Finite Group Databasebeta

← #17102471

Subgroups — #17102471

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531820C211yesmaximal, characteristic
2#14530099C211yesmaximal, characteristic, complemented
3#14529973C211yesmaximal, characteristic
4#14529927C211yesmaximal, characteristic, complemented
5#14529785C211yesmaximal, characteristic
6#14529742C211yesmaximal, characteristic, complemented
7#14529463C211yesmaximal, characteristic, complemented
8#14529415C211yesmaximal, characteristic, complemented
9#14529392C211yesmaximal, characteristic, complemented
10#14529369C211yesmaximal, characteristic, complemented
11#14529121C211yesmaximal, characteristic, complemented
12#14529097C211yesmaximal, characteristic, complemented
13#14528876C211yesmaximal, characteristic, complemented
14C4×#10587480C211yesmaximal, characteristic, complemented
15C17×#2489C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×#10587616C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31#11762711C2211yescharacteristic
32#11762709—21no—
33#11762707C2211yescharacteristic
34#11762698—21no—
35#11762685—21no—
36C4×C4⋊Dic17C2211yescharacteristic
37#11762656—21no—
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#125C2211yescharacteristic
49C2×C17×C22⋊Q8C2211yescharacteristic
50C17×#400—21no—
51C17×#397—21no—
52C17×#391C2211yescharacteristic
53C17×#391C2211yescharacteristic
54C17×#388—21no—
55C17×#382—21no—
56C17×#380C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C22×C22⋊D17—21no—
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C17×#125—22no—
65C17×#125—41no—
66C17×C22⋊Q8—22no—
67C17×C22⋊Q8—41no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17D411yescharacteristic
77C2×C22⋊C4×C17D411yescharacteristic
78C2×C42×C17—21no—
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C22⋊Dic17—21no—
82C2×#63230—22no—
83C2×#63230—41no—
84C2×#63230—41no—
85#10587616—22no—
86#10587616—22no—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91#10587612—22no—
92#10587612—41no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C4×D68—22no—
104C4×D68—41no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95D411yescharacteristic
112C17×#95D411yescharacteristic
113#10587506—22no—
114#10587506—22no—
115#10587506—41no—
116#10587502—21no—
117#10587502—21no—
118C22⋊#3622—41no—
119C22⋊#3622—41no—
120#10587480—22no—
121#10587480—22no—
122#10587480—41no—
123#10587479—22no—
124C4⋊#3622—41no—
125C4×#3622—41no—
126C24×C17C2×D411yescharacteristic
127C23×D17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C2×C22⋊D17—24no—
151C2×C22⋊D17—81no—
152C2×C22⋊D17—81no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C2×D68—21no—
157C2×D68—21no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68#4112yes—
162C2×C4×D17—42no—
163C2×C4×D17—42no—
164C2×C4×D17—81no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—42no—
171C4⋊C4×C17—42no—
172C4⋊C4×C17—42no—
173C4⋊C4×C17#4112yes—
174C22⋊C4×C17—22no—
175C22⋊C4×C17—41no—
176C22⋊C4×C17—42no—
177C22⋊C4×C17—42no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—42no—
183C22⋊Dic17—42no—
184#63230—44no—
185#63227—24no—
186#63227—42no—
187#63227—42no—
188#63227—81no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17#4112yes—
194C4⋊Dic17#4112yes—
195#63225—81no—
196C4×Dic17—42no—
197C4×Dic17—81no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210#26728—171nomaximal, Sylow
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17C22×D411yescharacteristic
219C22×D17—22no—
220C22×D17—41no—
221C22×D17—41no—
222C22×D17—41no—
223C22×D17—41no—
224C22×D17—41no—
225C22×D17—41no—
226Q8×C17—41no—
227Q8×C17—41no—
228Q8×C17—41no—
229Q8×C17—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—22no—
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×C17—21no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17#13512yes—
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×C4×C17H(5,𝔽2)11yescharacteristic
266C2×C4×C17H(5,𝔽2)11yescharacteristic
267C22⋊D17—84no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277D68—22no—
278D68—22no—
279D68—41no—
280D68—41no—
281C4×D17—84no—
282#3622—41no—
283#3622—41no—
284#3622—41no—
285#3622—41no—
286#2489D1711yescharacteristic, complemented
287#2363—171no—
288#2055—171no—
289#2034—171no—
290#2025—171no—
291#2000—171no—
292#1989—171no—
293#1933—171no—
294#1927—171no—
295#1923—171no—
296#1920—171no—
297#1877—171no—
298#1873—171no—
299#1825—171no—
300C4×#102—171no—
301C22×C17—41no—
302C22×C17—41no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17#52511yescharacteristic
309C22×C17#54611yescharacteristic
310C22×C17#54611yescharacteristic
311C22×C17C2×#13511yescharacteristic
312C22×C17#57511yescharacteristic
313C22×C17#57711yescharacteristic
314C22×C17#57711yescharacteristic
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—42no—
318C2×D17—81no—
319C2×D17—81no—
320C2×D17—81no—
321C2×D17—81no—
322C4×C17—22no—
323C4×C17—41no—
324C4×C17—41no—
325C4×C17—42no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17—81no—
329C4×C17—81no—
330C4×C17—81no—
331C4×C17#52911yescharacteristic
332C4×C17#53811yescharacteristic
333C4×C17#55511yescharacteristic
334C4×C17#55911yescharacteristic
335Dic17—42no—
336Dic17—42no—
337Dic17—81no—
338C22×SD16—171no—
339C2×#125C2×D1711yescharacteristic
340C2×C22⋊Q8C2×D1711yescharacteristic
341C2×#121—171no—
342C2×C4×D4—171no—
343C2×#117—171no—
344C2×#106—171no—
345C2×C4⋊C8—171no—
346C2×#103—171no—
347C2×#103—171no—
348C2×#103—171no—
349C2×#102—171no—
350C2×#102—171no—
351C2×#102—171no—
352C2×C22⋊C8—171no—
353C2×C22⋊C8—171no—
354C2×C4×C8—171no—
355#400—21no—
356#397—21no—
357#391C2×D1711yescharacteristic
358#391C2×D1711yescharacteristic
359#390—171no—
360#390—341no—
361#389—171no—
362#388—21no—
363#388—341no—
364#386—341no—
365#382—21no—
366#380C2×D1711yescharacteristic
367C4×C4⋊C4—171no—
368C4×C4⋊C4C2×D1711yescharacteristic
369C4×C22⋊C4—341no—
370C4×C22⋊C4C2×D1711yescharacteristic
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#336—171no—
380#336—171no—
381C2×C17—81no—
382C2×C17#245911yescharacteristic
383C2×C17#310211yescharacteristic
384C2×C17#312311yescharacteristic
385C2×C17#329611yescharacteristic
386C2×C17#330311yescharacteristic
387C2×C17#331711yescharacteristic
388C2×C17#332611yescharacteristic
389D17—82no—
390C22×Q8C22×D1711yescharacteristic
391C22×D4—171no—
392C22×D4—341no—
393C23×C4—341no—
394C23×C4C22×D1711yescharacteristic
395C2×SD16—342no—
396C2×SD16—681no—
397C2×SD16—681no—
398C22×C8—171no—
399C22×C8—171no—
400C22×C8—171no—
401C22×C8—171no—
402#125—22no—
403#125—41no—
404C22⋊Q8—22no—
405C22⋊Q8—41no—
406#121—342no—
407#121—342no—
408C4×D4—342no—
409C4×D4—681no—
410#117—342no—
411#117—681no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—341no—
425C2×C22⋊C4—341no—
426C2×C22⋊C4—341no—
427C2×C22⋊C4—341no—
428C2×C22⋊C4—341no—
429C2×C22⋊C4C22⋊D1711yescharacteristic
430C2×C22⋊C4C22⋊D1711yescharacteristic
431C2×C42—21no—
432C2×C42—341no—
433C2×C42—341no—
434C2×C42C22×D1711yescharacteristic
435C2×C42C22×D1711yescharacteristic
436#106—342no—
437C4⋊C8—681no—
438#103—342no—
439#103—342no—
440#103—681no—
441#102—342no—
442#102—342no—
443#102—681no—
444C22⋊C8—681no—
445C22⋊C8—681no—
446C4×C8—681no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—341no—
454#95—341no—
455#95C22⋊D1711yescharacteristic
456#95C22⋊D1711yescharacteristic
457C17#2672811yescharacteristic, complemented, Sylow
458C24—341no—
459C24C2×C22⋊D1711yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×D4—172no—
467C2×D4—344no—
468C2×D4—341no—
469C2×D4—341no—
470C2×D4—341no—
471C2×D4—341no—
472C2×D4—341no—
473C2×D4—1361no—
474C2×D4—1361no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—682no—
489C22×C4—682no—
490C22×C4—1361no—
491C22×C4C2×C22⋊D1711yescharacteristic
492C22×C4C2×C22⋊D1711yescharacteristic
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C23×D1711yescharacteristic
497SD16—684no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—172no—
519C4⋊C4—172no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—1361no—
525C4⋊C4#6325412yes—
526C22⋊C4—22no—
527C22⋊C4—41no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—344no—
531C22⋊C4—682no—
532C22⋊C4—682no—
533C22⋊C4—682no—
534C22⋊C4—1361no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—682no—
541C42—1361no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—342no—
550C23—681no—
551C23—681no—
552C23—681no—
553C23—681no—
554C23—681no—
555C23—681no—
556C23C22×C22⋊D1711yescharacteristic
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561D4—342no—
562D4—342no—
563D4—681no—
564D4—681no—
565D4—1364no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—342no—
597C2×C4—342no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—1364no—
606C2×C4#1058763612yes—
607C2×C4C2×#6325411yescharacteristic
608C2×C4#1058769111yescharacteristic
609C2×C4#41×D1711yescharacteristic
610C2×C4#41×D1711yescharacteristic
611C2×C4#41×D1711yescharacteristic
612C2×C4#1058769611yescharacteristic
613C8—681no—
614C8—681no—
615C8—681no—
616C8—681no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—682no—
625C22—682no—
626C22—682no—
627C22—1361no—
628C22—1361no—
629C22—1361no—
630C22—1361no—
631C22#1176331611yescharacteristic
632C22#1176344711yescharacteristic
633C22#1176345311yescharacteristic
634C22C2×#1058763611yescharacteristic
635C22#1176360711yescharacteristic
636C22#1176361711yescharacteristic
637C22#1176361711yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—682no—
648C4—682no—
649C4—1361no—
650C4#1176333811yescharacteristic
651C4#1176339311yescharacteristic
652C4#1176352111yescharacteristic
653C4#1176354111yescharacteristic
654C2—81no—
655C2—1362no—
656C2#1453271711yescharacteristic
657C2#1453879611yescharacteristic
658C2#1453903111yescharacteristic
659C2#1454061311yescharacteristic
660C2#1454071211yescharacteristic
661C2#1454090411yescharacteristic
662C2#1454103711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2489C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#2489D17
hypocenterC17#26728
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #14530099C2semidirect
4. #14529927C2semidirect
6. #14529742C2semidirect
7. #14529463C2semidirect
8. #14529415C2semidirect
9. #14529392C2semidirect
10. #14529369C2semidirect
11. #14529121C2semidirect
12. #14529097C2semidirect
13. #14528876C2semidirect
14. C4×#10587480C2semidirect
15. C17×#2489C2semidirect
286. #2489D17semidirect
457. C17#26728semidirect

Maximal subgroups

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

subgroupindex
#145318202
#145300992
#145299732
#145299272
#145297852
#145297422
#145294632
#145294152
#145293922
#145293692
#145291212
#145290972
#145288762
C4×#105874802
C17×#24892
#2672817