Finite Group Databasebeta

← #17040680

Subgroups — #17040680

Order 4352.

980 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763032C211yesmaximal, characteristic
2C2×#11762917C211yesmaximal, characteristic, complemented
3C2×C4×#63219C211yesmaximal, characteristic
4#14531495C211yesmaximal, characteristic
5#14531147C211yesmaximal, characteristic, complemented
6#14529714C211yesmaximal, characteristic
7#14529548C211yesmaximal, characteristic, complemented
8#14529419C211yesmaximal, characteristic, complemented
9#14529412C211yesmaximal, characteristic
10#14529138C211yesmaximal, characteristic
11#14529124C211yesmaximal, characteristic, complemented
12#14528790C211yesmaximal, characteristic, complemented
13#14528610C211yesmaximal, characteristic
14C4×#10587487C211yesmaximal, characteristic, complemented
15C17×#1768C211yesmaximal, characteristic, complemented
16C22×Dic68C2211yescharacteristic
17C22×#63219C2211yescharacteristic
18C2×#10587561C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×Dic34C2211yescharacteristic
21C2×C4×Dic34C2211yescharacteristic
22C2×C4×Dic34C2211yescharacteristic
23C2×C4×D68C2211yescharacteristic
24#11763032C418yescomplemented
25#11762917—24no—
26C4×#63219—24no—
27C2×#10587488C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33C2×#10587487C2211yescharacteristic
34C2×#10587492C2211yescharacteristic
35C2×#10587492C2211yescharacteristic
36C2×#10587492C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762711C2211yescharacteristic
39#11762701—21no—
40#11762701—21no—
41#11762683—21no—
42#11762683—21no—
43#11762679C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47#11762660—21no—
48#11762660—21no—
49#11762660—21no—
50#11762654—21no—
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54#11762572C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×#97C2211yescharacteristic
58C2×C4×C8×C17C2211yescharacteristic
59C17×#376C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C17×#336C2211yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×Dic34C2311yescharacteristic
65C22×Dic34C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×#63227—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×Dic68—22no—
97C2×Dic68—22no—
98C2×Dic68C2×C414yescomplemented
99C2×#63219—22no—
100C2×#63219—22no—
101C2×#63219C2×C414yes—
102#10587561—22no—
103#10587561C2×C414yes—
104C4×D68—22no—
105C4×D68—22no—
106C4×Dic34—22no—
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—22no—
110C4×Dic34—22no—
111C4×Dic34—22no—
112C4×Dic34C2×C414yescomplemented
113C4×Dic34D414yes—
114C17×C4⋊C8—22no—
115C17×C4⋊C8C2×C414yescomplemented
116C17×#97—22no—
117C4×C8×C17—22no—
118C17×#95D411yescharacteristic
119C17×#95D411yescharacteristic
120C17×#95D411yescharacteristic
121C17×#95D411yescharacteristic
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587492—22no—
128#10587492—22no—
129#10587492C2×C414yes—
130#10587488—22no—
131#10587488—22no—
132#10587487—22no—
133#10587487—22no—
134#10587487—22no—
135#10587487—22no—
136#10587487C2×C414yescomplemented
137C23×D17—21no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68—21no—
155C2×D68—21no—
156C2×D68—21no—
157C2×D68C22×C412yes—
158C2×C4×D17—42no—
159C2×C4×D17—81no—
160C2×C4×D17—81no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34C22×C412yes—
170C2×Dic34C22×C412yes—
171C2×Dic34C2×D412yes—
172C2×Dic34#4112yes—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17C22×C412yes—
184C2×C8×C17C22×C412yes—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17C22×C412yes—
188C42×C17C2×D412yes—
189C42×C17#4112yes—
190C42×C17#4112yes—
191#63227—24no—
192#63227—24no—
193#63227—42no—
194#63227—81no—
195#63227—81no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—81no—
204C4⋊Dic17—81no—
205C4⋊Dic17C22×C412yes—
206C4⋊Dic17C2×D412yes—
207C4⋊Dic17#4112yes—
208#63225—24no—
209#63225—24no—
210#63225—24no—
211#63225—42no—
212C4×Dic17—24no—
213C4×Dic17—24no—
214C4×Dic17—24no—
215C4×Dic17—42no—
216C4×Dic17—81no—
217C4×Dic17—81no—
218C4×Dic17—81no—
219Dic68—28no—
220#63219—28no—
221#21511—171nomaximal, Sylow
222C23×C17C22×D411yescharacteristic
223C22×D17—22no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17C4×D412yes—
245C2×C4×C17C4×D412yes—
246C2×C4×C17#12112yes—
247C2×C4×C17#12112yes—
248C2×C4×C17C23×C411yescharacteristic
249C2×C4×C17C22×D411yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275D68—22no—
276D68—22no—
277D68—41no—
278D68—41no—
279C4×D17—84no—
280Dic34—22no—
281Dic34—22no—
282Dic34—22no—
283Dic34—22no—
284Dic34—22no—
285Dic34—41no—
286Dic34—41no—
287Dic34C4×D414yes—
288C8×C17—22no—
289C8×C17—22no—
290C8×C17—41no—
291C8×C17—41no—
292C2×#464—171no—
293C2×#441—171no—
294C2×C4×SD16—171no—
295#2326—171no—
296#2276—171no—
297#1983—171no—
298#1948—171no—
299#1927—171no—
300#1926—171no—
301#1880—171no—
302#1878—171no—
303#1803—171no—
304#1768D1711yescharacteristic, complemented
305#1762—171no—
306C4×#103—171no—
307C22×C17C2×C4×D411yescharacteristic
308C22×C17C2×#12111yescharacteristic
309C22×C17#52511yescharacteristic
310C22×C17#57711yescharacteristic
311C22×C17#57711yescharacteristic
312C22×C17#57711yescharacteristic
313C22×C17#57811yescharacteristic
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—81no—
318C2×D17—81no—
319C2×D17—81no—
320C2×D17—81no—
321C4×C17—22no—
322C4×C17—22no—
323C4×C17—22no—
324C4×C17—22no—
325C4×C17—41no—
326C4×C17—41no—
327C4×C17C2×C4×D411yescharacteristic
328C4×C17C4×#4111yescharacteristic
329C4×C17#52911yescharacteristic
330C4×C17#53211yescharacteristic
331Dic17—24no—
332Dic17—42no—
333Dic17—42no—
334Dic17—42no—
335Dic17—42no—
336Dic17—81no—
337Dic17—81no—
338C22×Q16—171no—
339C22×SD16—171no—
340C2×#124—171no—
341C2×C4×Q8—171no—
342C2×C4×Q8—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8—171no—
345C2×C4×D4—171no—
346#464—178no—
347#441—344no—
348C4×SD16—344no—
349C2×#106—171no—
350C2×#106—171no—
351C2×C4⋊C8C2×D1711yescharacteristic
352C2×C4⋊C8C2×D1711yescharacteristic
353C2×#103—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#102—171no—
359C2×#102—171no—
360C2×#102—171no—
361C2×#97C2×D1711yescharacteristic
362C2×C4×C8C2×D1711yescharacteristic
363#391—171no—
364#390—171no—
365#388—341no—
366#388—341no—
367#385—341no—
368#385—341no—
369#384—171no—
370C4×C4⋊C4—171no—
371C4×C4⋊C4—171no—
372C4×C4⋊C4—171no—
373C4×C4⋊C4—341no—
374C4×C4⋊C4—341no—
375C4×C4⋊C4—341no—
376#376—341no—
377#376C2×D1711yescharacteristic
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#336C2×D1711yescharacteristic
383#336C2×D1711yescharacteristic
384C2×C17C4×#12111yescharacteristic
385C2×C17#294711yescharacteristic
386C2×C17#295211yescharacteristic
387C2×C17#304311yescharacteristic
388C2×C17#304611yescharacteristic
389C2×C17#312111yescharacteristic
390C2×C17#312511yescharacteristic
391D17—82no—
392C22×Q8—171no—
393C22×Q8—171no—
394C22×D4—171no—
395C23×C4—341no—
396C2×Q16—174no—
397C2×Q16—342no—
398C2×Q16—342no—
399C2×SD16—174no—
400C2×SD16—342no—
401C2×SD16—342no—
402C22×C8C22×D1711yescharacteristic
403C22×C8C22×D1711yescharacteristic
404C22×C8C22×D1711yescharacteristic
405C22×C8C22×D1711yescharacteristic
406#124—174no—
407#124—342no—
408C4×Q8—174no—
409C4×Q8—174no—
410C4×Q8—342no—
411C4×Q8—342no—
412C4×Q8—342no—
413C4×Q8—342no—
414C4×Q8—342no—
415C4×Q8—342no—
416C4×D4—342no—
417C4×D4—342no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—341no—
435C2×C42—341no—
436C2×C42—341no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42—341no—
440C2×C42—341no—
441C2×C42C22×D1711yescharacteristic
442C2×C42C22×D1711yescharacteristic
443C2×C42C22×D1711yescharacteristic
444#106—342no—
445#106—342no—
446C4⋊C8—22no—
447C4⋊C8C4×D1714yescomplemented
448#103—174no—
449#103—342no—
450#103—342no—
451#103—342no—
452#103—342no—
453#102—174no—
454#102—342no—
455#102—342no—
456#97—22no—
457C4×C8—22no—
458#95—341no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95D6811yescharacteristic
464#95D6811yescharacteristic
465#95D6811yescharacteristic
466#95D6811yescharacteristic
467C17#2151111yescharacteristic, complemented, Sylow
468C24—341no—
469C2×Q8—172no—
470C2×Q8—172no—
471C2×Q8—172no—
472C2×Q8—172no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8—341no—
481C2×D4—172no—
482C2×D4—341no—
483C2×D4—341no—
484C2×D4—341no—
485C2×D4—341no—
486C2×D4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—682no—
495C22×C4—1361no—
496C22×C4—1361no—
497C22×C4C2×D6811yescharacteristic
498C22×C4C2×D6811yescharacteristic
499C22×C4C2×D6811yescharacteristic
500C22×C4C2×D6811yescharacteristic
501C22×C4C2×D6811yescharacteristic
502C22×C4C2×D6811yescharacteristic
503C22×C4C23×D1711yescharacteristic
504Q16—348no—
505SD16—348no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8C2×C4×D1712yes—
517C2×C8C2×C4×D1712yes—
518C4⋊C4—172no—
519C4⋊C4—172no—
520C4⋊C4—172no—
521C4⋊C4—344no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—344no—
526C4⋊C4—344no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—682no—
532C4⋊C4—1361no—
533C4⋊C4—1361no—
534C22⋊C4—344no—
535C22⋊C4—344no—
536C22⋊C4—682no—
537C22⋊C4—1361no—
538C22⋊C4—1361no—
539C42—21no—
540C42—21no—
541C42—344no—
542C42—344no—
543C42—344no—
544C42—682no—
545C42—1361no—
546C42—1361no—
547C42—1361no—
548C42C2×C4×D1712yes—
549C42#6325212yes—
550C42D4×D1712yes—
551C42#6325612yes—
552C23—342no—
553C23—681no—
554C23—681no—
555C23—681no—
556C23—681no—
557C23—681no—
558C23—681no—
559C23C22×D6811yescharacteristic
560Q8—174no—
561Q8—342no—
562Q8—342no—
563Q8—342no—
564Q8—342no—
565Q8—342no—
566Q8—681no—
567Q8—681no—
568D4—342no—
569D4—342no—
570D4—681no—
571D4—681no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—342no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—342no—
590C2×C4—342no—
591C2×C4—342no—
592C2×C4—342no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—1364no—
608C2×C4C4×D6812yes—
609C2×C4C4×D6812yes—
610C2×C4#1058758112yes—
611C2×C4#1058758112yes—
612C2×C4C22×C4×D1711yescharacteristic
613C2×C4C2×#6325211yescharacteristic
614C2×C4C2×D4×D1711yescharacteristic
615C2×C4C2×#6325611yescharacteristic
616C2×C4#41×D1711yescharacteristic
617C2×C4#41×D1711yescharacteristic
618C8—22no—
619C8—22no—
620C8—41no—
621C8—41no—
622C22—682no—
623C22—682no—
624C22—682no—
625C22—1361no—
626C22—1361no—
627C22—1361no—
628C22—1361no—
629C22C2×C4×D6811yescharacteristic
630C22C2×#1058758111yescharacteristic
631C22#1176332111yescharacteristic
632C22#1176362111yescharacteristic
633C22#1176362111yescharacteristic
634C22#1176362111yescharacteristic
635C22#1176362611yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—344no—
643C4—682no—
644C4—682no—
645C4—682no—
646C4—682no—
647C4—1361no—
648C4—1361no—
649C4C4×D4×D1711yescharacteristic
650C4C4×#6325611yescharacteristic
651C4#1176334611yescharacteristic
652C4#1176335611yescharacteristic
653C2—1362no—
654C2C4×#1058758111yescharacteristic
655C2#1453762911yescharacteristic
656C2#1453766911yescharacteristic
657C2#1453830611yescharacteristic
658C2#1453833611yescharacteristic
659C2#1453901511yescharacteristic
660C2#1453906311yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1768C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1768D17
hypocenterC17#21511
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
2. C2×#11762917C2semidirect
5. #14531147C2semidirect
7. #14529548C2semidirect
8. #14529419C2semidirect
11. #14529124C2semidirect
12. #14528790C2semidirect
14. C4×#10587487C2semidirect
15. C17×#1768C2semidirect
24. #11763032C4semidirect
98. C2×Dic68C2×C4semidirect
112. C4×Dic34C2×C4semidirect
115. C17×C4⋊C8C2×C4semidirect
136. #10587487C2×C4semidirect
304. #1768D17semidirect
447. C4⋊C8C4×D17semidirect
467. C17#21511semidirect

Maximal subgroups

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

subgroupindex
C2×#117630322
C2×#117629172
C2×C4×#632192
#145314952
#145311472
#145297142
#145295482
#145294192
#145294122
#145291382
#145291242
#145287902
#145286102
C4×#105874872
C17×#17682
#2151117