42C22×D1713yes—
342C23—173no—
343C23—173no—
344C23—171no—
345C23—3412no—
346C23—686no—
347C23—686no—
348C23C23×D1713yes—
349C23C23×D1711yescharacteristic
350Q8—26no—
351Q8—346no—
352Q8—342no—
353Q8—346no—
354Q8D4×D1714yes—
355D4—26no—
356D4—43no—
357D4—1712no—
358D4—346no—
359D4—346no—
360D4—346no—
361D4—342no—
362D4—346no—
363D4—346no—
364D4—346no—
365D4—683no—
366D4—683no—
367D4—683no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—171no—
371C2×C4—1712no—
372C2×C4—171no—
373C2×C4—173no—
374C2×C4—173no—
375C2×C4—171no—
376C2×C4—3412no—
377C2×C4—346no—
378C2×C4—346no—
379C2×C4—683no—
380C2×C4—683no—
381C2×C4C23×D1711yescharacteristic
382C2×C4C23×D1713yes—
383C2×C4C23×D1713yes—
384C2×C4C23×D1711yescharacteristic
385C2×C4C23×D1713yes—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—21no—
391C22—23no—
392C22—176no—
393C22—343no—
394C22—343no—
395C22—341no—
396C22—343no—
397C22—341no—
398C22—343no—
399C22—341no—
400C22—343no—
401C22—13612no—
402C22C24×D1711yescharacteristic
403C4—21no—
404C4—23no—
405C4—23no—
406C4—21no—
407C4—172no—
408C4—341no—
409C4—343no—
410C4—341no—
411C4—343no—
412C4C2×D4×D1716yes—
413C2—43no—
414C2—41no—
415C2—346no—
416C2—683no—
417C2—681no—
418C2C22×D4×D1711yescharacteristic
419C2#1176364511yescharacteristic
420C2#1176364611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#548C2
hypercenter#548D17
hypocenterC17#3500
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#10587694C2semidirect
2. C2×#10587696C2semidirect
3. #11763471C2semidirect
4. #11763470C2semidirect
5. #11763466C2semidirect
6. #11763456C2semidirect
7. #11763455C2semidirect
8. #11763452C2semidirect
9. #11763450C2semidirect
10. #11763382C2semidirect
11. #11763375C2semidirect
12. #11763356C2semidirect
13. #124×D17C2semidirect
14. C4×#63256C2semidirect
15. C17×#548C2semidirect
17. C22×Q8×C17C22semidirect
22. C2×#63256C22semidirect
23. C2×Q8×D17C22semidirect
24. C2×Q8×D17C22semidirect
28. C2×#63252C22semidirect
29. C2×#63252C22semidirect
31. C17×#124C22semidirect
32. C17×C22⋊Q8C22semidirect
33. C17×#121C22semidirect
34. C4×Q8×C17C22semidirect
35. C4×D4×C17C22semidirect
36. #10587633C22semidirect
37. #10587632C22semidirect
38. Q8×Dic17C22semidirect
39. #10587626C22semidirect
40. #10587624C22semidirect
41. #10587624C22semidirect
42. #10587623C22semidirect
43. #10587622C22semidirect
44. #10587616C22semidirect
45. C4×C22⋊D17C22semidirect
46. C4×C22⋊D17C22semidirect
47. #10587610C22semidirect
48. #10587583C22semidirect
49. #10587581C22semidirect
50. #10587581C22semidirect
51. #10587580C22semidirect
52. #10587579C22semidirect
53. #10587578C22semidirect
54. #10587574C22semidirect
55. #10587571C22semidirect
56. #10587571C22semidirect
57. #10587571C22semidirect
58. #10587570C22semidirect
59. #10587570C22semidirect
60. #10587569C22semidirect
61. #10587568C22semidirect
62. C22⋊C4×D17C22semidirect
63. C22⋊C4×D17C22semidirect
64. #10587561C22semidirect
65. C4×D68C22semidirect
66. C4×D68C22semidirect
67. C42×D17C22semidirect
85. #63256D4semidirect
208. #548D17semidirect
237. C22×Q8C2×D17semidirect
249. #124C2×D17semidirect
255. C22⋊Q8C2×D17semidirect
263. #121C2×D17semidirect
267. C4×Q8C2×D17semidirect
273. C4×D4C2×D17semidirect
278. C17#3500semidirect

Maximal subgroups

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

subgroupindex
C2×#105876942
C2×#105876962
#117634712
#117634702
#117634662
#117634562
#117634552
#117634522
#117634502
#117633822
#117633752
#117633562
#124×D172
C4×#632562
C17×#5482
#350017