Finite Group Databasebeta

← #17075721

Subgroups — #17075721

Order 4352.

802 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763099C211yesmaximal, characteristic, complemented
2C2×#11762870C211yesmaximal, characteristic, complemented
3#14534605C211yesmaximal, characteristic, complemented
4#14530221C211yesmaximal, characteristic, complemented
5#14530220C211yesmaximal, characteristic, complemented
6#14529970C211yesmaximal, characteristic
7#14529963C211yesmaximal, characteristic
8#14529736C211yesmaximal, characteristic, complemented
9#14529552C211yesmaximal, characteristic, complemented
10#14529519C211yesmaximal, characteristic, complemented
11#14529509C211yesmaximal, characteristic, complemented
12#14529484C211yesmaximal, characteristic, complemented
13#14529470C211yesmaximal, characteristic, complemented
14#14528794C211yesmaximal, characteristic, complemented
15C17×#2324C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17#11763099—24no—
18#11762870—24no—
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×C22⋊#3622C2211yescharacteristic
31C2×#10587474C2211yescharacteristic
32C2×C4×#3622C2211yescharacteristic
33#11762748—21no—
34#11762745—21no—
35#11762729—21no—
36#11762721—21no—
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762675C2211yescharacteristic
40#11762669C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50C2×C17×#121C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C17×#117C2211yescharacteristic
53C17×#390—21no—
54C17×#390C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#388—21no—
57C17×#386—21no—
58C17×#385—21no—
59C17×#380C2211yescharacteristic
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×D4×C17—21no—
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17C2311yescharacteristic
65C22×Dic34—21no—
66C17×#121—22no—
67C17×#121—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C17×#117—22no—
71C17×#117—22no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17D411yescharacteristic
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C22⋊Dic17—21no—
89C2×C22⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63225—21no—
96C2×#63225—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587555—22no—
102#10587555—22no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106#10587503—22no—
107#10587503—22no—
108#10587503—22no—
109#10587503—22no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116C22⋊#3622—22no—
117C22⋊#3622—22no—
118#10587479—22no—
119#10587479—22no—
120#10587479—22no—
121#10587478—22no—
122C4⋊#3622—22no—
123C4⋊#3622D414yescomplemented
124#10587474—22no—
125C4×#3622—22no—
126C24×C17—21no—
127C24×C17C2×D411yescharacteristic
128C2×D4×C17—24no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—81no—
135C2×D4×C17—81no—
136C2×D4×C17#4112yes—
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×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—42no—
147C22×C4×C17—81no—
148C22×C4×C17—81no—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2411yescharacteristic
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C2×Dic34—42no—
160C2×Dic34—81no—
161C2×Dic34—81no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—81no—
170C4⋊C4×C17#4112yes—
171C22⋊C4×C17—24no—
172C22⋊C4×C17—22no—
173C22⋊C4×C17—22no—
174C22⋊C4×C17—42no—
175C22⋊C4×C17—81no—
176C22⋊C4×C17—81no—
177C22⋊C4×C17—81no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—81no—
181C42×C17C2×D412yes—
182C42×C17#4112yes—
183C22⋊Dic17—81no—
184C22⋊Dic17—81no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—42no—
194#63225—81no—
195#63225—81no—
196C2×#3622—21no—
197C2×#3622—21no—
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×#3622C2×D412yes—
207C2×#3622C2×D412yes—
208#24716—171nomaximal, Sylow
209C23×C17—21no—
210C23×C17—22no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17C22×D411yescharacteristic
224D4×C17—22no—
225D4×C17—22no—
226D4×C17—41no—
227D4×C17—41no—
228D4×C17—84no—
229C2×C4×C17—21no—
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—22no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
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—84no—
256C2×C4×C17#12112yes—
257C2×C4×C17#12112yes—
258C2×C4×C17C22×D411yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C172−1+411yescharacteristic
263C2×C4×C172−1+411yescharacteristic
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—
275C2×Dic17—41no—
276Dic34—84no—
277#3622—22no—
278#3622—22no—
279#3622—41no—
280#3622—41no—
281C2×#476—171no—
282C2×#433—171no—
283#2628—171no—
284#2324D1711yescharacteristic
285#2077—171no—
286#2077—171no—
287#2034—171no—
288#2033—171no—
289#1988—171no—
290#1949—171no—
291#1943—171no—
292#1941—171no—
293#1936—171no—
294#1934—171no—
295#1804—171no—
296C22×C17—41no—
297C22×C17—42no—
298C22×C17—41no—
299C22×C17—42no—
300C22×C17—41no—
301C22×C17—42no—
302C22×C17—41no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17—81no—
307C22×C17—81no—
308C22×C17—81no—
309C22×C17—81no—
310C22×C17C2×#12111yescharacteristic
311C22×C17#52511yescharacteristic
312C22×C17#53611yescharacteristic
313C22×C17#57611yescharacteristic
314C22×C17#57711yescharacteristic
315C22×C17#57711yescharacteristic
316C22×C17#57711yescharacteristic
317C4×C17—22no—
318C4×C17—22no—
319C4×C17—42no—
320C4×C17—41no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17—81no—
324C4×C17—81no—
325C4×C17#54711yescharacteristic
326C4×C17#54811yescharacteristic
327C4×C17#55211yescharacteristic
328C4×C17#55611yescharacteristic
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332Dic17—81no—
333C2×C4⋊Q8—171no—
334C2×#121C2×D1711yescharacteristic
335C2×C4×D4C2×D1711yescharacteristic
336C2×#117C2×D1711yescharacteristic
337#476—344no—
338#433—344no—
339C2×#107—171no—
340C2×#106—171no—
341C2×#106—171no—
342C2×#106—171no—
343C2×C4⋊C8—171no—
344C2×C4⋊C8—171no—
345C2×#102—171no—
346C2×#102—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×C22⋊C8—171no—
350C2×C22⋊C8—171no—
351C2×#97—171no—
352C2×C4×C8—171no—
353#400—341no—
354#399—341no—
355#395—341no—
356#393—341no—
357#390—21no—
358#390C2×D1711yescharacteristic
359#389—171no—
360#389—171no—
361#389C2×D1711yescharacteristic
362#388—21no—
363#386—21no—
364#385—21no—
365#382—171no—
366#380—171no—
367#380—171no—
368#380C2×D1711yescharacteristic
369C4×C4⋊C4C2×D1711yescharacteristic
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340—171no—
378C2×C17—81no—
379C2×C17—82no—
380C2×C17#241511yescharacteristic
381C2×C17#305711yescharacteristic
382C2×C17#306211yescharacteristic
383C2×C17#311811yescharacteristic
384C2×C17#312511yescharacteristic
385C2×C17#324111yescharacteristic
386C2×C17#324411yescharacteristic
387C22×Q8—341no—
388C22×D4—21no—
389C22×D4C22×D1711yescharacteristic
390C23×C4—21no—
391C23×C4C22×D1711yescharacteristic
392C22×C8—171no—
393C22×C8—171no—
394C22×C8—171no—
395C22×C8—171no—
396C4⋊Q8—342no—
397C4⋊Q8—342no—
398#121—22no—
399#121—22no—
400C4×D4—22no—
401C4×D4—22no—
402#117—22no—
403#117—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—341no—
410C2×C4⋊C4—341no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4C22⋊D1711yescharacteristic
413C2×C4⋊C4C22⋊D1711yescharacteristic
414C2×C4⋊C4C22×D1711yescharacteristic
415C2×C4⋊C4C22×D1711yescharacteristic
416C2×C4⋊C4C22×D1711yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4C22⋊D1711yescharacteristic
425C2×C22⋊C4C22⋊D1711yescharacteristic
426C2×C42—21no—
427C2×C42C22×D1711yescharacteristic
428C2×C42C22×D1711yescharacteristic
429#107—342no—
430#106—342no—
431#106—342no—
432#106—342no—
433C4⋊C8—174no—
434C4⋊C8—342no—
435#102—342no—
436#102—342no—
437#102—342no—
438#102—342no—
439C22⋊C8—342no—
440C22⋊C8—342no—
441#97—342no—
442C4×C8—342no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452C17#2471611yescharacteristic, complemented, Sylow
453C24—21no—
454C24C2×C22⋊D1711yescharacteristic
455C2×Q8—682no—
456C2×Q8—1361no—
457C2×Q8—1361no—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—81no—
465C2×D4—81no—
466C2×D4#6325212yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4C2×C22⋊D1711yescharacteristic
484C22×C4C2×C22⋊D1711yescharacteristic
485C22×C4C2×C22⋊D1711yescharacteristic
486C22×C4C2×C22⋊D1711yescharacteristic
487C22×C4C2×C22⋊D1711yescharacteristic
488C22×C4C23×D1711yescharacteristic
489C2×C8—172no—
490C2×C8—172no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—81no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—341no—
513C4⋊C4—341no—
514C4⋊C4—341no—
515C4⋊C4—341no—
516C4⋊C4—341no—
517C4⋊C4—682no—
518C4⋊C4—1361no—
519C4⋊C4—1361no—
520C4⋊C4#6325212yes—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C22⋊C4—1361no—
529C22⋊C4—1361no—
530C42—21no—
531C42—21no—
532C42—81no—
533C42D4×D1712yes—
534C42#6325412yes—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C22×C22⋊D1711yescharacteristic
550Q8—1364no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4#1058762512yes—
596C2×C4#1058762512yes—
597C2×C4C2×#6325211yescharacteristic
598C2×C4C2×#6325211yescharacteristic
599C2×C4C2×D4×D1711yescharacteristic
600C2×C4C2×#6325411yescharacteristic
601C2×C4#1058769711yescharacteristic
602C2×C4#1058769711yescharacteristic
603C8—342no—
604C8—342no—
605C8—681no—
606C8—681no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#1058762511yescharacteristic
622C22#1176332011yescharacteristic
623C22#1176338011yescharacteristic
624C22#1176361211yescharacteristic
625C22#1176361711yescharacteristic
626C22#1176362011yescharacteristic
627C22#1176362011yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—1361no—
637C4—1361no—
638C4—1361no—
639C4—1361no—
640C4#1176346411yescharacteristic
641C4#1176346811yescharacteristic
642C4#1176349911yescharacteristic
643C4#1176352911yescharacteristic
644C2—81no—
645C2—82no—
646C2#1453221311yescharacteristic
647C2#1453842011yescharacteristic
648C2#1453846611yescharacteristic
649C2#1453896811yescharacteristic
650C2#1453906111yescharacteristic
651C2#1454003111yescharacteristic
652C2#1454007611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2324C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2324D17
hypocenterC17#24716
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. C2×#11763099C2semidirect
2. C2×#11762870C2semidirect
3. #14534605C2semidirect
4. #14530221C2semidirect
5. #14530220C2semidirect
8. #14529736C2semidirect
9. #14529552C2semidirect
10. #14529519C2semidirect
11. #14529509C2semidirect
12. #14529484C2semidirect
13. #14529470C2semidirect
14. #14528794C2semidirect
123. C4⋊#3622D4semidirect
452. C17#24716semidirect

Maximal subgroups

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

subgroupindex
C2×#117630992
C2×#117628702
#145346052
#145302212
#145302202
#145299702
#145299632
#145297362
#145295522
#145295192
#145295092
#145294842
#145294702
#145287942
C17×#23242
#2471617