Finite Group Databasebeta

← #17038696

Subgroups — #17038696

Order 4352.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762911C211yesmaximal, characteristic
2#14531430C211yesmaximal, characteristic, complemented
3#14531422C211yesmaximal, characteristic
4#14529663C211yesmaximal, characteristic
5#14529636C211yesmaximal, characteristic
6#14529520C211yesmaximal, characteristic, complemented
7#14529515C211yesmaximal, characteristic, complemented
8#14529479C211yesmaximal, characteristic
9#14529472C211yesmaximal, characteristic, complemented
10#14529465C211yesmaximal, characteristic
11#14529458C211yesmaximal, characteristic, complemented
12C4×#10587481C211yesmaximal, characteristic, complemented
13C4×#10587480C211yesmaximal, characteristic
14#14528472C211yesmaximal, characteristic, complemented
15C17×#1776C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19#11762911C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587487C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587492C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762714C2211yescharacteristic
33#11762714C2211yescharacteristic
34#11762710C2211yescharacteristic
35#11762710C2211yescharacteristic
36#11762705C2211yescharacteristic
37#11762705C2211yescharacteristic
38#11762705C2211yescharacteristic
39#11762705C2211yescharacteristic
40#11762700—21no—
41#11762700—21no—
42#11762693—21no—
43#11762693—21no—
44#11762676—21no—
45#11762673—21no—
46#11762671—21no—
47#11762668—21no—
48#11762655C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×Dic34C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C42×C17—21no—
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17C2311yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C22×#3622C2311yescharacteristic
88C22×#3622C2311yescharacteristic
89C2×#63219—22no—
90C2×#63219—22no—
91C2×#63219C2×C414yes—
92C4×D68—22no—
93C4×D68C2×C414yes—
94C4×Dic34—22no—
95C4×Dic34C2×C414yes—
96C17×#107—22no—
97C17×#106—22no—
98C17×#97C2×C414yes—
99C17×#95—21no—
100C17×#95—21no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502D411yescharacteristic
110#10587502D411yescharacteristic
111#10587502D411yescharacteristic
112#10587502D411yescharacteristic
113#10587492C2×C414yes—
114#10587489C2×C414yes—
115#10587487C2×C414yes—
116#10587486—22no—
117#10587486—22no—
118C8×Dic17—22no—
119C8×Dic17—22no—
120#10587481—22no—
121#10587481—22no—
122#10587480—22no—
123#10587480—22no—
124C4×#3622—22no—
125C23×D17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C2×D68—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68C22×C412yes—
146C2×C4×D17—24no—
147C2×C4×D17—81no—
148C2×C4×D17—81no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34C22×C412yes—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17C22×C412yes—
158C2×C8×C17C22×C412yes—
159C2×C8×C17#4112yes—
160C2×C8×C17#4112yes—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C42×C17—81no—
170C42×C17—81no—
171C42×C17C22×C412yes—
172C42×C17#4112yes—
173#63227—24no—
174#63227—81no—
175#63227—81no—
176#63227—81no—
177#63227—81no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17C22×C412yes—
181C4⋊Dic17C22×C412yes—
182#63225—24no—
183#63225—81no—
184#63225—81no—
185#63225—81no—
186#63225—81no—
187C4×Dic17—24no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622#4112yes—
197C2×#3622#4112yes—
198#63219—28no—
199#21291—171nomaximal, Sylow
200C23×C17C22×D411yescharacteristic
201C22×D17—22no—
202C22×D17—41no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17C4×D412yes—
225C2×C4×C17C4×D412yes—
226C2×C4×C17C23×C411yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×Dic17—22no—
233C2×Dic17—22no—
234C2×Dic17—22no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253D68—22no—
254D68—22no—
255D68—41no—
256D68—41no—
257C4×D17—84no—
258Dic34—22no—
259Dic34—22no—
260Dic34—41no—
261Dic34—41no—
262C8×C17—22no—
263C8×C17#11714yes—
264#3622—22no—
265#3622—22no—
266C2×#440—171no—
267#2320—171no—
268#2319—171no—
269#1973—171no—
270#1968—171no—
271#1943—171no—
272#1942—171no—
273#1936—171no—
274#1935—171no—
275#1934—171no—
276#1933—171no—
277#1776D1711yescharacteristic, complemented
278C4×#103—171no—
279C4×#102—171no—
280#1726—171no—
281C22×C17C2×C4×D411yescharacteristic
282C22×C17#53211yescharacteristic
283C22×C17#53211yescharacteristic
284C22×C17#57611yescharacteristic
285C22×C17#57711yescharacteristic
286C22×C17#57711yescharacteristic
287C22×C17#57811yescharacteristic
288C2×D17—42no—
289C2×D17—42no—
290C2×D17—42no—
291C2×D17—81no—
292C2×D17—81no—
293C2×D17—81no—
294C2×D17—81no—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—81no—
298C4×C17—81no—
299C4×C17—81no—
300C4×C17—81no—
301C4×C17C2×#11711yescharacteristic
302C4×C17C4×#4111yescharacteristic
303C4×C17#52911yescharacteristic
304C4×C17#52911yescharacteristic
305Dic17—41no—
306Dic17—41no—
307Dic17—41no—
308Dic17—41no—
309Dic17—42no—
310Dic17—42no—
311Dic17—42no—
312C22×SD16—171no—
313C2×C4×Q8—171no—
314C2×C4×D4—171no—
315#440—178no—
316C2×#107—171no—
317C2×#107C2×D1711yescharacteristic
318C2×#106C2×D1711yescharacteristic
319C2×#103—171no—
320C2×#103—171no—
321C2×#103—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×#102—171no—
325C2×#97—171no—
326C2×#97—171no—
327C2×#97C2×D1711yescharacteristic
328C2×C4×C8—171no—
329C2×C4×C8—171no—
330C2×C4×C8—171no—
331#391—171no—
332#391—171no—
333#390—171no—
334#390—171no—
335#389—171no—
336#389—171no—
337#389—171no—
338#389—171no—
339#388—341no—
340#388—341no—
341#387—341no—
342#387—341no—
343#383—341no—
344#382—341no—
345#381—341no—
346#380—341no—
347C4×C4⋊C4C2×D1711yescharacteristic
348C4×C4⋊C4C2×D1711yescharacteristic
349#376—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340C2×D1711yescharacteristic
355#340C2×D1711yescharacteristic
356C2×C17#229311yescharacteristic
357C2×C17#294611yescharacteristic
358C2×C17#294711yescharacteristic
359C2×C17#317411yescharacteristic
360C2×C17#317511yescharacteristic
361C2×C17#317811yescharacteristic
362C2×C17#317911yescharacteristic
363D17—82no—
364C22×Q8—171no—
365C22×D4—171no—
366C23×C4—341no—
367C2×SD16—174no—
368C2×SD16—342no—
369C2×SD16—342no—
370C22×C8—171no—
371C22×C8—171no—
372C22×C8C22×D1711yescharacteristic
373C22×C8C22×D1711yescharacteristic
374C4×Q8—174no—
375C4×Q8—342no—
376C4×D4—174no—
377C4×D4—342no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—341no—
399C2×C42C22×D1711yescharacteristic
400#107—22no—
401#107—174no—
402#106—22no—
403#103—174no—
404#103—342no—
405#103—342no—
406#102—174no—
407#102—342no—
408#102—342no—
409#97—342no—
410#97—342no—
411#97C4×D1714yes—
412C4×C8—342no—
413C4×C8—342no—
414C4×C8—342no—
415#95—21no—
416#95—21no—
417#95—171no—
418#95—171no—
419#95—171no—
420#95—171no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429C17#2129111yescharacteristic, complemented, Sylow
430C24—341no—
431C2×Q8—172no—
432C2×Q8—341no—
433C2×Q8—341no—
434C2×Q8—341no—
435C2×Q8—341no—
436C2×Q8—341no—
437C2×D4—172no—
438C2×D4—341no—
439C2×D4—341no—
440C2×D4—341no—
441C2×D4—341no—
442C2×D4—341no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—171no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—344no—
455C22×C4—1361no—
456C22×C4—1361no—
457C22×C4D4×D1711yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460SD16—348no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—172no—
464C2×C8—172no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8C2×C4×D1712yes—
470C2×C8C2×C4×D1712yes—
471C2×C8#6325212yes—
472C2×C8#6325212yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—172no—
482C4⋊C4—172no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—344no—
486C4⋊C4—1361no—
487C4⋊C4—1361no—
488C4⋊C4—1361no—
489C4⋊C4—1361no—
490C22⋊C4—344no—
491C22⋊C4—1361no—
492C22⋊C4—1361no—
493C22⋊C4—1361no—
494C22⋊C4—1361no—
495C42—81no—
496C42—81no—
497C42—341no—
498C42—341no—
499C42—341no—
500C42—341no—
501C42—344no—
502C42C2×C4×D1712yes—
503C42#6325612yes—
504C23—342no—
505C23—681no—
506C23—681no—
507C23—681no—
508C23—681no—
509C23—681no—
510C23—681no—
511C23C2×D4×D1711yescharacteristic
512Q8—342no—
513Q8—342no—
514Q8—681no—
515Q8—681no—
516D4—342no—
517D4—342no—
518D4—681no—
519D4—681no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—342no—
537C2×C4—342no—
538C2×C4—342no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—1364no—
558C2×C4#1058757912yes—
559C2×C4#1058757912yes—
560C2×C4C22×C4×D1711yescharacteristic
561C2×C4C2×#6325211yescharacteristic
562C2×C4C2×#6325211yescharacteristic
563C2×C4C2×#6325611yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C2×C4#41×D1711yescharacteristic
566C8—22no—
567C8—342no—
568C8—342no—
569C8#1058755814yes—
570C22—682no—
571C22—682no—
572C22—682no—
573C22—1361no—
574C22—1361no—
575C22—1361no—
576C22—1361no—
577C22C2×#1058757911yescharacteristic
578C22#1176335611yescharacteristic
579C22#1176335611yescharacteristic
580C22#1176361311yescharacteristic
581C22#1176361811yescharacteristic
582C22#1176361911yescharacteristic
583C22#1176362511yescharacteristic
584C4—22no—
585C4—22no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—682no—
591C4—682no—
592C4—681no—
593C4—681no—
594C4—682no—
595C4—681no—
596C4—681no—
597C4C2×#1058755811yescharacteristic
598C4#1176327711yescharacteristic
599C4#1176334611yescharacteristic
600C4#1176334611yescharacteristic
601C2—1362no—
602C2#1453124511yescharacteristic
603C2#1453761411yescharacteristic
604C2#1453762511yescharacteristic
605C2#1453941011yescharacteristic
606C2#1453942111yescharacteristic
607C2#1453945811yescharacteristic
608C2#1453947311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1776C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1776D17
hypocenterC17#21291
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. #14531430C2semidirect
6. #14529520C2semidirect
7. #14529515C2semidirect
9. #14529472C2semidirect
11. #14529458C2semidirect
12. C4×#10587481C2semidirect
14. #14528472C2semidirect
15. C17×#1776C2semidirect
19. #11762911C4semidirect
277. #1776D17semidirect
429. C17#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#117629112
#145314302
#145314222
#145296632
#145296362
#145295202
#145295152
#145294792
#145294722
#145294652
#145294582
C4×#105874812
C4×#105874802
#145284722
C17×#17762
#2129117