Finite Group Databasebeta

← #14540546

Subgroups — #14540546

Order 2176.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763617C211yesmaximal, characteristic, complemented
2#11763612C211yesmaximal, characteristic, complemented
3#11763611C211yesmaximal, characteristic, complemented
4#11763195C211yesmaximal, characteristic, complemented
5#11763079C211yesmaximal, characteristic, complemented
6#11763077C211yesmaximal, characteristic, complemented
7#11763074C211yesmaximal, characteristic, complemented
8#11763069C211yesmaximal, characteristic, complemented
9#11762985C211yesmaximal, characteristic, complemented
10#11762982C211yesmaximal, characteristic, complemented
11#11762977C211yesmaximal, characteristic, complemented
12#11762975C211yesmaximal, characteristic, complemented
13#11762931C211yesmaximal, characteristic, complemented
14#11762929C211yesmaximal, characteristic, complemented
15C17×#577C211yesmaximal, characteristic, complemented
16C17×2−1+4C2211yescharacteristic, complemented
17C17×H(5,𝔽2)C2211yescharacteristic, complemented
18#10587696C2211yescharacteristic
19#10587696C2211yescharacteristic, complemented
20C17×#137—21no—
21C17×#137C2211yescharacteristic, complemented
22C17×AGL(1,ℤ/8)—21no—
23C17×AGL(1,ℤ/8)C2211yescharacteristic, complemented
24C17×#135—21no—
25C17×#135C2211yescharacteristic
26C2×SD16×C17—21no—
27C2×SD16×C17C2211yescharacteristic, complemented
28C17×#131C2211yescharacteristic, complemented
29#10587636—21no—
30#10587636—21no—
31#10587636C2211yescharacteristic
32#10587635—21no—
33#10587635—21no—
34#10587635C2211yescharacteristic
35#10587635C2211yescharacteristic, complemented
36#10587635C2211yescharacteristic, complemented
37#10587635C2211yescharacteristic, complemented
38#10587634C2211yescharacteristic, complemented
39#10587634C2211yescharacteristic, complemented
40#10587633C2211yescharacteristic, complemented
41#10587628—21no—
42#10587628C2211yescharacteristic, complemented
43C2×#63230—21no—
44C2×#63230C2211yescharacteristic, complemented
45#10587626C2211yescharacteristic, complemented
46#10587618—21no—
47#10587618C2211yescharacteristic
48C2×#63228—21no—
49C2×#63228C2211yescharacteristic, complemented
50#10587605—21no—
51#10587600—21no—
52#10587600—21no—
53D8×D17—21no—
54C2×D136C2211yescharacteristic, complemented
55#10587587C2211yescharacteristic, complemented
56#10587508C2211yescharacteristic, complemented
57#10587508C2211yescharacteristic, complemented
58#10587508C2211yescharacteristic, complemented
59#10587507C2211yescharacteristic, complemented
60#10587504C2211yescharacteristic, complemented
61#10587496C2211yescharacteristic, complemented
62#10587494C2211yescharacteristic, complemented
63#10587494C2211yescharacteristic, complemented
64#10587493C2211yescharacteristic, complemented
65#10587490C2211yescharacteristic, complemented
66#10587486C2211yescharacteristic, complemented
67#41×C17—21no—
68#41×C17—21no—
69#41×C17—21no—
70#41×C17—21no—
71#41×C17—41no—
72#41×C17C2311yescharacteristic
73#41×C17C2311yescharacteristic
74#41×C17C2311yescharacteristic
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17C2311yescharacteristic
78C2×D4×C17—21no—
79C2×D4×C17—21no—
80C2×D4×C17—41no—
81C2×D4×C17C2311yescharacteristic
82C2×C22⋊D17—41no—
83#63256—21no—
84#63256—21no—
85D4×D17—21no—
86D4×D17—21no—
87D4×D17—41no—
88D4×D17—41no—
89#63252—21no—
90#63252—21no—
91#63252C2311yescharacteristic
92C2×D68—21no—
93C2×D68—21no—
94C2×D68C2311yescharacteristic
95C2×D68C2311yescharacteristic
96Q16×C17—21no—
97Q16×C17D411yescharacteristic, complemented
98SD16×C17—21no—
99SD16×C17—21no—
100SD16×C17—21no—
101SD16×C17—21no—
102SD16×C17D411yescharacteristic, complemented
103SD16×C17D411yescharacteristic, complemented
104D8×C17—21no—
105D8×C17D411yescharacteristic, complemented
106M16×C17—21no—
107M16×C17C2311yescharacteristic
108C2×C8×C17—21no—
109C2×C8×C17C2311yescharacteristic
110#63231—21no—
111#63231D411yescharacteristic, complemented
112#63230—21no—
113#63230—21no—
114#63230—21no—
115#63230—21no—
116#63230—21no—
117#63230—21no—
118#63230—21no—
119#63230D411yescharacteristic, complemented
120#63229—21no—
121#63229D411yescharacteristic, complemented
122#63228—21no—
123#63228—21no—
124#63228—21no—
125#63228—21no—
126#63228—21no—
127#63228—21no—
128#63228—21no—
129#63228D411yescharacteristic, complemented
130#63227—41no—
131C4×Dic17C2311yescharacteristic
132#63223—21no—
133#63223—21no—
134#63223C2311yescharacteristic
135#63223C2311yescharacteristic
136#63223C2311yescharacteristic
137C2×#3622—21no—
138C2×#3622—21no—
139C2×#3622C2311yescharacteristic
140D136—21no—
141D136—21no—
142#63218—21no—
143C8×D17—21no—
144C23×C17—21no—
145C23×C17—41no—
146C22×D17—21no—
147C22×D17—21no—
148C22×D17—41no—
149C22×D17—41no—
150Q8×C17—21no—
151Q8×C17—21no—
152Q8×C17—41no—
153Q8×C17C2×D411yescharacteristic
154Q8×C17C2×D411yescharacteristic
155Q8×C17C2×D411yescharacteristic
156D4×C17—21no—
157D4×C17—21no—
158D4×C17—21no—
159D4×C17—21no—
160D4×C17—21no—
161D4×C17—41no—
162D4×C17—41no—
163D4×C17—41no—
164D4×C17C2×D411yescharacteristic
165D4×C17C2×D411yescharacteristic
166D4×C17C2×D411yescharacteristic
167C2×C4×C17—21no—
168C2×C4×C17—21no—
169C2×C4×C17—21no—
170C2×C4×C17—21no—
171C2×C4×C17—41no—
172C2×C4×C17—41no—
173C2×C4×C17—41no—
174C2×C4×C17C2411yescharacteristic
175C22⋊D17—21no—
176C22⋊D17—41no—
177C22⋊D17—41no—
178C22⋊D17—82no—
179C2×Dic17—21no—
180D68—21no—
181D68—21no—
182D68—21no—
183D68—21no—
184D68—41no—
185D68—41no—
186D68C2×D411yescharacteristic
187C4×D17—21no—
188C4×D17—41no—
189C4×D17—41no—
190Dic34C2×D411yescharacteristic
191C8×C17—21no—
192C8×C17C2×D411yescharacteristic
193C8×C17C2×D411yescharacteristic
194#3622—21no—
195#3622—21no—
196#3622—21no—
197#3622C2×D411yescharacteristic
198#3622C2×D411yescharacteristic
199#3288—171nomaximal, Sylow
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—21no—
203C22×C17—41no—
204C22×C17—81no—
205C22×C17—161no—
206C22×C17H(5,𝔽2)11yescharacteristic
207C2×D17—21no—
208C2×D17—41no—
209C2×D17—41no—
210C2×D17—81no—
211C2×D17—81no—
212C2×D17—161no—
213C2×D17—161no—
214C4×C17—21no—
215C4×C17—21no—
216C4×C17—21no—
217C4×C17—41no—
218C4×C17C22×D411yescharacteristic
219C4×C17C22×D411yescharacteristic
220Dic17—21no—
221Dic17—41no—
222#577—171no—
223#577D1711yescharacteristic, complemented
224#576—171no—
225#576—171no—
226#496—171no—
227#472—171no—
228#472—171no—
229#471—171no—
230#471—171no—
231#455—171no—
232#454—171no—
233#453—171no—
234#453—171no—
235#444—171no—
236#444—171no—
237C2×C17—21no—
238C2×C17—41no—
239C2×C17—41no—
240C2×C17—41no—
241C2×C17—81no—
242C2×C17D4211yescharacteristic
243D17—41no—
244D17—81no—
245D17—81no—
2462−1+4C2×D1711yescharacteristic, complemented
247H(5,𝔽2)—171no—
248H(5,𝔽2)—171no—
249H(5,𝔽2)C2×D1711yescharacteristic, complemented
250#137—21no—
251#137—171no—
252#137—341no—
253#137C2×D1711yescharacteristic, complemented
254AGL(1,ℤ/8)—21no—
255AGL(1,ℤ/8)—171no—
256AGL(1,ℤ/8)—171no—
257AGL(1,ℤ/8)—171no—
258AGL(1,ℤ/8)—171no—
259AGL(1,ℤ/8)—171no—
260AGL(1,ℤ/8)—341no—
261AGL(1,ℤ/8)—341no—
262AGL(1,ℤ/8)—341no—
263AGL(1,ℤ/8)—341no—
264AGL(1,ℤ/8)—341no—
265AGL(1,ℤ/8)C2×D1711yescharacteristic, complemented
266#135—21no—
267#135—171no—
268#135—341no—
269#135—341no—
270#135—341no—
271#135C2×D1711yescharacteristic
272C2×SD16—21no—
273C2×SD16—171no—
274C2×SD16—341no—
275C2×SD16C2×D1711yescharacteristic, complemented
276C2×D8—171no—
277C2×D8—171no—
278C2×D8—341no—
279C2×D8—341no—
280#131—171no—
281#131—171no—
282#131—171no—
283#131C2×D1711yescharacteristic, complemented
284#127—171no—
285#124—171no—
286#108—171no—
287#108—171no—
288#104—171no—
289#104—171no—
290#104—171no—
291#104—171no—
292#101—171no—
293#100—171no—
294#100—171no—
295#100—171no—
296#97—171no—
297C17#328811yescharacteristic, complemented, Sylow
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—41no—
303#41—171no—
304#41—341no—
305#41—341no—
306#41—341no—
307#41—341no—
308#41C22×D1711yescharacteristic
309#41C22×D1711yescharacteristic
310#41C22×D1711yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C22×D1711yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—171no—
318C2×D4—171no—
319C2×D4—341no—
320C2×D4—341no—
321C2×D4—341no—
322C2×D4—341no—
323C2×D4—681no—
324C2×D4—681no—
325C2×D4—681no—
326C2×D4C22×D1711yescharacteristic
327Q16—21no—
328Q16—171no—
329Q16—341no—
330Q16C22⋊D1711yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—171no—
336SD16—171no—
337SD16—341no—
338SD16—341no—
339SD16—341no—
340SD16—341no—
341SD16—341no—
342SD16—341no—
343SD16—341no—
344SD16—341no—
345SD16C22⋊D1711yescharacteristic, complemented
346SD16C22⋊D1711yescharacteristic, complemented
347D8—21no—
348D8—171no—
349D8—341no—
350D8—341no—
351D8—341no—
352D8—341no—
353D8—341no—
354D8—341no—
355D8—341no—
356D8—341no—
357D8—341no—
358D8C22⋊D1711yescharacteristic, complemented
359M16—21no—
360M16—171no—
361M16—171no—
362M16—171no—
363M16—341no—
364M16—341no—
365M16—341no—
366M16C22×D1711yescharacteristic
367C2×C8—21no—
368C2×C8—171no—
369C2×C8—341no—
370C2×C8—341no—
371C2×C8—341no—
372C2×C8C22×D1711yescharacteristic
373C22⋊C4—681no—
374C42—171no—
375C23—21no—
376C23—41no—
377C23—341no—
378C23—341no—
379C23—681no—
380C23—681no—
381Q8—21no—
382Q8—21no—
383Q8—41no—
384Q8—171no—
385Q8D4×D1711yescharacteristic
386Q8C2×C22⋊D1711yescharacteristic
387Q8C2×C22⋊D1711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—171no—
397D4—341no—
398D4—341no—
399D4—341no—
400D4—341no—
401D4—341no—
402D4—681no—
403D4—681no—
404D4—681no—
405D4—681no—
406D4—1362no—
407D4D4×D1711yescharacteristic
408D4C2×C22⋊D1711yescharacteristic
409D4C2×C22⋊D1711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—341no—
418C2×C4—341no—
419C2×C4—681no—
420C2×C4—681no—
421C2×C4C23×D1711yescharacteristic
422C8—21no—
423C8—171no—
424C8—171no—
425C8—341no—
426C8—341no—
427C8—341no—
428C8C2×C22⋊D1711yescharacteristic
429C8C2×C22⋊D1711yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—81no—
435C22—161no—
436C22—2721no—
437C22—2721no—
438C22—341no—
439C22—681no—
440C22—681no—
441C22—1361no—
442C22—1361no—
443C22#1058769111yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—41no—
448C4—341no—
449C4—681no—
450C4C2×D4×D1711yescharacteristic
451C4C22×C22⋊D1711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—681no—
458C2—1361no—
459C2—1361no—
460C2D4×C22⋊D1711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#577C2
socleC2×C17D42
hypercenter#577D17
hypocenterC17#3288
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #11763617C2semidirect
2. #11763612C2semidirect
3. #11763611C2semidirect
4. #11763195C2semidirect
5. #11763079C2semidirect
6. #11763077C2semidirect
7. #11763074C2semidirect
8. #11763069C2semidirect
9. #11762985C2semidirect
10. #11762982C2semidirect
11. #11762977C2semidirect
12. #11762975C2semidirect
13. #11762931C2semidirect
14. #11762929C2semidirect
15. C17×#577C2semidirect
16. C17×2−1+4C22semidirect
17. C17×H(5,𝔽2)C22semidirect
19. #10587696C22semidirect
21. C17×#137C22semidirect
23. C17×AGL(1,ℤ/8)C22semidirect
27. C2×SD16×C17C22semidirect
28. C17×#131C22semidirect
35. #10587635C22semidirect
36. #10587635C22semidirect
37. #10587635C22semidirect
38. #10587634C22semidirect
39. #10587634C22semidirect
40. #10587633C22semidirect
42. #10587628C22semidirect
44. C2×#63230C22semidirect
45. #10587626C22semidirect
49. C2×#63228C22semidirect
54. C2×D136C22semidirect
55. #10587587C22semidirect
56. #10587508C22semidirect
57. #10587508C22semidirect
58. #10587508C22semidirect
59. #10587507C22semidirect
60. #10587504C22semidirect
61. #10587496C22semidirect
62. #10587494C22semidirect
63. #10587494C22semidirect
64. #10587493C22semidirect
65. #10587490C22semidirect
66. #10587486C22semidirect
97. Q16×C17D4semidirect
102. SD16×C17D4semidirect
103. SD16×C17D4semidirect
105. D8×C17D4semidirect
111. #63231D4semidirect
119. #63230D4semidirect
121. #63229D4semidirect
129. #63228D4semidirect
223. #577D17semidirect
246. 2−1+4C2×D17semidirect
249. H(5,𝔽2)C2×D17semidirect
253. #137C2×D17semidirect
265. AGL(1,ℤ/8)C2×D17semidirect
275. C2×SD16C2×D17semidirect
283. #131C2×D17semidirect
297. C17#3288semidirect
330. Q16C22⋊D17semidirect
345. SD16C22⋊D17semidirect
346. SD16C22⋊D17semidirect
358. D8C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
#117636172
#117636122
#117636112
#117631952
#117630792
#117630772
#117630742
#117630692
#117629852
#117629822
#117629772
#117629752
#117629312
#117629292
C17×#5772
#328817