Finite Group Databasebeta

← #17040725

Subgroups — #17040725

Order 4352.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763015C211yesmaximal, characteristic
2C2×#11762911C211yesmaximal, characteristic
3C2×C4×#63219C211yesmaximal, characteristic
4#14531503C211yesmaximal, characteristic, complemented
5#14531146C211yesmaximal, characteristic
6#14529706C211yesmaximal, characteristic, complemented
7#14529554C211yesmaximal, characteristic, complemented
8#14529426C211yesmaximal, characteristic, complemented
9#14529419C211yesmaximal, characteristic, complemented
10#14529117C211yesmaximal, characteristic
11#14529117C211yesmaximal, characteristic
12#14528795C211yesmaximal, characteristic
13#14528615C211yesmaximal, characteristic, complemented
14C4×#10587487C211yesmaximal, characteristic, complemented
15C17×#1768C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C22×#63219C2211yescharacteristic
18C2×#10587555C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×Dic34C2211yescharacteristic
21C2×C4×D68C2211yescharacteristic
22C2×C4×D68C2211yescharacteristic
23C2×C4×D68C2211yescharacteristic
24#11763015C418yescomplemented
25#11762911—24no—
26C4×#63219—24no—
27C2×#10587489C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33C2×#10587487C2211yescharacteristic
34C2×#10587487C2211yescharacteristic
35C2×#10587492C2211yescharacteristic
36C2×#10587492C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762694—21no—
40#11762694—21no—
41#11762694—21no—
42#11762694—21no—
43#11762675C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47#11762656—21no—
48#11762656—21no—
49#11762656—21no—
50#95×D17—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×C4×D17—21no—
65C22×C4×D17—21no—
66C22×Dic34C2311yescharacteristic
67C22×Dic34C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
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×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×#63219—22no—
93C2×#63219—22no—
94C2×#63219—22no—
95C2×#63219—22no—
96C2×#63219C2×C414yes—
97C2×#63219C2×C414yes—
98C4×D68—22no—
99C4×D68—22no—
100C4×D68—22no—
101C4×D68—22no—
102C4×D68C2×C414yes—
103C4×D68D414yes—
104#10587555—22no—
105#10587555C2×C414yescomplemented
106C4×Dic34—22no—
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—22no—
110C17×C4⋊C8—22no—
111C17×C4⋊C8C2×C414yescomplemented
112C17×#97—22no—
113C4×C8×C17—22no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116C17×#95D411yescharacteristic
117C17×#95D411yescharacteristic
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587492—22no—
128#10587492—22no—
129#10587489—22no—
130#10587488—22no—
131#10587487—22no—
132#10587487—22no—
133#10587487—22no—
134#10587487—22no—
135#10587487C2×C414yescomplemented
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×D68C22×C412yes—
156C2×D68C2×D412yes—
157C2×D68#4112yes—
158C2×C4×D17—24no—
159C2×C4×D17—24no—
160C2×C4×D17—24no—
161C2×C4×D17—81no—
162C2×C4×D17—81no—
163C2×C4×D17—81no—
164C2×C4×D17—81no—
165C2×C4×D17—81no—
166C2×C4×D17—81no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34C22×C412yes—
178C2×Dic34C22×C412yes—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17—21no—
186C2×C8×C17—21no—
187C2×C8×C17—21no—
188C2×C8×C17—21no—
189C2×C8×C17C22×C412yes—
190C2×C8×C17C22×C412yes—
191C42×C17—21no—
192C42×C17—21no—
193C42×C17C22×C412yes—
194C42×C17C2×D412yes—
195C42×C17#4112yes—
196C42×C17#4112yes—
197#63227—24no—
198#63227—24no—
199#63227—24no—
200C4⋊Dic17—24no—
201C4⋊Dic17—24no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17C22×C412yes—
210C4⋊Dic17C2×D412yes—
211C4⋊Dic17#4112yes—
212#63225—42no—
213#63225—42no—
214C4×Dic17—42no—
215C4×Dic17—42no—
216C4×Dic17—81no—
217C4×Dic17—81no—
218C4×Dic17—81no—
219#63219—28no—
220#63219—28no—
221#21514—171nomaximal, Sylow
222C23×C17C22×D411yescharacteristic
223C22×D17—22no—
224C22×D17—22no—
225C22×D17—22no—
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—41no—
260C2×Dic17—41no—
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—22no—
278D68C4×D414yes—
279C4×D17—84no—
280C4×D17—84no—
281C4×D17—84no—
282Dic34—22no—
283Dic34—22no—
284Dic34—22no—
285Dic34—22no—
286Dic34—41no—
287Dic34—41no—
288Dic34—41no—
289Dic34—41no—
290C8×C17—22no—
291C8×C17—22no—
292C8×C17—41no—
293C8×C17—41no—
294C2×#461—171no—
295C2×#440—171no—
296C2×C4×SD16—171no—
297#2327—171no—
298#2275—171no—
299#1981—171no—
300#1949—171no—
301#1928—171no—
302#1927—171no—
303#1877—171no—
304#1877—171no—
305#1804—171no—
306#1768D1711yescharacteristic, complemented
307#1763—171no—
308C4×#103—171no—
309C22×C17C2×C4×D411yescharacteristic
310C22×C17C2×#12111yescharacteristic
311C22×C17#52511yescharacteristic
312C22×C17#57711yescharacteristic
313C22×C17#57711yescharacteristic
314C22×C17#57711yescharacteristic
315C22×C17#57811yescharacteristic
316C2×D17—24no—
317C2×D17—42no—
318C2×D17—42no—
319C2×D17—42no—
320C2×D17—42no—
321C2×D17—42no—
322C2×D17—42no—
323C4×C17—22no—
324C4×C17—22no—
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17C2×C4×D411yescharacteristic
330C4×C17C4×#4111yescharacteristic
331C4×C17#52911yescharacteristic
332C4×C17#53211yescharacteristic
333Dic17—42no—
334Dic17—42no—
335Dic17—42no—
336Dic17—42no—
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340C22×SD16—171no—
341C22×SD16—171no—
342C2×C4⋊Q8—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8—171no—
345C2×C4×D4—171no—
346C2×C4×D4—171no—
347C2×C4×D4—171no—
348#461—178no—
349#440—344no—
350C4×SD16—344no—
351C2×#107—171no—
352C2×#106—171no—
353C2×C4⋊C8C2×D1711yescharacteristic
354C2×C4⋊C8C2×D1711yescharacteristic
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×#102—171no—
362C2×#102—171no—
363C2×#97C2×D1711yescharacteristic
364C2×C4×C8C2×D1711yescharacteristic
365#391—171no—
366#391—171no—
367#387—341no—
368#387—341no—
369#387—341no—
370#387—341no—
371#382—171no—
372C4×C4⋊C4—171no—
373C4×C4⋊C4—171no—
374C4×C4⋊C4—171no—
375C4×C22⋊C4—341no—
376C4×C22⋊C4—341no—
377C4×C22⋊C4—341no—
378#376C2×D1711yescharacteristic
379C2×#95—341no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#336C2×D1711yescharacteristic
385#336C2×D1711yescharacteristic
386C2×C17C4×#12111yescharacteristic
387C2×C17#294711yescharacteristic
388C2×C17#294911yescharacteristic
389C2×C17#304211yescharacteristic
390C2×C17#304611yescharacteristic
391C2×C17#312011yescharacteristic
392C2×C17#312311yescharacteristic
393D17—44no—
394C22×Q8—171no—
395C22×Q8—171no—
396C22×D4—171no—
397C23×C4—341no—
398C23×C4—341no—
399C23×C4—341no—
400C2×SD16—174no—
401C2×SD16—174no—
402C2×SD16—342no—
403C2×SD16—342no—
404C2×SD16—342no—
405C2×SD16—342no—
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408C22×C8C22×D1711yescharacteristic
409C22×C8C22×D1711yescharacteristic
410C4⋊Q8—174no—
411C4⋊Q8—342no—
412C4×Q8—342no—
413C4×Q8—342no—
414C4×Q8—342no—
415C4×Q8—342no—
416C4×D4—174no—
417C4×D4—174no—
418C4×D4—342no—
419C4×D4—342no—
420C4×D4—342no—
421C4×D4—342no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊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×C42C22×D1711yescharacteristic
440C2×C42C22×D1711yescharacteristic
441C2×C42C22×D1711yescharacteristic
442#107—342no—
443#106—342no—
444C4⋊C8—22no—
445C4⋊C8C4×D1714yescomplemented
446#103—174no—
447#103—174no—
448#103—342no—
449#103—342no—
450#103—342no—
451#103—342no—
452#102—342no—
453#102—342no—
454#97—22no—
455C4×C8—22no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95D6811yescharacteristic
466#95D6811yescharacteristic
467#95D6811yescharacteristic
468#95D6811yescharacteristic
469C17#2151411yescharacteristic, complemented, Sylow
470C24—341no—
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×Q8—341no—
482C2×Q8—341no—
483C2×D4—172no—
484C2×D4—172no—
485C2×D4—172no—
486C2×D4—341no—
487C2×D4—341no—
488C2×D4—341no—
489C22×C4—344no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—341no—
495C22×C4—344no—
496C22×C4—344no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—1361no—
500C22×C4—1361no—
501C22×C4—1361no—
502C22×C4—1361no—
503C22×C4—1361no—
504C22×C4—1361no—
505C22×C4C2×D6811yescharacteristic
506C22×C4C2×D6811yescharacteristic
507C22×C4C2×D6811yescharacteristic
508C22×C4C2×D6811yescharacteristic
509C22×C4C2×D6811yescharacteristic
510C22×C4C2×D6811yescharacteristic
511C22×C4C23×D1711yescharacteristic
512SD16—348no—
513SD16—348no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8C2×C4×D1712yes—
525C2×C8C2×C4×D1712yes—
526C4⋊C4—172no—
527C4⋊C4—172no—
528C4⋊C4—172no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—344no—
533C4⋊C4—344no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—682no—
539C4⋊C4—682no—
540C22⋊C4—344no—
541C22⋊C4—344no—
542C22⋊C4—344no—
543C42—21no—
544C42—21no—
545C42—682no—
546C42—682no—
547C42—1361no—
548C42—1361no—
549C42—1361no—
550C42C2×C4×D1712yes—
551C42#6325212yes—
552C42D4×D1712yes—
553C42#6325612yes—
554C23—342no—
555C23—342no—
556C23—342no—
557C23—681no—
558C23—681no—
559C23—681no—
560C23—681no—
561C23C22×D6811yescharacteristic
562Q8—342no—
563Q8—342no—
564Q8—342no—
565Q8—342no—
566Q8—681no—
567Q8—681no—
568Q8—681no—
569Q8—681no—
570D4—174no—
571D4—342no—
572D4—342no—
573D4—342no—
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—21no—
587C2×C4—21no—
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—681no—
608C2×C4—681no—
609C2×C4—1364no—
610C2×C4—1364no—
611C2×C4—1364no—
612C2×C4C4×D6812yes—
613C2×C4C4×D6812yes—
614C2×C4#1058758112yes—
615C2×C4#1058758112yes—
616C2×C4C22×C4×D1711yescharacteristic
617C2×C4C2×#6325211yescharacteristic
618C2×C4C2×D4×D1711yescharacteristic
619C2×C4C2×#6325611yescharacteristic
620C2×C4#41×D1711yescharacteristic
621C2×C4#41×D1711yescharacteristic
622C8—22no—
623C8—22no—
624C8—41no—
625C8—41no—
626C22—344no—
627C22—682no—
628C22—682no—
629C22—682no—
630C22—682no—
631C22—682no—
632C22—682no—
633C22C2×C4×D6811yescharacteristic
634C22C2×#1058758111yescharacteristic
635C22#1176332111yescharacteristic
636C22#1176362111yescharacteristic
637C22#1176362111yescharacteristic
638C22#1176362111yescharacteristic
639C22#1176362611yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—682no—
647C4—682no—
648C4—682no—
649C4—682no—
650C4—682no—
651C4—1361no—
652C4—1361no—
653C4C4×D4×D1711yescharacteristic
654C4C4×#6325611yescharacteristic
655C4#1176334611yescharacteristic
656C4#1176335611yescharacteristic
657C2—684no—
658C2C4×#1058758111yescharacteristic
659C2#1453762911yescharacteristic
660C2#1453764511yescharacteristic
661C2#1453829711yescharacteristic
662C2#1453833611yescharacteristic
663C2#1453900011yescharacteristic
664C2#1453903711yescharacteristic
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#21514
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
4. #14531503C2semidirect
6. #14529706C2semidirect
7. #14529554C2semidirect
8. #14529426C2semidirect
9. #14529419C2semidirect
13. #14528615C2semidirect
14. C4×#10587487C2semidirect
15. C17×#1768C2semidirect
24. #11763015C4semidirect
105. #10587555C2×C4semidirect
111. C17×C4⋊C8C2×C4semidirect
135. #10587487C2×C4semidirect
136. #10587487C2×C4semidirect
306. #1768D17semidirect
445. C4⋊C8C4×D17semidirect
469. C17#21514semidirect

Maximal subgroups

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

subgroupindex
C2×#117630152
C2×#117629112
C2×C4×#632192
#145315032
#145311462
#145297062
#145295542
#145294262
#145294192
#145291172
#145287952
#145286152
C4×#105874872
C17×#17682
#2151417