Finite Group Databasebeta

← #17005228

Subgroups — #17005228

Order 4352.

618 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538802C211yesmaximal, characteristic
2#14537798C211yesmaximal, characteristic, complemented
3#14537782C211yesmaximal, characteristic, complemented
4#14527678C211yesmaximal, characteristic, complemented
5#14527671C211yesmaximal, characteristic, complemented
6#14527637C211yesmaximal, characteristic, complemented
7#14527623C211yesmaximal, characteristic, complemented
8#14527617C211yesmaximal, characteristic, complemented
9#14527610C211yesmaximal, characteristic, complemented
10#14527587C211yesmaximal, characteristic, complemented
11#14527576C211yesmaximal, characteristic, complemented
12#14527562C211yesmaximal, characteristic, complemented
13#14527245C211yesmaximal, characteristic, complemented
14C8×#63223C211yesmaximal, characteristic, complemented
15C17×#1476C211yesmaximal, characteristic, complemented
16#11763343C2211yescharacteristic
17#11763150C411yescharacteristic
18#11763133C411yescharacteristic
19#11763116C411yescharacteristic
20#11763109C411yescharacteristic
21#11763031C411yescharacteristic
22#11763024C411yescharacteristic
23#11762971C411yescharacteristic
24#11762954C411yescharacteristic
25#11762940C2211yescharacteristic
26#11762940C2211yescharacteristic
27#11762936C2211yescharacteristic
28#11762933C2211yescharacteristic
29C4×#63231C2211yescharacteristic
30C4×#63230C2211yescharacteristic
31C4×#63229C2211yescharacteristic
32C4×#63228C2211yescharacteristic
33#11762886C2211yescharacteristic
34#11762884—21no—
35#11762883—21no—
36#11762881—21no—
37#11762881—21no—
38D4×#3622C2211yescharacteristic
39D4×#3622C2211yescharacteristic
40#11762874C2211yescharacteristic
41#11762868—21no—
42#11762866—21no—
43#11762866C2211yescharacteristic
44#11762862C2211yescharacteristic
45#11762763—21no—
46C4×#63223C2211yescharacteristic
47C2×#10587474—21no—
48#11762561C2211yescharacteristic
49#11762561C2211yescharacteristic
50#11762559C2211yescharacteristic
51#11762559C2211yescharacteristic
52#11762541C2211yescharacteristic
53#11762541C2211yescharacteristic
54#11762538C2211yescharacteristic
55#11762538C2211yescharacteristic
56#11762531C2211yescharacteristic
57#11762531C2211yescharacteristic
58C8×#3622C2211yescharacteristic
59C8×#3622C2211yescharacteristic
60C4×#41×C17C2211yescharacteristic
61C17×#431C2211yescharacteristic
62C2×C4×C8×C17C2211yescharacteristic
63C17×Q8⋊C8C2211yescharacteristic
64C17×Q8⋊C8C2211yescharacteristic
65C17×D4⋊C8C2211yescharacteristic
66C17×D4⋊C8C2211yescharacteristic
67C2×#41×C17C2×C411yescharacteristic
68C22×C8×C17—21no—
69C4×Q8×C17C2×C411yescharacteristic
70C4×Q8×C17C2311yescharacteristic
71C4×D4×C17—21no—
72C4×D4×C17—21no—
73C4×D4×C17C2×C411yescharacteristic
74C4×D4×C17C2311yescharacteristic
75C17×#117—21no—
76C17×#117C2×C411yescharacteristic
77C2×C42×C17—21no—
78C2×C42×C17C2311yescharacteristic
79C2×#63231C2×C411yescharacteristic
80C2×#63230C2×C411yescharacteristic
81C2×#63229C2×C411yescharacteristic
82C2×#63228C2×C411yescharacteristic
83#10587616C2×C411yescharacteristic
84#10587615—21no—
85C4×C22⋊D17—21no—
86#10587610C2×C411yescharacteristic
87C2×#63223D411yescharacteristic
88C2×#63223D411yescharacteristic
89C22×#3622—21no—
90C22×#3622—21no—
91C2×#63218—21no—
92C2×C8×D17—21no—
93#10587562—21no—
94#10587561C2×C411yescharacteristic
95C4×D68C2311yescharacteristic
96#10587558—21no—
97#10587556C2×C411yescharacteristic
98C4×Dic34C2311yescharacteristic
99C17×C4⋊C8C2311yescharacteristic
100C17×C4⋊C8C2311yescharacteristic
101C17×C22⋊C8—21no—
102C4×C8×C17—21no—
103C4×C8×C17—21no—
104C4×C8×C17C2311yescharacteristic
105C4×C8×C17C2311yescharacteristic
106#10587506C2×C411yescharacteristic
107#10587506C2×C411yescharacteristic
108#10587503C2×C411yescharacteristic
109#10587503C2×C411yescharacteristic
110C22⋊#3622—21no—
111C22⋊#3622—21no—
112C22⋊#3622—21no—
113C22⋊#3622C2×C411yescharacteristic
114C22⋊#3622C2×C411yescharacteristic
115#10587491—21no—
116#10587491—21no—
117#10587486—21no—
118#10587485—21no—
119#10587485—21no—
120C8×Dic17—21no—
121#10587481C2×C411yescharacteristic
122#10587481C2×C411yescharacteristic
123#10587480C2×C411yescharacteristic
124#10587480C2×C411yescharacteristic
125#10587479C2×C411yescharacteristic
126#10587479C2×C411yescharacteristic
127#10587478C2×C411yescharacteristic
128#10587478C2×C411yescharacteristic
129C4⋊#3622—21no—
130C4⋊#3622—21no—
131C4⋊#3622C2×C411yescharacteristic
132C4⋊#3622C2×C411yescharacteristic
133C4⋊#3622C2311yescharacteristic
134C4⋊#3622C2311yescharacteristic
135#10587474—21no—
136#10587474—21no—
137#10587474—21no—
138#10587474D411yescharacteristic
139#10587474D411yescharacteristic
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144#41×C17—22no—
145#41×C17—41no—
146C2×Q8×C17C22×C411yescharacteristic
147C2×D4×C17—21no—
148C2×D4×C17C22×C411yescharacteristic
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17C22×C411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17#4111yescharacteristic
155C2×C22⋊D17—21no—
156C2×D68C22×C411yescharacteristic
157C2×C4×D17—21no—
158C2×Dic34C22×C411yescharacteristic
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—42no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17C22×C411yescharacteristic
167C4⋊C4×C17C22×C411yescharacteristic
168C22⋊C4×C17—21no—
169C22⋊C4×C17—21no—
170C22⋊C4×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17C22×C411yescharacteristic
175C42×C17C2×D411yescharacteristic
176C42×C17#4111yescharacteristic
177C42×C17C2411yescharacteristic
178C22⋊Dic17—21no—
179#63231—41no—
180#63230—41no—
181#63229—22no—
182#63228—22no—
183#63227—21no—
184#63227—21no—
185#63227—21no—
186#63227—21no—
187C4⋊Dic17C22×C411yescharacteristic
188C4⋊Dic17C22×C411yescharacteristic
189#63225—21no—
190#63225—21no—
191#63225—21no—
192#63225—21no—
193C4×Dic17—21no—
194#63223—22no—
195#63223—22no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—42no—
199C2×#3622—42no—
200C2×#3622C22×C411yescharacteristic
201C2×#3622C22×C411yescharacteristic
202C2×#3622C22×C411yescharacteristic
203C2×#3622C22×C411yescharacteristic
204C2×#3622C2×D411yescharacteristic
205C2×#3622C2×D411yescharacteristic
206C2×#3622C2×D411yescharacteristic
207C2×#3622C2×D411yescharacteristic
208#63218—42no—
209C8×D17—42no—
210#17996—171nomaximal, Sylow
211C23×C17—21no—
212C23×C17C4×D411yescharacteristic
213C22×D17—21no—
214Q8×C17—21no—
215Q8×C17—21no—
216D4×C17—21no—
217D4×C17—42no—
218D4×C17#13112yes—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—41no—
227C2×C4×C17—42no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—81no—
231C2×C4×C17C4×D411yescharacteristic
232C2×C4×C17C4×D411yescharacteristic
233C2×C4×C17C4×D411yescharacteristic
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C22⋊D17—81no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241D68—21no—
242D68—21no—
243C4×D17—41no—
244C4×D17—41no—
245Dic34—21no—
246Dic34—21no—
247C8×C17—22no—
248C8×C17—41no—
249C8×C17—41no—
250C8×C17—41no—
251#3622—21no—
252#3622—21no—
253#3622—21no—
254#3622—21no—
255#3622—21no—
256#3622—21no—
257#3622—41no—
258#3622—41no—
259#3622#13512yes—
260#3622#13512yes—
261#3102—171no—
262#2970—171no—
263#2969—171no—
264#1591—171no—
265#1590—171no—
266#1584—171no—
267#1582—171no—
268#1582—171no—
269#1581—171no—
270C8×SD16—171no—
271C8×D8—171no—
272#1572—171no—
273#1494—171no—
274#1476D1711yescharacteristic, complemented
275C8×M16—171no—
276C22×C17—21no—
277C22×C17—22no—
278C22×C17—21no—
279C22×C17—21no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17C2×C4×D411yescharacteristic
283C2×D17—41no—
284C2×D17—41no—
285C2×D17—41no—
286C4×C17—21no—
287C4×C17—21no—
288C4×C17—21no—
289C4×C17—21no—
290C4×C17—41no—
291C4×C17—41no—
292C4×C17—41no—
293C4×C17C2×#13111yescharacteristic
294C4×C17#56811yescharacteristic
295C4×C17C2×#13511yescharacteristic
296C4×C17C2×#13511yescharacteristic
297Dic17—41no—
298Dic17—41no—
299Dic17—41no—
300#529—171no—
301C4×#41C2×D1711yescharacteristic
302#485—171no—
303#482—171no—
304#479—171no—
305#478—171no—
306#464—171no—
307#463—171no—
308#452—171no—
309#449—171no—
310#446—171no—
311#446—171no—
312#446—171no—
313C8×Q8—171no—
314C4×Q16—171no—
315C4×SD16—171no—
316C4×SD16—171no—
317C4×D8—171no—
318#436—171no—
319#436—341no—
320#435—341no—
321#435—341no—
322#435—341no—
323C8×D4—171no—
324C8×D4—171no—
325C8×D4—171no—
326#433—341no—
327#432—171no—
328#432—341no—
329#431—171no—
330#431C2×D1711yescharacteristic
331#405—341no—
332C4×M16—171no—
333C2×#97—341no—
334C2×C4×C8C2×D1711yescharacteristic
335#335—171no—
336#335—171no—
337#334—171no—
338#334—171no—
339Q8⋊C8—171no—
340Q8⋊C8—171no—
341Q8⋊C8C2×D1711yescharacteristic
342Q8⋊C8C2×D1711yescharacteristic
343D4⋊C8—171no—
344D4⋊C8—171no—
345D4⋊C8C2×D1711yescharacteristic
346D4⋊C8C2×D1711yescharacteristic
347#322—171no—
348#322—171no—
349C82—171no—
350C82—171no—
351C2×C17—41no—
352C2×C17—42no—
353C2×C17#293311yescharacteristic
354C2×C17C4×#13511yescharacteristic
355C2×C17#295611yescharacteristic
356D17—81no—
357C2×#41C4×D1711yescharacteristic
358C2×Q16—171no—
359C2×SD16—171no—
360C2×SD16—171no—
361C2×D8—171no—
362C2×M16—171no—
363C2×M16—171no—
364C2×M16—341no—
365C22×C8—21no—
366C22×C8—341no—
367C22×C8—341no—
368C22×C8—341no—
369#126—341no—
370#125—171no—
371#124—171no—
372#123—341no—
373C22⋊Q8—171no—
374#121—171no—
375C4×Q8—171no—
376C4×Q8C4×D1711yescharacteristic
377C4×Q8C22×D1711yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—171no—
381C4×D4—341no—
382C4×D4C4×D1711yescharacteristic
383C4×D4C22×D1711yescharacteristic
384#117—21no—
385#117—341no—
386#117C4×D1711yescharacteristic
387C2×C42—21no—
388C2×C42C22×D1711yescharacteristic
389#107—171no—
390#107—171no—
391#106—171no—
392#106—171no—
393C4⋊C8—171no—
394C4⋊C8—171no—
395C4⋊C8—171no—
396C4⋊C8—171no—
397C4⋊C8—341no—
398C4⋊C8—341no—
399C4⋊C8—341no—
400C4⋊C8—341no—
401C4⋊C8C22×D1711yescharacteristic
402C4⋊C8C22×D1711yescharacteristic
403#103—171no—
404#103—171no—
405#103—171no—
406#103—171no—
407#102—171no—
408#102—171no—
409#102—171no—
410#102—171no—
411C22⋊C8—21no—
412C22⋊C8—171no—
413C22⋊C8—171no—
414C22⋊C8—341no—
415C22⋊C8—341no—
416C22⋊C8—341no—
417C22⋊C8—341no—
418C22⋊C8—341no—
419#97—171no—
420#97—171no—
421#97—341no—
422#97—341no—
423#97—341no—
424#97—341no—
425C4×C8—21no—
426C4×C8—21no—
427C4×C8—171no—
428C4×C8—171no—
429C4×C8—171no—
430C4×C8—171no—
431C4×C8—341no—
432C4×C8C22×D1711yescharacteristic
433C4×C8C22×D1711yescharacteristic
434C17#1799611yescharacteristic, complemented, Sylow
435#41—22no—
436#41—41no—
437C2×Q8—171no—
438C2×Q8C2×C4×D1711yescharacteristic
439C2×D4—21no—
440C2×D4—171no—
441C2×D4—341no—
442C2×D4C2×C4×D1711yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—341no—
447C22×C4C2×C4×D1711yescharacteristic
448C22×C4D4×D1711yescharacteristic
449C22×C4#6325411yescharacteristic
450Q16—681no—
451SD16—342no—
452SD16—681no—
453D8—342no—
454M16—342no—
455M16—342no—
456M16—682no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—42no—
462C2×C8—171no—
463C2×C8—171no—
464C2×C8—171no—
465C2×C8—171no—
466C2×C8—171no—
467C2×C8—171no—
468C2×C8—171no—
469C2×C8—171no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—682no—
473C2×C8—682no—
474C2×C8—682no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—171no—
478C4⋊C4—171no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4C2×C4×D1711yescharacteristic
484C4⋊C4C2×C4×D1711yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—341no—
489C22⋊C4—341no—
490C22⋊C4—341no—
491C22⋊C4—341no—
492C22⋊C4—341no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—341no—
497C42C2×C4×D1711yescharacteristic
498C42D4×D1711yescharacteristic
499C42#6325411yescharacteristic
500C42C23×D1711yescharacteristic
501C23—21no—
502C23—341no—
503C23#1058756711yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—341no—
507Q8—341no—
508D4—21no—
509D4—42no—
510D4—341no—
511D4—341no—
512D4—1361no—
513D4#1058758712yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4#1058756711yescharacteristic
532C2×C4#1058756711yescharacteristic
533C2×C4#1058756711yescharacteristic
534C2×C4C22×C4×D1711yescharacteristic
535C2×C4C2×D4×D1711yescharacteristic
536C2×C4C2×#6325411yescharacteristic
537C8—22no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—172no—
542C8—172no—
543C8—341no—
544C8—341no—
545C8—341no—
546C8—341no—
547C8—341no—
548C8—341no—
549C8—681no—
550C8—681no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—681no—
558C22—681no—
559C22—681no—
560C22C2×#1058756711yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—681no—
569C4—681no—
570C4—681no—
571C4C2×#1058758711yescharacteristic
572C4#1176357511yescharacteristic
573C4#135×D1711yescharacteristic
574C4#135×D1711yescharacteristic
575C2—41no—
576C2—42no—
577C2—1361no—
578C2#1453750511yescharacteristic
579C2#1453755211yescharacteristic
580C2#1453769011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1476C2
socleC22×C17C2×C4×D4
prime radical#11762954C4
cosocleC42×C17C24
hypercenter#1476D17
hypocenterC17#17996
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #14537798C2semidirect
3. #14537782C2semidirect
4. #14527678C2semidirect
5. #14527671C2semidirect
6. #14527637C2semidirect
7. #14527623C2semidirect
8. #14527617C2semidirect
9. #14527610C2semidirect
10. #14527587C2semidirect
11. #14527576C2semidirect
12. #14527562C2semidirect
13. #14527245C2semidirect
14. C8×#63223C2semidirect
15. C17×#1476C2semidirect
274. #1476D17semidirect
434. C17#17996semidirect

Maximal subgroups

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

subgroupindex
#145388022
#145377982
#145377822
#145276782
#145276712
#145276372
#145276232
#145276172
#145276102
#145275872
#145275762
#145275622
#145272452
C8×#632232
C17×#14762
#1799617