Finite Group Databasebeta

← #17067422

Subgroups — #17067422

Order 4352.

1410 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763125C211yesmaximal, characteristic, complemented
2C2×#11762810C211yesmaximal, characteristic, complemented
3#14533092C211yesmaximal, characteristic, complemented
4#14530015C212yesmaximal, complemented
5#14529970C212yesmaximal, complemented
6#14529223C211yesmaximal, characteristic, complemented
7#14529082C211yesmaximal, characteristic, complemented
8#14529052C212yesmaximal, complemented
9#14528724C211yesmaximal, characteristic, complemented
10#14528724C212yesmaximal, complemented
11C2×C17×#550C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C2×#10587621—21no—
14C22×C4⋊Dic17C2211yescharacteristic
15#11763125—24no—
16C2×#10587479C2212yes—
17#11762810—24no—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587503C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×#10587503C2212yes—
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2212yes—
26#11762739—21no—
27#11762734—21no—
28#11762707C2212yes—
29#11762670—21no—
30#11762669C2212yes—
31#11762669C2211yescharacteristic
32#11762571C2212yes—
33#11762564C2212yes—
34C22×#41×C17C2211yescharacteristic
35C22×#41×C17C2211yescharacteristic
36C17×#550—24no—
37C17×#550—24no—
38C2×C17×#127—21no—
39C2×C17×#127C2211yescharacteristic
40C2×C17×#121—21no—
41C2×C17×#121—21no—
42C2×C17×#121C2211yescharacteristic
43C2×C17×#121C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17—21no—
46C2×C4×D4×C17C2211yescharacteristic
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17—24no—
51C2×#41×C17—24no—
52C2×#41×C17—42no—
53C2×#41×C17—42no—
54C2×#41×C17D414yes—
55C22×Q8×C17C2311yescharacteristic
56C22×D4×C17—21no—
57C22×D4×C17—21no—
58C22×D4×C17—21no—
59C22×D4×C17—21no—
60C22×D4×C17—21no—
61C22×D4×C17—21no—
62C22×D4×C17D411yescharacteristic
63C22×D4×C17C2311yescharacteristic
64C22×D4×C17C2311yescharacteristic
65C23×C4×C17—21no—
66C23×C4×C17—21no—
67C23×C4×C17C2311yescharacteristic
68C23×C4×C17C2311yescharacteristic
69C23×Dic17—21no—
70C17×#127—28no—
71C17×#127—22no—
72C17×#127—22no—
73C17×#121—28no—
74C17×#121—28no—
75C17×#121—22no—
76C17×#121—22no—
77C17×#121—22no—
78C17×#121D414yes—
79C4×Q8×C17—22no—
80C4×Q8×C17—22no—
81C4×D4×C17—28no—
82C4×D4×C17—22no—
83C4×D4×C17—22no—
84C2×C4⋊C4×C17—21no—
85C2×C4⋊C4×C17D411yescharacteristic
86C2×C4⋊C4×C17C2311yescharacteristic
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17D411yescharacteristic
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C42×C17—21no—
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—22no—
94C2×C22⋊Dic17—21no—
95#10587621—44no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—22no—
98C2×C4⋊Dic17—22no—
99C2×C4⋊Dic17C2312yes—
100C2×C4⋊Dic17C2312yes—
101C2×#63225—22no—
102C2×#63223—22no—
103C2×#63223—42no—
104C22×#3622C2312yes—
105C22×#3622C2312yes—
106#10587506—24no—
107#10587506—24no—
108#10587503—24no—
109#10587503—24no—
110#10587503—42no—
111#10587503—42no—
112#10587502—22no—
113#10587502—21no—
114#10587502—22no—
115C22⋊#3622—22no—
116C22⋊#3622—42no—
117#10587479—24no—
118C24×C17—21no—
119C24×C17—21no—
120C24×C17C2×D411yescharacteristic
121C24×C17C2×D411yescharacteristic
122#41×C17—28no—
123#41×C17—44no—
124#41×C17—48no—
125#41×C17—48no—
126#41×C17—44no—
127#41×C17—44no—
128C2×Q8×C17—21no—
129C2×Q8×C17—22no—
130C2×Q8×C17C2×D412yes—
131C2×Q8×C17#4114yes—
132C2×D4×C17—24no—
133C2×D4×C17—24no—
134C2×D4×C17—22no—
135C2×D4×C17—24no—
136C2×D4×C17—24no—
137C2×D4×C17—24no—
138C2×D4×C17—24no—
139C2×D4×C17—22no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17—22no—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—22no—
146C2×D4×C17—41no—
147C2×D4×C17—44no—
148C2×D4×C17—44no—
149C2×D4×C17—42no—
150C2×D4×C17—42no—
151C2×D4×C17—82no—
152C2×D4×C17—82no—
153C2×D4×C17—82no—
154C2×D4×C17C2×D412yes—
155C2×D4×C17C2×D412yes—
156C22×C4×C17—21no—
157C22×C4×C17—22no—
158C22×C4×C17—22no—
159C22×C4×C17—24no—
160C22×C4×C17—21no—
161C22×C4×C17—24no—
162C22×C4×C17—21no—
163C22×C4×C17—21no—
164C22×C4×C17—22no—
165C22×C4×C17—22no—
166C22×C4×C17—44no—
167C22×C4×C17—44no—
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2×D411yescharacteristic
171C22×C4×C17C2×D412yes—
172C22×C4×C17C2×D412yes—
173C22×C4×C17C2×D411yescharacteristic
174C22×C4×C17C2411yescharacteristic
175C22×Dic17—22no—
176C22×Dic17—22no—
177C22×Dic17—42no—
178C22×Dic17—42no—
179C22×Dic17—42no—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—22no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17C2×D412yes—
184C22⋊C4×C17—24no—
185C22⋊C4×C17—22no—
186C22⋊C4×C17—24no—
187C22⋊C4×C17—22no—
188C42×C17—24no—
189C42×C17—21no—
190C42×C17—21no—
191C22⋊Dic17—42no—
192C22⋊Dic17—44no—
193C4⋊Dic17—22no—
194C4⋊Dic17—22no—
195C4⋊Dic17—22no—
196C4⋊Dic17—22no—
197C4⋊Dic17—42no—
198#63225—44no—
199#63223—44no—
200C2×#3622—22no—
201C2×#3622—22no—
202C2×#3622—22no—
203C2×#3622—22no—
204C2×#3622—22no—
205C2×#3622—22no—
206#24045—171nomaximal, Sylow
207C23×C17—22no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—22no—
211C23×C17—21no—
212C23×C17—22no—
213C23×C17—22no—
214C23×C17—22no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—22no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—42no—
225C23×C17—42no—
226C23×C17#12012yes—
227C23×C17C22×D411yescharacteristic
228Q8×C17—22no—
229Q8×C17—24no—
230Q8×C17#12314yescomplemented
231D4×C17—22no—
232D4×C17—22no—
233D4×C17—24no—
234D4×C17—44no—
235D4×C17—48no—
236D4×C17—48no—
237D4×C17—48no—
238D4×C17—41no—
239D4×C17—41no—
240D4×C17—42no—
241D4×C17—84no—
242D4×C17—84no—
243D4×C17—84no—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—24no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—24no—
252C2×C4×C17—21no—
253C2×C4×C17—22no—
254C2×C4×C17—21no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—48no—
258C2×C4×C17—48no—
259C2×C4×C17—42no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—42no—
263C2×C4×C17#12012yes—
264C2×C4×C17AGL(1,ℤ/8)12yes—
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×#550D1711yescharacteristic
279C2×#481—171no—
280C2×#417—171no—
281#2486—171no—
282#2041—172no—
283#2034—172no—
284#1895—171no—
285#1870—171no—
286#1865—172no—
287#1788—172no—
288#1788—171no—
289C22×C17—22no—
290C22×C17—22no—
291C22×C17—22no—
292C22×C17—42no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17—42no—
296C22×C17—42no—
297C22×C17—41no—
298C22×C17—41no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—41no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17—42no—
305C22×C17—82no—
306C22×C17—82no—
307C22×C17—82no—
308C22×C17—82no—
309C22×C17C2×#12011yescharacteristic
310C22×C17#54612yes—
311C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C17#57511yescharacteristic
313C22×C17#57712yes—
314C4×C17—22no—
315C4×C17—24no—
316C4×C17—22no—
317C4×C17—41no—
318C4×C17—41no—
319C4×C17—42no—
320C4×C17C2×#12311yescharacteristic
321C4×C17#53411yescharacteristic
322C4×C17#54612yes—
323Dic17—82no—
324Dic17—82no—
325C22×#41C2×D1711yescharacteristic
326C22×#41C2×D1711yescharacteristic
327C22×M16—171no—
328#550—24no—
329#550—24no—
330C2×#127—21no—
331C2×#127C2×D1711yescharacteristic
332C2×#123—341no—
333C2×#121—21no—
334C2×#121—21no—
335C2×#121C2×D1711yescharacteristic
336C2×#121C2×D1711yescharacteristic
337C2×C4×Q8C2×D1711yescharacteristic
338C2×C4×D4—21no—
339C2×C4×D4C2×D1711yescharacteristic
340C22×C4⋊C4—171no—
341#481—344no—
342C2×#106—172no—
343#417—344no—
344C2×#103—172no—
345C2×#103—172no—
346C2×#102—172no—
347C2×#102—172no—
348C2×#102—172no—
349C2×#102—172no—
350C2×C22⋊C8—171no—
351C2×C22⋊C8—172no—
352#398—341no—
353#397—341no—
354#389—172no—
355#381—341no—
356#380—171no—
357#380—172no—
358#340—172no—
359#336—172no—
360C2×C17—42no—
361C2×C17—81no—
362C2×C17—82no—
363C2×C17—82no—
364C2×C17#240511yescharacteristic
365C2×C17#300611yescharacteristic
366C2×C17#300911yescharacteristic
367C2×C17#331712yes—
368C2×C17#333212yes—
369C2×#41—22no—
370C2×#41—22no—
371C2×#41—22no—
372C2×#41—24no—
373C2×#41—24no—
374C2×#41—42no—
375C2×#41—42no—
376C2×#41C22⋊D1714yes—
377C22×Q8C22×D1711yescharacteristic
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—21no—
382C22×D4—21no—
383C22×D4—21no—
384C22×D4C22⋊D1711yescharacteristic
385C22×D4C22×D1711yescharacteristic
386C22×D4C22×D1711yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—341no—
390C23×C4C22×D1711yescharacteristic
391C23×C4C22×D1711yescharacteristic
392C2×M16—342no—
393C2×M16—682no—
394C22×C8—172no—
395C22×C8—172no—
396#127—28no—
397#127—22no—
398#127—22no—
399#123—684no—
400#121—28no—
401#121—28no—
402#121—22no—
403#121—22no—
404#121—22no—
405#121C22⋊D1714yes—
406C4×Q8—22no—
407C4×Q8—22no—
408C4×D4—28no—
409C4×D4—22no—
410C4×D4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—172no—
413C2×C4⋊C4—172no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4—342no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—342no—
418C2×C4⋊C4C22⋊D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—342no—
424C2×C22⋊C4C22⋊D1711yescharacteristic
425C2×C22⋊C4C22⋊D1711yescharacteristic
426C2×C42—21no—
427C2×C42C22×D1711yescharacteristic
428#106—344no—
429#103—344no—
430#103—344no—
431#102—344no—
432#102—344no—
433#102—682no—
434#102—682no—
435C22⋊C8—342no—
436C22⋊C8—682no—
437#95—342no—
438#95—341no—
439#95—342no—
440C17#2404511yescharacteristic, complemented, Sylow
441C24—21no—
442C24—21no—
443C24C2×C22⋊D1711yescharacteristic
444C24C2×C22⋊D1711yescharacteristic
445#41—28no—
446#41—44no—
447#41—48no—
448#41—48no—
449#41—44no—
450#41—44no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8#6325414yes—
454C2×Q8C2×C22⋊D1712yes—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—41no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—82no—
475C2×D4—82no—
476C2×D4—82no—
477C2×D4C2×C22⋊D1712yes—
478C2×D4C2×C22⋊D1712yes—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—682no—
494C22×C4—682no—
495C22×C4—682no—
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1712yes—
499C22×C4C2×C22⋊D1712yes—
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C23×D1711yescharacteristic
503M16—684no—
504C2×C8—342no—
505C2×C8—342no—
506C2×C8—342no—
507C2×C8—342no—
508C2×C8—342no—
509C2×C8—342no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—342no—
514C4⋊C4—342no—
515C4⋊C4—342no—
516C4⋊C4—342no—
517C4⋊C4—684no—
518C4⋊C4—682no—
519C4⋊C4C2×C22⋊D1712yes—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—682no—
525C22⋊C4—684no—
526C42—24no—
527C42—21no—
528C42—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23#1058763912yes—
549C23C22×C22⋊D1711yescharacteristic
550Q8—22no—
551Q8—24no—
552Q8#1058762114yescomplemented
553D4—22no—
554D4—22no—
555D4—24no—
556D4—44no—
557D4—48no—
558D4—48no—
559D4—48no—
560D4—41no—
561D4—41no—
562D4—42no—
563D4—84no—
564D4—84no—
565D4—84no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—1364no—
592C2×C4#1058763512yes—
593C2×C4#1058763912yes—
594C2×C4C2×#6325412yes—
595C2×C4#1058769112yes—
596C2×C4C22×C22⋊D1711yescharacteristic
597C2×C4C22×C22⋊D1711yescharacteristic
598C8—682no—
599C8—682no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C2×#1058763911yescharacteristic
621C22#1176345312yes—
622C22C2×#1058763511yescharacteristic
623C22#1176360711yescharacteristic
624C22#1176362012yes—
625C4—22no—
626C4—24no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—1362no—
632C4—1362no—
633C4C2×#1058762111yescharacteristic
634C4#1176336811yescharacteristic
635C4#1176345312yes—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—82no—
640C2#1453210711yescharacteristic
641C2#1453803611yescharacteristic
642C2#1453806111yescharacteristic
643C2#1454091012yes—
644C2#1454111112yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#550C2
socleC23×C17C22×D4
hypercenterC2×#550D17
hypocenterC17#24045
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×#11763125C2semidirect
2. C2×#11762810C2semidirect
3. #14533092C2semidirect
4. #14530015C2semidirect
5. #14529970C2semidirect
6. #14529223C2semidirect
7. #14529082C2semidirect
8. #14529052C2semidirect
9. #14528724C2semidirect
10. #14528724C2semidirect
230. Q8×C17#123semidirect
440. C17#24045semidirect
552. Q8#10587621semidirect

Maximal subgroups

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

subgroupindex
C2×#117631252
C2×#117628102
#145330922
#145300152
#145299702
#145292232
#145290822
#145290522
#145287242
C2×C17×#5502
#2404517