Finite Group Databasebeta

← #17978641

Subgroups — #17978641

Order 4352.

646 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545563C211yesmaximal, characteristic, complemented
2#14545547C211yesmaximal, characteristic, complemented
3#14545195C212yesmaximal, complemented
4#14545194C212yesmaximal, complemented
5#14545192C212yesmaximal, complemented
6#14545189C212yesmaximal, complemented
7C2×#11763734C211yesmaximal, characteristic, complemented
8#14545112C211yesmaximal, characteristic, complemented
9D17⋊#360C211yesmaximal, characteristic, complemented
10#14545003C211yesmaximal, characteristic, complemented
11#14541917C211yesmaximal, characteristic
12#11763888—21no—
13#11763878—21no—
14#11763865C2211yescharacteristic
15#11763856C2212yes—
16#11763855C2212yes—
17#11763852—22no—
18#11763850C2212yescomplemented
19#11763849C2212yescomplemented
20SD16×#595—22no—
21C2×#10587535C2211yescharacteristic
22C2×#10587533C2211yescharacteristic
23#11763806—21no—
24#11763800—21no—
25#11763772C2211yescharacteristic
26#11763734C2212yescomplemented
27#11763734C2212yescomplemented
28#11763728C2212yescomplemented
29#11763727C2212yescomplemented
30#11763726C2212yes—
31#11763725C2212yes—
32#11763700C2212yescomplemented
33#11763699C2212yescomplemented
34#135×D17C2211yescharacteristic
35#135×D17C2211yescharacteristic
36#11763251C411yescharacteristic, complemented
37#11763251C2211yescharacteristic
38#11763248C411yescharacteristic, complemented
39#11763248C2211yescharacteristic
40#11763246C411yescharacteristic, complemented
41#11763246C2211yescharacteristic
42#11763243C411yescharacteristic, complemented
43#11763243C2211yescharacteristic
44#11763241C411yescharacteristic, complemented
45#11763239C411yescharacteristic, complemented
46#11763239C411yescharacteristic, complemented
47C2×#10587469C2211yescharacteristic
48C17×#508C411yescharacteristic, complemented
49#41×D17—21no—
50#41×D17—21no—
51Q8×#595—22no—
52#10587680—22no—
53D4×#595—22no—
54D4×#595—42no—
55#10587678—22no—
56#10587678—42no—
57#10587677—41no—
58#10587677D411yescharacteristic
59C2×C4⋊#595—21no—
60C2×C4⋊#595—41no—
61D17⋊M16—41no—
62D17⋊M16D411yescharacteristic
63C2×#63243—21no—
64C2×#63243—41no—
65C17×#135C2×C411yescharacteristic
66C17×#135C2×C411yescharacteristic, complemented
67#10587636—21no—
68#10587636—21no—
69#10587605C2×C411yescharacteristic
70#10587605C2311yescharacteristic
71Q16×D17C2×C411yescharacteristic, complemented
72Q16×D17C2311yescharacteristic
73#10587602—21no—
74#10587602—21no—
75SD16×D17—21no—
76SD16×D17—21no—
77#10587598C2×C411yescharacteristic
78#10587598C2311yescharacteristic
79D8×D17C2×C411yescharacteristic, complemented
80D8×D17C2311yescharacteristic
81#10587590C2×C411yescharacteristic
82#10587590C2×C411yescharacteristic, complemented
83C2×C8×D17C2311yescharacteristic
84#10587550—22no—
85#10587549—22no—
86#10587548—22no—
87#10587547—22no—
88#10587538—21no—
89#10587535—21no—
90#10587535C2312yes—
91#10587533—21no—
92#10587533C2312yes—
93#10587532C2312yes—
94C8×#595D412yes—
95#10587530C2312yes—
96#10587529—21no—
97C17×Q32C2×C411yescharacteristic, complemented
98C17×SD32C2×C411yescharacteristic, complemented
99C17×SD32C2×C411yescharacteristic, complemented
100C17×D16C2×C411yescharacteristic, complemented
101C2×C16×C17C2×C411yescharacteristic, complemented
102#10587500C2×C411yescharacteristic, complemented
103#10587500C2×C411yescharacteristic, complemented
104#10587499C2×C411yescharacteristic, complemented
105#10587499C2×C411yescharacteristic, complemented
106#10587498C2×C411yescharacteristic, complemented
107#10587498C2×C411yescharacteristic, complemented
108#10587497C2×C411yescharacteristic, complemented
109#10587497C2×C411yescharacteristic, complemented
110C2×#63214C2×C411yescharacteristic, complemented
111Dic136C2×C411yescharacteristic, complemented
112#10587471C2×C411yescharacteristic, complemented
113#10587471C2×C411yescharacteristic, complemented
114D272C2×C411yescharacteristic, complemented
115#10587469C2×C411yescharacteristic, complemented
116#10587469C2×C411yescharacteristic, complemented
117#10587469C2311yescharacteristic
118#10587469C2311yescharacteristic
119C22×#595—42no—
120C22×#595—41no—
121#41×C17—21no—
122#41×C17—21no—
123#63256—21no—
124#63256—21no—
125Q8×D17—21no—
126Q8×D17—21no—
127#63254—21no—
128#63254—21no—
129#63254—41no—
130#63254—41no—
131D4×D17—21no—
132D4×D17—21no—
133D4×D17—41no—
134D4×D17—41no—
135#63252—21no—
136#63252—21no—
137#63252—41no—
138#63252—41no—
139C2×C4×D17—41no—
140C2×C4×D17—41no—
141C2×C4×D17C2×D411yescharacteristic
142C22⋊#595—21no—
143C22⋊#595—42no—
144#63247—21no—
145#63247—42no—
146C2×#3624—42no—
147C2×#3624—41no—
148C4⋊#595—21no—
149C4⋊#595—22no—
150C4⋊#595—42no—
151C4⋊#595—41no—
152C4×#595—42no—
153C4×#595C2×D412yes—
154#63243—21no—
155#63243—22no—
156#63243—42no—
157#63243—41no—
158#63242—42no—
159#63242C2×D412yes—
160Q16×C17C22×C411yescharacteristic
161Q16×C17C22×C411yescharacteristic
162SD16×C17—21no—
163SD16×C17—21no—
164D8×C17C22×C411yescharacteristic
165D8×C17C22×C411yescharacteristic
166C2×C8×C17C22×C411yescharacteristic
167#63231—21no—
168#63231—21no—
169#63230—21no—
170#63230—21no—
171#63229—21no—
172#63229—21no—
173#63228—21no—
174#63228—21no—
175C2×#3622#4111yescharacteristic
176Dic68C22×C411yescharacteristic
177Dic68C22×C411yescharacteristic
178D136C22×C411yescharacteristic
179D136C22×C411yescharacteristic
180#63219—21no—
181#63219—21no—
182C8×D17C22×C411yescharacteristic
183C8×D17C2×D411yescharacteristic
184C8×D17#4111yescharacteristic
185C8×D17C2411yescharacteristic
186#63216—22no—
187C16×C17C22×C411yescharacteristic
188C16×C17C22×C411yescharacteristic
189#63214C22×C411yescharacteristic
190#63214C22×C411yescharacteristic
191#33469—171nomaximal, Sylow
192C22×D17—41no—
193C22×D17—41no—
194C22×D17AGL(1,ℤ/8)11yescharacteristic
195C2×#595—22no—
196C2×#595—42no—
197C2×#595—162no—
198C2×#595—161no—
199Q8×C17—21no—
200Q8×C17—21no—
201D4×C17—21no—
202D4×C17—21no—
203D4×C17—41no—
204D4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—41no—
207C2×C4×C17C4×D411yescharacteristic
208C22⋊D17—41no—
209C22⋊D17—41no—
210C22⋊D17—41no—
211C22⋊D17—41no—
212C22⋊D17—41no—
213C22⋊D17—41no—
214C2×Dic17—41no—
215C2×Dic17—41no—
216C2×Dic17AGL(1,ℤ/8)11yescharacteristic
217D68—21no—
218D68—21no—
219D68—41no—
220D68—41no—
221C4×D17—41no—
222C4×D17—41no—
223C4×D17—41no—
224C4×D17—41no—
225C4×D17—41no—
226C4×D17—41no—
227C4×D17C4×D411yescharacteristic
228C4×D17AGL(1,ℤ/8)11yescharacteristic
229C4×D17AGL(1,ℤ/8)11yescharacteristic
230C4×D17C22×D411yescharacteristic
231Dic34—21no—
232Dic34—21no—
233Dic34—41no—
234Dic34—41no—
235#3624—22no—
236#3624—42no—
237C8×C17C4×D411yescharacteristic
238C8×C17C23×C411yescharacteristic
239#3622C4×D411yescharacteristic
240#3622C2×#4111yescharacteristic
241#3414—171no—
242#2957—171no—
243#2952—171no—
244AGL(1,ℤ/16)—172no—
245AGL(1,ℤ/16)—172no—
246#2180—172no—
247#2180—172no—
248C2×#365—171no—
249#2148—171no—
250C2×#360—171no—
251#2110—171no—
252C22×C17—41no—
253C22×C17—41no—
254C22×C17#44211yescharacteristic
255C2×D17—41no—
256C2×D17—41no—
257C2×D17—41no—
258C2×D17—81no—
259C2×D17—81no—
260C2×D17—161no—
261C2×D17#44211yescharacteristic
262C2×D17C2×AGL(1,ℤ/8)11yescharacteristic
263#595—42no—
264#595—82no—
265C4×C17—41no—
266C4×C17—41no—
267C4×C17#44211yescharacteristic
268C4×C17C2×C4×D411yescharacteristic
269Dic17—41no—
270Dic17—41no—
271Dic17#44211yescharacteristic
272Dic17C2×AGL(1,ℤ/8)11yescharacteristic
273C2×#135—171no—
274C2×#135—171no—
275#568—341no—
276#520—341no—
277#510—171no—
278#510—171no—
279#510—171no—
280#510—171no—
281#509—171no—
282#509—171no—
283#509—171no—
284#509—171no—
285#508—171no—
286#508—171no—
287#508—171no—
288#508#59511yescharacteristic, complemented
289C2×M32—171no—
290C2×M32—171no—
291#444—172no—
292#444—172no—
293#443—342no—
294#442—172no—
295#441—172no—
296C4×SD16—342no—
297C2×#108—171no—
298C2×#106—171no—
299#421—341no—
300#417—341no—
301#405—171no—
302#365—172no—
303#365—172no—
304#362—172no—
305#361—172no—
306#360—172no—
307#360—172no—
308#347—172no—
309#347—172no—
310C2×C17—21no—
311C2×C17—81no—
312C2×C17—81no—
313C2×C17C2×#44211yescharacteristic
314D17—21no—
315D17—21no—
316D17—81no—
317D17—81no—
318C2×#41—341no—
319C2×#41—341no—
320#135—171no—
321#135—171no—
322#135—171no—
323#135—171no—
324#135—171no—
325#135—171no—
326#135—341no—
327#135—341no—
328#135—341no—
329#135—341no—
330#135C2×#59511yescharacteristic
331#135C2×#59511yescharacteristic, complemented
332C2×Q16—171no—
333C2×Q16—171no—
334C2×SD16—341no—
335C2×SD16—341no—
336C2×D8—171no—
337C2×D8—171no—
338#131—342no—
339#131—342no—
340#131—682no—
341C2×M16—171no—
342C2×M16—341no—
343C2×M16—681no—
344C2×M16—681no—
345C22×C8—171no—
346C4×Q8—342no—
347C4×D4—342no—
348C4×D4—682no—
349#117—171no—
350#117—681no—
351C2×C4⋊C4—341no—
352C2×C4⋊C4—681no—
353Q32—171no—
354Q32—171no—
355Q32—171no—
356Q32C2×#59511yescharacteristic, complemented
357SD32—171no—
358SD32—171no—
359SD32—171no—
360SD32—171no—
361SD32—171no—
362SD32—171no—
363SD32C2×#59511yescharacteristic, complemented
364SD32C2×#59511yescharacteristic, complemented
365D16—171no—
366D16—171no—
367D16—171no—
368D16C2×#59511yescharacteristic, complemented
369M32—171no—
370M32—171no—
371M32—171no—
372M32—171no—
373M32—172no—
374M32—341no—
375C2×C16—171no—
376C2×C16—341no—
377C2×C16C2×#59511yescharacteristic, complemented
378#108—172no—
379#108—341no—
380#106—172no—
381#106—341no—
382#104—342no—
383#104—342no—
384#103—342no—
385#102—342no—
386#97—172no—
387C4×C8—172no—
388C17#3346911yescharacteristic, complemented, Sylow
389#41—21no—
390#41—21no—
391#41—341no—
392#41—341no—
393#41—341no—
394#41—341no—
395#41—341no—
396#41—341no—
397#41—681no—
398#41—681no—
399#41—681no—
400#41—681no—
401C2×Q8—341no—
402C2×Q8—341no—
403C2×D4—341no—
404C2×D4—341no—
405C2×D4—681no—
406C2×D4—681no—
407C22×C4—171no—
408C22×C4—681no—
409C22×C4—681no—
410C22×C4—682no—
411C22×C4—681no—
412Q16—171no—
413Q16—171no—
414Q16—341no—
415Q16—341no—
416Q16C22×#59511yescharacteristic
417Q16C22×#59511yescharacteristic
418SD16—21no—
419SD16—21no—
420SD16—341no—
421SD16—341no—
422SD16—341no—
423SD16—341no—
424SD16—341no—
425SD16—341no—
426D8—171no—
427D8—171no—
428D8—341no—
429D8—341no—
430D8C22×#59511yescharacteristic
431D8C22×#59511yescharacteristic
432M16—341no—
433M16—341no—
434M16—342no—
435M16—682no—
436M16—682no—
437M16—681no—
438C2×C8—171no—
439C2×C8—171no—
440C2×C8—171no—
441C2×C8—171no—
442C2×C8—171no—
443C2×C8—172no—
444C2×C8—682no—
445C2×C8—682no—
446C2×C8—681no—
447C2×C8C22×#59511yescharacteristic
448C4⋊C4—341no—
449C4⋊C4—342no—
450C4⋊C4—681no—
451C4⋊C4—682no—
452C22⋊C4—341no—
453C22⋊C4—682no—
454C42—172no—
455C42—682no—
456C16—171no—
457C16—171no—
458C16—342no—
459C16C22×#59511yescharacteristic
460C16C22×#59511yescharacteristic
461C23—171no—
462C23—681no—
463C23—681no—
464Q8—21no—
465Q8—21no—
466Q8—341no—
467Q8—341no—
468Q8—681no—
469Q8—681no—
470D4—21no—
471D4—21no—
472D4—41no—
473D4—41no—
474D4—341no—
475D4—341no—
476D4—681no—
477D4—681no—
478D4—681no—
479D4—681no—
480D4—681no—
481D4—681no—
482D4—681no—
483D4—681no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—2722no—
487C2×C4—2721no—
488C2×C4—171no—
489C2×C4—171no—
490C2×C4—171no—
491C2×C4—171no—
492C2×C4—171no—
493C2×C4—342no—
494C2×C4—681no—
495C2×C4—681no—
496C2×C4—681no—
497C2×C4—681no—
498C2×C4—681no—
499C2×C4—681no—
500C2×C4—681no—
501C2×C4—681no—
502C2×C4—682no—
503C2×C4D4×#59511yescharacteristic
504C8—171no—
505C8—171no—
506C8—342no—
507C8—682no—
508C8D4×#59511yescharacteristic
509C8C23×#59511yescharacteristic
510C22—41no—
511C22—41no—
512C22—2721no—
513C22—171no—
514C22—171no—
515C22—681no—
516C22—681no—
517C22—681no—
518C22—1361no—
519C22—1361no—
520C22#1176385011yescharacteristic
521C4—41no—
522C4—41no—
523C4—171no—
524C4—171no—
525C4—682no—
526C4—681no—
527C4—681no—
528C4—1362no—
529C4#1176385011yescharacteristic
530C4C2×D4×#59511yescharacteristic
531C2—21no—
532C2—81no—
533C2—81no—
534C2—341no—
535C2—341no—
536C2—1361no—
537C2—1361no—
538C2C2×#1176385011yescharacteristic
subgroupG/H
centerC2C2×#11763850
commutatorC8×C17C23×C4
FrattiniC8C23×#595
FittingC17×#508C4
socleC2×C17C2×#442
prime radical#14541917C2
cosocleC8×D17C24
hypercenter#508#595
hypocenterC17#33469
LCSG ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #14545563C2semidirect
2. #14545547C2semidirect
3. #14545195C2semidirect
4. #14545194C2semidirect
5. #14545192C2semidirect
6. #14545189C2semidirect
7. C2×#11763734C2semidirect
8. #14545112C2semidirect
9. D17⋊#360C2semidirect
10. #14545003C2semidirect
18. #11763850C22semidirect
19. #11763849C22semidirect
26. #11763734C22semidirect
27. #11763734C22semidirect
28. #11763728C22semidirect
29. #11763727C22semidirect
32. #11763700C22semidirect
33. #11763699C22semidirect
36. #11763251C4semidirect
38. #11763248C4semidirect
40. #11763246C4semidirect
42. #11763243C4semidirect
44. #11763241C4semidirect
45. #11763239C4semidirect
46. #11763239C4semidirect
48. C17×#508C4semidirect
66. C17×#135C2×C4semidirect
71. Q16×D17C2×C4semidirect
79. D8×D17C2×C4semidirect
82. #10587590C2×C4semidirect
97. C17×Q32C2×C4semidirect
98. C17×SD32C2×C4semidirect
99. C17×SD32C2×C4semidirect
100. C17×D16C2×C4semidirect
101. C2×C16×C17C2×C4semidirect
102. #10587500C2×C4semidirect
103. #10587500C2×C4semidirect
104. #10587499C2×C4semidirect
105. #10587499C2×C4semidirect
106. #10587498C2×C4semidirect
107. #10587498C2×C4semidirect
108. #10587497C2×C4semidirect
109. #10587497C2×C4semidirect
110. C2×#63214C2×C4semidirect
111. Dic136C2×C4semidirect
112. #10587471C2×C4semidirect
113. #10587471C2×C4semidirect
114. D272C2×C4semidirect
115. #10587469C2×C4semidirect
116. #10587469C2×C4semidirect
288. #508#595semidirect
331. #135C2×#595semidirect
356. Q32C2×#595semidirect
363. SD32C2×#595semidirect
364. SD32C2×#595semidirect
368. D16C2×#595semidirect
377. C2×C16C2×#595semidirect
388. C17#33469semidirect

Maximal subgroups

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

subgroupindex
#145455632
#145455472
#145451952
#145451942
#145451922
#145451892
C2×#117637342
#145451122
D17⋊#3602
#145450032
#145419172
#3346917