Finite Group Databasebeta

← #17067344

Subgroups — #17067344

Order 4352.

1484 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763142C211yesmaximal, characteristic, complemented
2C22×#10587503C211yesmaximal, characteristic, complemented
3#14533101C211yesmaximal, characteristic, complemented
4#14530020C212yesmaximal, complemented
5#14529963C212yesmaximal, complemented
6#14529208C211yesmaximal, characteristic, complemented
7#14529077C211yesmaximal, characteristic, complemented
8#14529031C212yesmaximal, complemented
9#14528724C212yesmaximal, complemented
10#14528715C211yesmaximal, characteristic, complemented
11C2×C17×#546C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14#11763142—24no—
15C2×#10587478C2212yescomplemented
16C2×#10587506C2212yescomplemented
17C2×#10587506C2212yescomplemented
18C2×#10587503—24no—
19C2×#10587503C2212yes—
20C2×#10587503C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2212yescomplemented
25#11762748—21no—
26#11762745—21no—
27#11762707C2212yescomplemented
28#11762689—21no—
29#11762681—21no—
30#11762669C2211yescharacteristic
31#11762669C2212yescomplemented
32#11762571C2212yescomplemented
33#11762564C2212yescomplemented
34C22×#41×C17C2211yescharacteristic
35C23×D4×C17C2211yescharacteristic
36C17×#546—24no—
37C17×#546—24no—
38C2×C17×#124C2211yescharacteristic
39C2×C17×#123—21no—
40C2×C17×C22⋊Q8C2211yescharacteristic
41C2×C17×#121—21no—
42C2×C17×#121C2211yescharacteristic
43C2×C17×#120—21no—
44C2×C4×D4×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C22×C22⋊C4×C17—21no—
47C25×C17—21no—
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17—24no—
51C2×#41×C17—42no—
52C22×Q8×C17C2311yescharacteristic
53C22×D4×C17—24no—
54C22×D4×C17—21no—
55C22×D4×C17—21no—
56C22×D4×C17—21no—
57C22×D4×C17—22no—
58C22×D4×C17—42no—
59C22×D4×C17D411yescharacteristic
60C22×D4×C17D414yes—
61C22×D4×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C23×C4×C17C2311yescharacteristic
66C23×C4×C17C2311yescharacteristic
67C23×Dic17—21no—
68C17×#124—22no—
69C17×#124—22no—
70C17×#123—28no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8D414yes—
73C17×#121—28no—
74C17×#121—22no—
75C17×#121—22no—
76C17×#120—28no—
77C4×D4×C17—22no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C4×D4×C17—22no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17D411yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C22⋊C4×C17—28no—
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C22⋊C4×C17D411yescharacteristic
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—21no—
94C2×C22⋊Dic17—22no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—22no—
97C2×C4⋊Dic17—22no—
98C2×C4⋊Dic17C2312yes—
99C2×C4⋊Dic17C2312yes—
100C2×#63225—22no—
101C22×#3622—22no—
102C22×#3622—42no—
103C22×#3622C2312yes—
104C22×#3622C2312yes—
105#10587506—24no—
106#10587506—42no—
107#10587503—24no—
108#10587503—24no—
109#10587503—24no—
110#10587503—42no—
111#10587502—22no—
112#10587502—21no—
113#10587502—22no—
114C22⋊#3622—22no—
115C22⋊#3622—42no—
116#10587478—24no—
117C24×C17—21no—
118C24×C17—21no—
119C24×C17—24no—
120C24×C17—44no—
121C24×C17—42no—
122C24×C17—42no—
123C24×C17—82no—
124C24×C17C2×D411yescharacteristic
125C24×C17C2×D411yescharacteristic
126#41×C17—48no—
127#41×C17—44no—
128#41×C17—44no—
129C2×Q8×C17—21no—
130C2×Q8×C17—22no—
131C2×Q8×C17—22no—
132C2×Q8×C17C2×D412yes—
133C2×D4×C17—28no—
134C2×D4×C17—24no—
135C2×D4×C17—22no—
136C2×D4×C17—22no—
137C2×D4×C17—22no—
138C2×D4×C17—24no—
139C2×D4×C17—24no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17—22no—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—44no—
146C2×D4×C17—48no—
147C2×D4×C17—44no—
148C2×D4×C17—44no—
149C2×D4×C17—82no—
150C2×D4×C17C2×D412yes—
151C2×D4×C17C2×D412yes—
152C2×D4×C17#4114yes—
153C22×C4×C17—22no—
154C22×C4×C17—22no—
155C22×C4×C17—24no—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—24no—
159C22×C4×C17—21no—
160C22×C4×C17—21no—
161C22×C4×C17—22no—
162C22×C4×C17—22no—
163C22×C4×C17—44no—
164C22×C4×C17—82no—
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D412yes—
170C22×C4×C17C2×D412yes—
171C22×C4×C17C2411yescharacteristic
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—42no—
175C22×Dic17—82no—
176C4⋊C4×C17—24no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17C2×D412yes—
180C22⋊C4×C17—24no—
181C22⋊C4×C17—24no—
182C22⋊C4×C17—24no—
183C22⋊C4×C17—24no—
184C22⋊C4×C17—24no—
185C22⋊C4×C17—22no—
186C22⋊C4×C17—22no—
187C42×C17—21no—
188C42×C17—21no—
189C22⋊Dic17—42no—
190C22⋊Dic17—82no—
191C4⋊Dic17—22no—
192C4⋊Dic17—22no—
193C4⋊Dic17—22no—
194C4⋊Dic17—22no—
195C4⋊Dic17—42no—
196#63225—82no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—22no—
201C2×#3622—22no—
202C2×#3622—22no—
203C2×#3622—44no—
204#24039—171nomaximal, Sylow
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—22no—
208C23×C17—24no—
209C23×C17—22no—
210C23×C17—21no—
211C23×C17—22no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—22no—
215C23×C17—48no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—42no—
223C23×C17—48no—
224C23×C17—84no—
225C23×C17—88no—
226C23×C17—88no—
227C23×C17#12012yes—
228C23×C17#13512yes—
229C23×C17#13712yes—
230C23×C17C22×D411yescharacteristic
231Q8×C17—24no—
232Q8×C17—41no—
233Q8×C17—41no—
234D4×C17—22no—
235D4×C17—22no—
236D4×C17—22no—
237D4×C17—24no—
238D4×C17—48no—
239D4×C17—48no—
240D4×C17—44no—
241D4×C17—42no—
242D4×C17—84no—
243D4×C17#12314yes—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—24no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—22no—
251C2×C4×C17—21no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—22no—
257C2×C4×C17—48no—
258C2×C4×C17—42no—
259C2×C4×C17—42no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—42no—
263C2×C4×C17—84no—
264C2×C4×C17#12012yes—
265C2×C4×C17C22×D411yescharacteristic
266C2×C4×C17C22×D411yescharacteristic
267C2×C4×C17C2×#4112yes—
268C2×C4×C17H(5,𝔽2)12yes—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—84no—
276#3622—42no—
277#3622—42no—
278C2×#546D1711yescharacteristic
279C2×#484—171no—
280C22×#102—171no—
281#2487—171no—
282#2042—172no—
283#2033—172no—
284#1892—171no—
285#1869—171no—
286#1862—172no—
287#1788—172no—
288#1786—171no—
289C22×C17—22no—
290C22×C17—22no—
291C22×C17—24no—
292C22×C17—24no—
293C22×C17—24no—
294C22×C17—42no—
295C22×C17—42no—
296C22×C17—44no—
297C22×C17—44no—
298C22×C17—41no—
299C22×C17—42no—
300C22×C17—41no—
301C22×C17—42no—
302C22×C17—41no—
303C22×C17—42no—
304C22×C17—42no—
305C22×C17—42no—
306C22×C17—816no—
307C22×C17—82no—
308C22×C17—82no—
309C22×C17#45214yes—
310C22×C17C2×#12011yescharacteristic
311C22×C17#54612yes—
312C22×C17C2×#13511yescharacteristic
313C22×C17C2×#13711yescharacteristic
314C22×C17#57612yes—
315C4×C17—22no—
316C4×C17—41no—
317C4×C17—42no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—42no—
321C4×C17C2×#12311yescharacteristic
322C4×C17#53411yescharacteristic
323C4×C17#54612yes—
324Dic17—82no—
325Dic17—82no—
326C22×#41C2×D1711yescharacteristic
327C23×D4C2×D1711yescharacteristic
328C23×C8—171no—
329#546—24no—
330#546—24no—
331C2×#124C2×D1711yescharacteristic
332C2×#123—21no—
333C2×C22⋊Q8C2×D1711yescharacteristic
334C2×#121—21no—
335C2×#121C2×D1711yescharacteristic
336C2×#120—21no—
337C2×C4×D4C2×D1711yescharacteristic
338C2×C4×D4C2×D1711yescharacteristic
339C22×C4⋊C4—171no—
340C22×C22⋊C4—21no—
341#484—344no—
342C2×#107—172no—
343C2×#103—172no—
344C2×#103—172no—
345C2×#102—172no—
346C2×#102—172no—
347C2×#102—172no—
348C2×#102—172no—
349C2×#102—344no—
350C2×C22⋊C8—171no—
351C2×C22⋊C8—172no—
352#400—341no—
353#399—341no—
354#389—172no—
355#386—341no—
356#385—341no—
357#380—171no—
358#380—172no—
359#340—172no—
360#336—172no—
361C2×C17—24no—
362C2×C17—44no—
363C2×C17—81no—
364C2×C17—82no—
365C2×C17#240511yescharacteristic
366C2×C17C2×#45211yescharacteristic
367C2×C17#300811yescharacteristic
368C2×C17#329712yes—
369C2×C17#331412yes—
370C25—21no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—24no—
374C2×#41—42no—
375C22×Q8C22×D1711yescharacteristic
376C22×D4—24no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—22no—
381C22×D4—42no—
382C22×D4C22⋊D1711yescharacteristic
383C22×D4C22⋊D1714yes—
384C22×D4C22×D1711yescharacteristic
385C22×D4C22×D1711yescharacteristic
386C23×C4—21no—
387C23×C4—21no—
388C23×C4—341no—
389C23×C4C22×D1711yescharacteristic
390C23×C4C22×D1711yescharacteristic
391C22×C8—172no—
392C22×C8—172no—
393C22×C8—342no—
394C22×C8—682no—
395#124—22no—
396#124—22no—
397#123—28no—
398C22⋊Q8—22no—
399C22⋊Q8C22⋊D1714yes—
400#121—28no—
401#121—22no—
402#121—22no—
403#120—28no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—172no—
410C2×C4⋊C4—172no—
411C2×C4⋊C4—342no—
412C2×C4⋊C4—342no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4C22⋊D1711yescharacteristic
416C2×C4⋊C4C22×D1711yescharacteristic
417C2×C22⋊C4—28no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4—342no—
425C2×C22⋊C4C22⋊D1711yescharacteristic
426C2×C22⋊C4C22⋊D1711yescharacteristic
427C2×C42C22×D1711yescharacteristic
428#107—344no—
429#103—344no—
430#103—682no—
431#102—344no—
432#102—344no—
433#102—344no—
434#102—682no—
435C22⋊C8—342no—
436C22⋊C8—682no—
437#95—342no—
438#95—341no—
439#95—342no—
440C17#2403911yescharacteristic, complemented, Sylow
441C24—21no—
442C24—21no—
443C24—24no—
444C24—44no—
445C24—42no—
446C24—42no—
447C24—82no—
448C24C2×C22⋊D1711yescharacteristic
449C24C2×C22⋊D1711yescharacteristic
450#41—48no—
451#41—44no—
452#41—44no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8C2×C22⋊D1712yes—
457C2×D4—28no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—24no—
469C2×D4—44no—
470C2×D4—48no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—82no—
474C2×D4#6325414yes—
475C2×D4C2×C22⋊D1712yes—
476C2×D4C2×C22⋊D1712yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—682no—
492C22×C4—1362no—
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C2×C22⋊D1712yes—
497C22×C4C2×C22⋊D1712yes—
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C23×D1711yescharacteristic
500C2×C8—342no—
501C2×C8—342no—
502C2×C8—342no—
503C2×C8—342no—
504C2×C8—342no—
505C2×C8—342no—
506C2×C8—684no—
507C4⋊C4—21no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—342no—
511C4⋊C4—342no—
512C4⋊C4—342no—
513C4⋊C4—342no—
514C4⋊C4—682no—
515C4⋊C4—1362no—
516C4⋊C4C2×C22⋊D1712yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—682no—
525C22⋊C4—1362no—
526C42—21no—
527C42—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—48no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—48no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—84no—
548C23—88no—
549C23—88no—
550C23#1058763612yes—
551C23#1058763712yes—
552C23#1058763912yes—
553C23C22×C22⋊D1711yescharacteristic
554Q8—24no—
555Q8—41no—
556Q8—41no—
557D4—22no—
558D4—22no—
559D4—22no—
560D4—24no—
561D4—48no—
562D4—48no—
563D4—44no—
564D4—42no—
565D4—84no—
566D4#1058762114yes—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—1364no—
594C2×C4#1058763912yes—
595C2×C4C2×#6325412yes—
596C2×C4#1058769112yes—
597C2×C4C22×C22⋊D1711yescharacteristic
598C2×C4C22×C22⋊D1711yescharacteristic
599C8—682no—
600C8—682no—
601C22—22no—
602C22—24no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—816no—
619C22—82no—
620C22—82no—
621C22#1176297114yes—
622C22C2×#1058763911yescharacteristic
623C22#1176345312yes—
624C22C2×#1058763611yescharacteristic
625C22C2×#1058763711yescharacteristic
626C22#1176361212yes—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—1362no—
634C4—1362no—
635C4C2×#1058762111yescharacteristic
636C4#1176336811yescharacteristic
637C4#1176345312yes—
638C2—24no—
639C2—44no—
640C2—81no—
641C2—82no—
642C2#1453210711yescharacteristic
643C2C2×#1176297111yescharacteristic
644C2#1453805211yescharacteristic
645C2#1454063412yes—
646C2#1454086512yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#546C2
socleC23×C17C22×D4
hypercenterC2×#546D17
hypocenterC17#24039
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#11763142C2semidirect
2. C22×#10587503C2semidirect
3. #14533101C2semidirect
4. #14530020C2semidirect
5. #14529963C2semidirect
6. #14529208C2semidirect
7. #14529077C2semidirect
8. #14529031C2semidirect
9. #14528724C2semidirect
10. #14528715C2semidirect
15. C2×#10587478C22semidirect
16. C2×#10587506C22semidirect
17. C2×#10587506C22semidirect
24. C2×C22⋊#3622C22semidirect
27. #11762707C22semidirect
31. #11762669C22semidirect
32. #11762571C22semidirect
33. #11762564C22semidirect
440. C17#24039semidirect

Maximal subgroups

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

subgroupindex
C2×#117631422
C22×#105875032
#145331012
#145300202
#145299632
#145292082
#145290772
#145290312
#145287242
#145287152
C2×C17×#5462
#2403917