Finite Group Databasebeta

← #17005160

Subgroups — #17005160

Order 4352.

566 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539391C211yesmaximal, characteristic
2#14537933C211yesmaximal, characteristic
3#14537881C211yesmaximal, characteristic, complemented
4#14527665C211yesmaximal, characteristic, complemented
5#14527658C211yesmaximal, characteristic, complemented
6#14527652C211yesmaximal, characteristic, complemented
7#14527638C211yesmaximal, characteristic, complemented
8#14527637C211yesmaximal, characteristic, complemented
9#14527631C211yesmaximal, characteristic, complemented
10#14527630C211yesmaximal, characteristic, complemented
11#14527617C211yesmaximal, characteristic, complemented
12#14527552C211yesmaximal, characteristic, complemented
13#14527240C211yesmaximal, characteristic, complemented
14#14527073C211yesmaximal, characteristic, complemented
15C17×#1494C211yesmaximal, characteristic
16#11763366C2211yescharacteristic
17#11763219C411yescharacteristic
18#11763204C411yescharacteristic
19#11763179C411yescharacteristic
20#11763168C411yescharacteristic
21#11763065C411yescharacteristic
22#11763065C411yescharacteristic
23#11763053C411yescharacteristic
24#11763053C411yescharacteristic
25#11762940C2211yescharacteristic
26#11762937—21no—
27#11762937—21no—
28#11762936C2211yescharacteristic
29#11762936C2211yescharacteristic
30Q8×#3622C2211yescharacteristic
31#11762933C2211yescharacteristic
32#11762933C2211yescharacteristic
33C4×#63231C2211yescharacteristic
34C4×#63231C2211yescharacteristic
35C4×#63229C2211yescharacteristic
36C4×#63229C2211yescharacteristic
37#11762887—21no—
38#11762887C2211yescharacteristic
39#11762881—21no—
40D4×#3622C2211yescharacteristic
41#11762870—21no—
42#11762867—21no—
43#11762867—21no—
44#11762862C2211yescharacteristic
45#11762836—21no—
46#11762833C2211yescharacteristic
47C4×#63223C2211yescharacteristic
48#11762561C2211yescharacteristic
49#11762561C2211yescharacteristic
50#11762559C2211yescharacteristic
51#11762559C2211yescharacteristic
52#11762541C2211yescharacteristic
53#11762541C2211yescharacteristic
54#11762541C2211yescharacteristic
55#11762541C2211yescharacteristic
56#11762531C2211yescharacteristic
57#11762531C2211yescharacteristic
58#11762530C2211yescharacteristic
59#11762530C2211yescharacteristic
60C17×#529C2211yescharacteristic
61C17×#432C2211yescharacteristic
62C2×C4×C8×C17C2211yescharacteristic
63C17×Q8⋊C8C2211yescharacteristic
64C17×Q8⋊C8C2211yescharacteristic
65C17×D4⋊C8C2211yescharacteristic
66C17×D4⋊C8C2211yescharacteristic
67C22×C8×C17—21no—
68C17×#126—21no—
69C17×#125C2×C411yescharacteristic
70C17×#124C2×C411yescharacteristic
71C17×#123—21no—
72C17×C22⋊Q8C2×C411yescharacteristic
73C17×#121C2×C411yescharacteristic
74C4×Q8×C17C2311yescharacteristic
75C4×D4×C17—21no—
76C4×D4×C17C2311yescharacteristic
77C17×#117—21no—
78C2×C42×C17C2311yescharacteristic
79C2×#63231C2×C411yescharacteristic
80C2×#63231C2×C411yescharacteristic
81C2×#63229C2×C411yescharacteristic
82C2×#63229C2×C411yescharacteristic
83#10587612—21no—
84#10587610—21no—
85#10587610C2×C411yescharacteristic
86#10587610C2×C411yescharacteristic
87C2×#63223—21no—
88C2×#63223—21no—
89C2×#63223C2×C411yescharacteristic
90C2×#63223C2×C411yescharacteristic
91C22×#3622—21no—
92#10587556—21no—
93#10587555C2×C411yescharacteristic
94#10587555C2×C411yescharacteristic
95C4×Dic34—21no—
96C4×Dic34C2311yescharacteristic
97C4×Dic34C2311yescharacteristic
98C17×C4⋊C8C2311yescharacteristic
99C17×C4⋊C8C2311yescharacteristic
100C17×C22⋊C8—21no—
101C17×#97—21no—
102C4×C8×C17—21no—
103C4×C8×C17C2311yescharacteristic
104C4×C8×C17C2311yescharacteristic
105#10587506C2×C411yescharacteristic
106#10587506C2×C411yescharacteristic
107#10587503C2×C411yescharacteristic
108#10587503C2×C411yescharacteristic
109C22⋊#3622—21no—
110C22⋊#3622—21no—
111C22⋊#3622D411yescharacteristic
112C22⋊#3622D411yescharacteristic
113#10587486—21no—
114#10587486—21no—
115#10587485—21no—
116#10587485—21no—
117#10587485—21no—
118#10587485—21no—
119C8×Dic17—21no—
120C8×Dic17—21no—
121#10587481C2×C411yescharacteristic
122#10587481C2×C411yescharacteristic
123#10587481C2×C411yescharacteristic
124#10587481C2×C411yescharacteristic
125#10587479C2×C411yescharacteristic
126#10587479C2×C411yescharacteristic
127#10587478C2×C411yescharacteristic
128#10587478C2×C411yescharacteristic
129C4⋊#3622—21no—
130C4⋊#3622—21no—
131C4⋊#3622—21no—
132C4⋊#3622D411yescharacteristic
133C4⋊#3622D411yescharacteristic
134C4⋊#3622C2311yescharacteristic
135C4⋊#3622C2311yescharacteristic
136#10587474—21no—
137#10587474C2×C411yescharacteristic
138#10587474C2×C411yescharacteristic
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C2×Q8×C17C22×C411yescharacteristic
145C2×D4×C17—21no—
146C2×D4×C17C22×C411yescharacteristic
147C22×C4×C17—21no—
148C22×C4×C17C22×C411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17#4111yescharacteristic
151C2×Dic34—21no—
152C2×Dic34C22×C411yescharacteristic
153C2×Dic34C22×C411yescharacteristic
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—42no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17C22×C411yescharacteristic
164C4⋊C4×C17C22×C411yescharacteristic
165C22⋊C4×C17—21no—
166C22⋊C4×C17—21no—
167C22⋊C4×C17—21no—
168C22⋊C4×C17—21no—
169C22⋊C4×C17—21no—
170C42×C17—21no—
171C42×C17C22×C411yescharacteristic
172C42×C17C2×D411yescharacteristic
173C42×C17#4111yescharacteristic
174C42×C17C2411yescharacteristic
175C22⋊Dic17—21no—
176C22⋊Dic17—21no—
177#63231—41no—
178#63231—41no—
179#63229—41no—
180#63229—41no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17C22×C411yescharacteristic
185C4⋊Dic17C22×C411yescharacteristic
186#63225—21no—
187#63225—21no—
188#63225—21no—
189#63225—21no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192#63223—41no—
193#63223—41no—
194#63223—81no—
195#63223—81no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—81no—
199C2×#3622C22×C411yescharacteristic
200C2×#3622C22×C411yescharacteristic
201C2×#3622C22×C411yescharacteristic
202C2×#3622C22×C411yescharacteristic
203C2×#3622C2×D411yescharacteristic
204C2×#3622C2×D411yescharacteristic
205C2×#3622C2×D411yescharacteristic
206C2×#3622C2×D411yescharacteristic
207#17988—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17C4×D411yescharacteristic
210Q8×C17—21no—
211Q8×C17—21no—
212D4×C17—21no—
213D4×C17—21no—
214D4×C17—81no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17C4×D411yescharacteristic
226C2×C4×C17C4×D411yescharacteristic
227C2×C4×C17C4×D411yescharacteristic
228C2×C4×C17C23×C411yescharacteristic
229C2×C4×C17C22×D411yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235Dic34—21no—
236Dic34—21no—
237Dic34—21no—
238Dic34—21no—
239Dic34—81no—
240C8×C17—22no—
241C8×C17—41no—
242C8×C17—41no—
243C8×C17—41no—
244#3622—21no—
245#3622—21no—
246#3622—21no—
247#3622—21no—
248#3622—21no—
249#3622—21no—
250#3622—21no—
251#3622—21no—
252#3622—41no—
253#3622—41no—
254#3172—171no—
255#2993—171no—
256#2985—171no—
257#1588—171no—
258#1587—171no—
259#1587—171no—
260#1585—171no—
261#1584—171no—
262#1584—171no—
263#1583—171no—
264#1582—171no—
265#1569—171no—
266#1494D1711yescharacteristic
267#1493—171no—
268#1451—171no—
269C22×C17—21no—
270C22×C17—21no—
271C22×C17—21no—
272C22×C17—41no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17C2×C4×D411yescharacteristic
276C4×C17—21no—
277C4×C17—21no—
278C4×C17—21no—
279C4×C17—21no—
280C4×C17—41no—
281C4×C17—41no—
282C4×C17—41no—
283C4×C17#56811yescharacteristic
284C4×C17#56811yescharacteristic
285C4×C17#57711yescharacteristic
286C4×C17#57811yescharacteristic
287Dic17—41no—
288Dic17—41no—
289Dic17—41no—
290Dic17—41no—
291#533—171no—
292#529C2×D1711yescharacteristic
293#501—171no—
294#497—171no—
295#491—171no—
296#489—171no—
297#470—171no—
298#470—171no—
299#468—171no—
300#468—171no—
301#446—171no—
302#446—171no—
303#446—171no—
304#446—341no—
305#446—341no—
306C8×Q8—171no—
307C8×Q8—171no—
308C8×Q8—171no—
309C4×Q16—171no—
310C4×Q16—171no—
311C4×SD16—171no—
312C4×SD16—171no—
313#436—171no—
314#436—341no—
315#435—341no—
316C8×D4—171no—
317#433—341no—
318#433—341no—
319#433—341no—
320#432C2×D1711yescharacteristic
321#431—171no—
322#424—341no—
323#423—171no—
324C4×M16—171no—
325C2×C4×C8C2×D1711yescharacteristic
326#335—171no—
327#335—171no—
328#334—171no—
329#334—171no—
330Q8⋊C8—171no—
331Q8⋊C8—171no—
332Q8⋊C8—171no—
333Q8⋊C8—171no—
334Q8⋊C8C2×D1711yescharacteristic
335Q8⋊C8C2×D1711yescharacteristic
336D4⋊C8C2×D1711yescharacteristic
337D4⋊C8C2×D1711yescharacteristic
338#322—171no—
339#322—171no—
340#322—171no—
341#322—171no—
342C2×C17—41no—
343C2×C17—81no—
344C2×C17#293211yescharacteristic
345C2×C17#294711yescharacteristic
346C2×C17#295611yescharacteristic
347C2×Q16—171no—
348C2×Q16—171no—
349C2×SD16—171no—
350C2×SD16—171no—
351C2×M16—171no—
352C2×M16—171no—
353C2×M16—341no—
354C2×M16—341no—
355C22×C8—21no—
356C22×C8—341no—
357C4⋊Q8—171no—
358C4⋊Q8—171no—
359#126—21no—
360#125—341no—
361#125C4×D1711yescharacteristic
362#124C4×D1711yescharacteristic
363#123—21no—
364C22⋊Q8—171no—
365C22⋊Q8—171no—
366C22⋊Q8—341no—
367C22⋊Q8C4×D1711yescharacteristic
368#121C4×D1711yescharacteristic
369C4×Q8—171no—
370C4×Q8—171no—
371C4×Q8—341no—
372C4×Q8C22×D1711yescharacteristic
373C4×D4—21no—
374C4×D4C22×D1711yescharacteristic
375#117—21no—
376#117—341no—
377C2×C42C22×D1711yescharacteristic
378#107—171no—
379#107—171no—
380#106—171no—
381#106—171no—
382C4⋊C8—171no—
383C4⋊C8—171no—
384C4⋊C8—171no—
385C4⋊C8—171no—
386C4⋊C8—341no—
387C4⋊C8—341no—
388C4⋊C8—341no—
389C4⋊C8—341no—
390C4⋊C8—341no—
391C4⋊C8—341no—
392C4⋊C8—341no—
393C4⋊C8C22×D1711yescharacteristic
394C4⋊C8C22×D1711yescharacteristic
395#103—171no—
396#103—171no—
397#103—171no—
398#103—171no—
399#103—171no—
400#103—171no—
401#102—171no—
402#102—171no—
403C22⋊C8—21no—
404C22⋊C8—171no—
405C22⋊C8—171no—
406C22⋊C8—341no—
407C22⋊C8—341no—
408#97—21no—
409#97—171no—
410#97—171no—
411#97—341no—
412#97—341no—
413#97—341no—
414C4×C8—21no—
415C4×C8—171no—
416C4×C8—171no—
417C4×C8—171no—
418C4×C8—171no—
419C4×C8—341no—
420C4×C8—341no—
421C4×C8—341no—
422C4×C8C22×D1711yescharacteristic
423C4×C8C22×D1711yescharacteristic
424C17#1798811yescharacteristic, complemented, Sylow
425C2×Q8—171no—
426C2×Q8—171no—
427C2×Q8—341no—
428C2×Q8C2×C4×D1711yescharacteristic
429C2×D4—21no—
430C2×D4C2×C4×D1711yescharacteristic
431C22×C4—21no—
432C22×C4C2×C4×D1711yescharacteristic
433C22×C4D4×D1711yescharacteristic
434C22×C4#6325411yescharacteristic
435Q16—681no—
436Q16—681no—
437SD16—681no—
438SD16—681no—
439M16—681no—
440M16—681no—
441M16—1361no—
442M16—1361no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—171no—
449C2×C8—171no—
450C2×C8—171no—
451C2×C8—171no—
452C2×C8—171no—
453C2×C8—171no—
454C2×C8—171no—
455C2×C8—171no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—1361no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—171no—
464C4⋊C4—171no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4C2×C4×D1711yescharacteristic
473C4⋊C4C2×C4×D1711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—341no—
480C22⋊C4—341no—
481C42—21no—
482C42—341no—
483C42—341no—
484C42C2×C4×D1711yescharacteristic
485C42D4×D1711yescharacteristic
486C42#6325411yescharacteristic
487C42C23×D1711yescharacteristic
488C23—21no—
489C23#1058756711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—341no—
493Q8—341no—
494Q8—341no—
495Q8—341no—
496Q8—1361no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—341no—
511C2×C4—341no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4#1058756711yescharacteristic
515C2×C4#1058756711yescharacteristic
516C2×C4#1058756711yescharacteristic
517C2×C4C22×C4×D1711yescharacteristic
518C2×C4C2×D4×D1711yescharacteristic
519C2×C4C2×#6325411yescharacteristic
520C8—22no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—341no—
525C8—341no—
526C8—341no—
527C8—341no—
528C8—341no—
529C8—341no—
530C8—341no—
531C8—341no—
532C8—681no—
533C8—681no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×#1058756711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—681no—
549C4—681no—
550C4—681no—
551C4—681no—
552C4#1176357411yescharacteristic
553C4#1176357411yescharacteristic
554C4#1176361611yescharacteristic
555C4#1176362311yescharacteristic
556C2—41no—
557C2—81no—
558C2#1453749911yescharacteristic
559C2#1453763011yescharacteristic
560C2#1453769011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1494C2
socleC22×C17C2×C4×D4
prime radicalC17×#121C2×C4
cosocleC42×C17C24
hypercenter#1494D17
hypocenterC17#17988
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
3. #14537881C2semidirect
4. #14527665C2semidirect
5. #14527658C2semidirect
6. #14527652C2semidirect
7. #14527638C2semidirect
8. #14527637C2semidirect
9. #14527631C2semidirect
10. #14527630C2semidirect
11. #14527617C2semidirect
12. #14527552C2semidirect
13. #14527240C2semidirect
14. #14527073C2semidirect
424. C17#17988semidirect

Maximal subgroups

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

subgroupindex
#145393912
#145379332
#145378812
#145276652
#145276582
#145276522
#145276382
#145276372
#145276312
#145276302
#145276172
#145275522
#145272402
#145270732
C17×#14942
#1798817