Finite Group Databasebeta

← #16940757

Subgroups — #16940757

Order 4352.

966 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762827C211yesmaximal, characteristic
2C2×#11762811C211yesmaximal, characteristic
3C2×#11762782C211yesmaximal, characteristic
4#14531140C211yesmaximal, characteristic, complemented
5C2×#11762580C211yesmaximal, characteristic, complemented
6#14528505C211yesmaximal, characteristic, complemented
7C2×C17×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#63252C2211yescharacteristic
9C22×#63223C2211yescharacteristic
10C2×#10587612C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12#11762827—22no—
13#11762827C414yescomplemented
14C2×#10587476—22no—
15#11762811—22no—
16#11762811—22no—
17C2×#10587487C412yescomplemented
18C2×#10587492C412yescomplemented
19#11762782—22no—
20#11762782C414yescomplemented
21C2×#10587477—22no—
22C4⋊C4×Dic17—22no—
23C4×C4⋊Dic17—22no—
24C22⋊C4×Dic17—22no—
25#11762580—22no—
26#11762571—22no—
27C22×M16×C17C2211yescharacteristic
28C2×C17×#117C2211yescharacteristic
29C2×C17×#117C2211yescharacteristic
30C17×H(3,ℤ/4)—22no—
31C23×C4×C17C2311yescharacteristic
32C22×C22⋊D17—21no—
33C23×Dic17—21no—
34C2×#63252—21no—
35C2×#63252—21no—
36C2×#63252—21no—
37C2×#63252—21no—
38C2×#63252—21no—
39C2×#63252—21no—
40C2×#63252—21no—
41C2×#63252—21no—
42C2×#63252—21no—
43C2×#63252—21no—
44C2×#63252C2×C412yes—
45C2×#63252C2×C412yes—
46C22×D68C2×C411yescharacteristic
47C22×C4×D17—21no—
48C22×Dic34C2×C411yescharacteristic
49C2×M16×C17—21no—
50C2×M16×C17—21no—
51C2×M16×C17—21no—
52C2×M16×C17—21no—
53C2×M16×C17—21no—
54C2×M16×C17—21no—
55C22×C8×C17C2×C412yes—
56C17×#117—21no—
57C17×#117—21no—
58C17×#117—21no—
59C17×#117—21no—
60C17×#117—21no—
61C17×#117C2×C412yes—
62C17×#117C2×C412yes—
63C17×#117D412yes—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C22⋊C4×C17—21no—
67C2×C22⋊C4×C17—21no—
68C2×C42×C17—21no—
69C2×C42×C17D412yes—
70C2×C22⋊Dic17D412yes—
71#10587612—21no—
72#10587612—21no—
73#10587612C2×C412yes—
74#10587612D412yes—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2×C412yes—
80C2×C4⋊Dic17D412yes—
81C2×C4⋊Dic17D412yes—
82C2×C4×Dic17—24no—
83C2×C4×Dic17—21no—
84C2×#63223—21no—
85C2×#63223—21no—
86C2×#63223—21no—
87C2×#63223—21no—
88C2×#63223—21no—
89C2×#63223C2×C412yes—
90C22×#3622—21no—
91#10587502—24no—
92#10587492—24no—
93#10587487—24no—
94#10587477—28no—
95#10587476—28no—
96C24×C17C2×D411yescharacteristic
97C23×D17—21no—
98C22×C4×C17—22no—
99C22×C4×C17—41no—
100C22×C4×C17C22×C411yescharacteristic
101C22×C4×C17C22×C411yescharacteristic
102C22×C4×C17C22×C411yescharacteristic
103C22×C4×C17C2×D411yescharacteristic
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17C2×D411yescharacteristic
107C22×C4×C17C2×D411yescharacteristic
108C22×C4×C17#4111yescharacteristic
109C22×C4×C17#4111yescharacteristic
110C22×C4×C17#4111yescharacteristic
111C22×C4×C17#4111yescharacteristic
112C22×C4×C17#4111yescharacteristic
113C22×C4×C17#4111yescharacteristic
114C2×C22⋊D17—22no—
115C2×C22⋊D17—22no—
116C2×C22⋊D17—41no—
117C2×C22⋊D17—41no—
118C2×C22⋊D17—41no—
119C2×C22⋊D17—41no—
120C22×Dic17—21no—
121C22×Dic17—22no—
122C22×Dic17—22no—
123C22×Dic17—22no—
124C22×Dic17—22no—
125C22×Dic17—82no—
126#63252—22no—
127#63252—22no—
128#63252—22no—
129#63252—22no—
130#63252—22no—
131#63252—22no—
132#63252—22no—
133#63252—22no—
134#63252—41no—
135#63252—41no—
136#63252—41no—
137#63252—41no—
138#63252—41no—
139#63252—41no—
140#63252—41no—
141#63252—41no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68C4212yescomplemented
148C2×C4×D17—22no—
149C2×C4×D17—22no—
150C2×C4×D17—41no—
151C2×C4×D17—41no—
152C2×C4×D17—41no—
153C2×C4×D17—41no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34C4212yescomplemented
160M16×C17—41no—
161M16×C17—41no—
162M16×C17—41no—
163M16×C17—41no—
164C2×C8×C17—22no—
165C2×C8×C17—22no—
166C2×C8×C17—22no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—41no—
171C22⋊C4×C17—22no—
172C22⋊C4×C17—22no—
173C22⋊C4×C17—22no—
174C22⋊C4×C17—41no—
175C42×C17—22no—
176C42×C17—22no—
177C42×C17—41no—
178C42×C17C22⋊C414yes—
179C22⋊Dic17—22no—
180C22⋊Dic17C22⋊C414yes—
181C4⋊Dic17—22no—
182C4⋊Dic17—22no—
183C4⋊Dic17—22no—
184C4⋊Dic17—22no—
185C4⋊Dic17—22no—
186C4⋊Dic17C22⋊C414yes—
187C4×Dic17—22no—
188C4×Dic17—22no—
189C4×Dic17—84no—
190#63223—22no—
191#63223—22no—
192#63223—41no—
193#63223—41no—
194C2×#3622—22no—
195C2×#3622—41no—
196C2×#3622—41no—
197#10954—171nomaximal, Sylow
198C23×C17—41no—
199C23×C17—41no—
200C23×C17C2×C22⋊C411yescharacteristic
201C23×C17C4×D411yescharacteristic
202C23×C17C4×D411yescharacteristic
203C23×C17#12011yescharacteristic
204C23×C17#12111yescharacteristic
205C23×C17#12111yescharacteristic
206C23×C17#12311yescharacteristic
207C22×D17—22no—
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—81no—
211C22×D17—81no—
212C2×C4×C17—24no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—81no—
217C2×C4×C17—81no—
218C2×C4×C17C2×C4211yescharacteristic
219C2×C4×C17C2×C22⋊C411yescharacteristic
220C2×C4×C17C2×C22⋊C411yescharacteristic
221C2×C4×C17#11711yescharacteristic
222C2×C4×C17#11711yescharacteristic
223C2×C4×C17#11711yescharacteristic
224C2×C4×C17C4×D411yescharacteristic
225C2×C4×C17C4×D411yescharacteristic
226C2×C4×C17C4×D411yescharacteristic
227C2×C4×C17C4×D411yescharacteristic
228C2×C4×C17C4×D411yescharacteristic
229C2×C4×C17C4×D411yescharacteristic
230C2×C4×C17C4×D411yescharacteristic
231C2×C4×C17C4×D411yescharacteristic
232C2×C4×C17C4×D411yescharacteristic
233C2×C4×C17C4×D411yescharacteristic
234C2×C4×C17#12111yescharacteristic
235C2×C4×C17#12111yescharacteristic
236C2×C4×C17#12111yescharacteristic
237C2×C4×C17#12111yescharacteristic
238C2×C4×C17#12311yescharacteristic
239C2×C4×C17#12311yescharacteristic
240C2×C4×C17#12411yescharacteristic
241C2×C4×C17#12411yescharacteristic
242C2×C4×C17#12411yescharacteristic
243C2×C4×C17#12611yescharacteristic
244C2×C4×C17#12611yescharacteristic
245C2×C4×C17#12711yescharacteristic
246C22⋊D17—42no—
247C22⋊D17—42no—
248C22⋊D17—42no—
249C22⋊D17—42no—
250C2×Dic17—22no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—42no—
254C2×Dic17—44no—
255C2×Dic17—42no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—82no—
259C2×Dic17—82no—
260D68—22no—
261D68—22no—
262D68—41no—
263D68—41no—
264C4×D17—42no—
265C4×D17—42no—
266C4×D17—42no—
267C4×D17—42no—
268Dic34—22no—
269Dic34—22no—
270Dic34—41no—
271Dic34—41no—
272C8×C17—42no—
273#3622—42no—
274C2×#421—171no—
275C2×#417—171no—
276C2×#410—171no—
277#2274—171no—
278C2×#344—171no—
279#1736—171no—
280C2×H(3,ℤ/4)D1711yescharacteristic, complemented
281C22×C17—21no—
282C22×C17—21no—
283C22×C17—21no—
284C22×C17—21no—
285C22×C17—21no—
286C22×C17—21no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17C4×C22⋊C411yescharacteristic
292C22×C17#38611yescharacteristic
293C22×C17#38611yescharacteristic
294C22×C17#38611yescharacteristic
295C22×C17#38811yescharacteristic
296C22×C17#38811yescharacteristic
297C22×C17#39011yescharacteristic
298C2×D17—42no—
299C2×D17—82no—
300C2×D17—81no—
301C2×D17—81no—
302C2×D17—82no—
303C4×C17—44no—
304C4×C17—82no—
305C4×C17C4×C22⋊C411yescharacteristic
306C4×C17C4×C22⋊C411yescharacteristic
307C4×C17#38811yescharacteristic
308C4×C17#38811yescharacteristic
309C4×C17#38811yescharacteristic
310C4×C17#38811yescharacteristic
311C4×C17#39011yescharacteristic
312C4×C17#39011yescharacteristic
313Dic17—44no—
314Dic17—44no—
315Dic17—42no—
316C22×#41—171no—
317C22×M16—171no—
318C22×M16C2×D1711yescharacteristic
319C2×#117—171no—
320C2×#117C2×D1711yescharacteristic
321C2×#117C2×D1711yescharacteristic
322C22×C4⋊C4—171no—
323#421—174no—
324#421—342no—
325C2×#104—342no—
326#417—342no—
327#417—342no—
328C2×#103—172no—
329C2×#102—172no—
330#410—174no—
331#410—342no—
332C2×#99—342no—
333C4×C4⋊C4—342no—
334C4×C4⋊C4—342no—
335C4×C22⋊C4—342no—
336#344—342no—
337#340—342no—
338H(3,ℤ/4)—22no—
339C2×C17—21no—
340C2×C17—21no—
341C2×C17—21no—
342C2×C17—21no—
343C2×C17—21no—
344C2×C17—21no—
345C2×C17#143711yescharacteristic
346C2×C17#190411yescharacteristic
347C2×C17#190511yescharacteristic
348D17—82no—
349C2×#41—172no—
350C2×#41—172no—
351C2×#41—341no—
352C2×#41—341no—
353C2×#41—341no—
354C2×#41—341no—
355C2×#41—341no—
356C2×#41—341no—
357C2×#41—341no—
358C2×#41—341no—
359C2×#41—341no—
360C2×#41—341no—
361C22×Q8—171no—
362C22×D4—171no—
363C22×D4—341no—
364C23×C4—341no—
365C23×C4—341no—
366C23×C4C22×D1711yescharacteristic
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—172no—
374C2×M16—341no—
375C2×M16—341no—
376C2×M16—341no—
377C2×M16—341no—
378C2×M16—341no—
379C22×C8—341no—
380C22×C8C4×D1712yes—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—172no—
387#117—172no—
388#117—341no—
389#117—341no—
390#117C4×D1712yes—
391#117C4×D1712yes—
392#117C22⋊D1712yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—172no—
396C2×C4⋊C4—172no—
397C2×C4⋊C4—172no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—172no—
405C2×C42—21no—
406C2×C42—341no—
407C2×C42—344no—
408C2×C42D6812yes—
409#104—348no—
410#103—344no—
411#102—344no—
412#99—348no—
413#95—344no—
414C17#1095411yescharacteristic, complemented, Sylow
415C24—341no—
416C24D4×D1711yescharacteristic
417#41—342no—
418#41—342no—
419#41—342no—
420#41—342no—
421#41—342no—
422#41—342no—
423#41—342no—
424#41—342no—
425#41—681no—
426#41—681no—
427#41—681no—
428#41—681no—
429#41—681no—
430#41—681no—
431#41—681no—
432#41—681no—
433C2×Q8—172no—
434C2×Q8—341no—
435C2×Q8—341no—
436C2×Q8—341no—
437C2×Q8—341no—
438C2×Q8—341no—
439C2×D4—172no—
440C2×D4—342no—
441C2×D4—342no—
442C2×D4—341no—
443C2×D4—341no—
444C2×D4—341no—
445C2×D4—341no—
446C2×D4—341no—
447C2×D4—681no—
448C2×D4—681no—
449C2×D4—681no—
450C2×D4—681no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—341no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—681no—
461C22×C4—681no—
462C22×C4—681no—
463C22×C4—681no—
464C22×C4—1362no—
465C22×C4C2×C4×D1711yescharacteristic
466C22×C4C2×C4×D1711yescharacteristic
467C22×C4C2×C4×D1711yescharacteristic
468C22×C4C2×D6811yescharacteristic
469C22×C4#6325211yescharacteristic
470C22×C4#6325211yescharacteristic
471C22×C4D4×D1711yescharacteristic
472C22×C4D4×D1711yescharacteristic
473C22×C4D4×D1711yescharacteristic
474C22×C4#6325411yescharacteristic
475C22×C4#6325411yescharacteristic
476C22×C4#6325611yescharacteristic
477C22×C4#6325611yescharacteristic
478C22×C4C2×C22⋊D1711yescharacteristic
479M16—41no—
480M16—41no—
481M16—41no—
482M16—41no—
483M16—342no—
484M16—342no—
485M16—681no—
486M16—681no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—342no—
491C2×C8—681no—
492C2×C8—681no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—174no—
498C4⋊C4—342no—
499C4⋊C4—342no—
500C4⋊C4—342no—
501C4⋊C4—342no—
502C4⋊C4—342no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—41no—
507C22⋊C4—174no—
508C22⋊C4—342no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—342no—
513C42—342no—
514C42—1364no—
515C42#6322714yes—
516C23—41no—
517C23—41no—
518C23—342no—
519C23—681no—
520C23—681no—
521C23—1361no—
522C23—1361no—
523C23C22⋊C4×D1711yescharacteristic
524C23#1058756711yescharacteristic
525C23#1058756811yescharacteristic
526C23#1058757011yescharacteristic
527C23#1058757911yescharacteristic
528C23#1058758011yescharacteristic
529C23#1058762511yescharacteristic
530Q8—342no—
531Q8—342no—
532Q8—681no—
533Q8—681no—
534D4—342no—
535D4—342no—
536D4—682no—
537D4—682no—
538D4—682no—
539D4—682no—
540D4—681no—
541D4—681no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—342no—
549C2×C4—344no—
550C2×C4—344no—
551C2×C4—682no—
552C2×C4—684no—
553C2×C4—682no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4—1362no—
561C2×C4—1362no—
562C2×C4C42×D1711yescharacteristic
563C2×C4#1058755811yescharacteristic
564C2×C4C4×D6811yescharacteristic
565C2×C4C4×D6811yescharacteristic
566C2×C4C22⋊C4×D1711yescharacteristic
567C2×C4#1058756711yescharacteristic
568C2×C4#1058756711yescharacteristic
569C2×C4#1058756711yescharacteristic
570C2×C4#1058757011yescharacteristic
571C2×C4#1058757111yescharacteristic
572C2×C4#1058757111yescharacteristic
573C2×C4#1058757211yescharacteristic
574C2×C4#1058757811yescharacteristic
575C2×C4#1058757811yescharacteristic
576C2×C4#1058757911yescharacteristic
577C2×C4#1058757911yescharacteristic
578C2×C4#1058757911yescharacteristic
579C2×C4#1058758011yescharacteristic
580C2×C4#1058758111yescharacteristic
581C2×C4#1058758111yescharacteristic
582C2×C4#1058758411yescharacteristic
583C2×C4#1058758411yescharacteristic
584C2×C4C2×#6322711yescharacteristic
585C2×C4C4×C22⋊D1711yescharacteristic
586C2×C4C4×C22⋊D1711yescharacteristic
587C2×C4#1058762511yescharacteristic
588C2×C4#1058762611yescharacteristic
589C2×C4#1058763311yescharacteristic
590C8—42no—
591C8—682no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—682no—
603C22—1362no—
604C22—1362no—
605C22—1361no—
606C22—1361no—
607C22#1176265611yescharacteristic
608C22#1176268611yescharacteristic
609C22#1176268711yescharacteristic
610C22#1176268711yescharacteristic
611C22#1176269811yescharacteristic
612C22#1176269911yescharacteristic
613C22#1176270911yescharacteristic
614C4—44no—
615C4—82no—
616C4—684no—
617C4—682no—
618C4—684no—
619C4#1176265611yescharacteristic
620C4C4×#6322711yescharacteristic
621C4#1176269811yescharacteristic
622C4#1176269911yescharacteristic
623C4#1176270111yescharacteristic
624C4#1176270111yescharacteristic
625C4#1176270911yescharacteristic
626C4#1176271011yescharacteristic
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—1362no—
634C2#1452702011yescharacteristic
635C2#1452927211yescharacteristic
636C2#1452927911yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC2×C4×C17C2×C42
FrattiniC23×C4C22×D17
FittingC2×C17×H(3,ℤ/4)C2
socleC22×C17#386
prime radicalC22×#63252C22
cosocleC23×C4×C17C23
hypercenterC2×H(3,ℤ/4)D17
hypocenterC17#10954
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. #14531140C2semidirect
5. C2×#11762580C2semidirect
6. #14528505C2semidirect
7. C2×C17×H(3,ℤ/4)C2semidirect
13. #11762827C4semidirect
17. C2×#10587487C4semidirect
18. C2×#10587492C4semidirect
20. #11762782C4semidirect
147. C2×D68C42semidirect
159. C2×Dic34C42semidirect
280. C2×H(3,ℤ/4)D17semidirect
414. C17#10954semidirect

Maximal subgroups

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

subgroupindex
C2×#117628272
C2×#117628112
C2×#117627822
#145311402
C2×#117625802
#145285052
C2×C17×H(3,ℤ/4)2
#1095417