Finite Group Databasebeta

← #17102425

Subgroups — #17102425

Order 4352.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533127C211yesmaximal, characteristic
2#14531793C211yesmaximal, characteristic
3#14530104C211yesmaximal, characteristic, complemented
4#14529977C211yesmaximal, characteristic
5#14529933C211yesmaximal, characteristic, complemented
6#14529738C211yesmaximal, characteristic, complemented
7#14529462C211yesmaximal, characteristic, complemented
8#14529421C211yesmaximal, characteristic, complemented
9#14529387C211yesmaximal, characteristic
10#14529372C211yesmaximal, characteristic, complemented
11#14529127C211yesmaximal, characteristic, complemented
12#14529092C211yesmaximal, characteristic
13#14528871C211yesmaximal, characteristic
14#103×Dic17C211yesmaximal, characteristic, complemented
15C17×#2001C211yesmaximal, characteristic
16C2×#10587576C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762747—21no—
28#11762734—21no—
29#11762716C2211yescharacteristic
30#11762716C2211yescharacteristic
31#11762716C2211yescharacteristic
32#11762708—21no—
33#11762708C2211yescharacteristic
34#11762703—21no—
35#11762695—21no—
36#11762684—21no—
37#11762674—21no—
38#11762669C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C22⋊C4×Dic17—21no—
42C4×C22⋊Dic17C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×Q16×C17C2211yescharacteristic
53C2×C17×C22⋊Q8C2211yescharacteristic
54C2×C17×C22⋊Q8C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C2×Q16×C17—22no—
64C2×Q16×C17—41no—
65C2×Q16×C17—41no—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C17×C22⋊Q8—22no—
69C17×C22⋊Q8—22no—
70C17×C22⋊Q8—22no—
71C17×C22⋊Q8—41no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17D411yescharacteristic
80C2×C22⋊Dic17D411yescharacteristic
81Q8×Dic17—22no—
82Q8×Dic17—41no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17—41no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17C2311yescharacteristic
98C2×C4×Dic17C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587576—22no—
102#10587576—41no—
103C17×#103—22no—
104C17×#103—22no—
105C17×C22⋊C8—41no—
106#10587506—22no—
107#10587506—22no—
108#10587506—41no—
109#10587506—41no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502D411yescharacteristic
121#10587502D411yescharacteristic
122C22⋊#3622—41no—
123#10587485—41no—
124C8×Dic17—41no—
125#10587479—22no—
126C24×C17C2×D411yescharacteristic
127C2×Q8×C17—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—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17#4112yes—
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—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—81no—
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162Q16×C17—44no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C4⋊C4×C17—24no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—42no—
173C4⋊C4×C17#4112yes—
174C4⋊C4×C17#4112yes—
175C22⋊C4×C17—24no—
176C22⋊C4×C17—42no—
177C22⋊Dic17—22no—
178C22⋊Dic17—22no—
179C22⋊Dic17—42no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—42no—
188C4⋊Dic17—42no—
189C4⋊Dic17—42no—
190C4⋊Dic17—81no—
191C4⋊Dic17#4112yes—
192#63225—42no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—42no—
198C4×Dic17—42no—
199C4×Dic17—81no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206#26725—171nomaximal, Sylow
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17#13512yes—
214C23×C17C22×D411yescharacteristic
215Q8×C17—22no—
216Q8×C17—22no—
217Q8×C17—41no—
218Q8×C17—41no—
219Q8×C17—41no—
220Q8×C17—41no—
221Q8×C17—41no—
222Q8×C17—41no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—21no—
226C2×C4×C17—22no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C17H(5,𝔽2)11yescharacteristic
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—22no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—84no—
272C2×Dic17#13712yes—
273C8×C17—41no—
274C8×C17—41no—
275#3622—41no—
276#3622—41no—
277#2489—171no—
278#2360—171no—
279#2056—171no—
280#2035—171no—
281#2026—171no—
282#2001D1711yescharacteristic
283#1989—171no—
284#1933—171no—
285#1927—171no—
286#1923—171no—
287#1920—171no—
288#1878—171no—
289#1872—171no—
290#1823—171no—
291C4×#103—171no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—22no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—41no—
298C22×C17—41no—
299C22×C17#52511yescharacteristic
300C22×C17#54611yescharacteristic
301C22×C17#54611yescharacteristic
302C22×C17C2×#13511yescharacteristic
303C22×C17C2×#13711yescharacteristic
304C22×C17#57711yescharacteristic
305C22×C17#57811yescharacteristic
306C4×C17—42no—
307C4×C17—42no—
308C4×C17—41no—
309C4×C17—42no—
310C4×C17—81no—
311C4×C17#52911yescharacteristic
312C4×C17#53811yescharacteristic
313C4×C17#55511yescharacteristic
314C4×C17#55911yescharacteristic
315Dic17—22no—
316Dic17—41no—
317Dic17—41no—
318Dic17—42no—
319Dic17—41no—
320Dic17—81no—
321Dic17—81no—
322Dic17—81no—
323C22×Q16C2×D1711yescharacteristic
324C2×#125—171no—
325C2×C22⋊Q8C2×D1711yescharacteristic
326C2×C22⋊Q8C2×D1711yescharacteristic
327C2×C4×Q8—171no—
328C22×C4⋊C4—171no—
329C2×#106—171no—
330C2×C4⋊C8—171no—
331C2×#103—171no—
332C2×#103—171no—
333C2×#103—171no—
334C2×#103—171no—
335C2×#103C2×D1711yescharacteristic
336C2×#103C2×D1711yescharacteristic
337C2×C22⋊C8—171no—
338C2×C22⋊C8C2×D1711yescharacteristic
339C2×C4×C8—171no—
340#400—341no—
341#397—341no—
342#391—171no—
343#391—171no—
344#391—171no—
345#389—171no—
346#389—341no—
347#388—341no—
348#387—341no—
349#385—341no—
350#382—341no—
351#380—171no—
352C4×C4⋊C4—171no—
353C4×C4⋊C4—171no—
354C4×C22⋊C4—171no—
355C4×C22⋊C4—341no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340C2×D1711yescharacteristic
364#336—171no—
365#336—171no—
366C2×C17—42no—
367C2×C17#245911yescharacteristic
368C2×C17#310811yescharacteristic
369C2×C17#312011yescharacteristic
370C2×C17#329611yescharacteristic
371C2×C17#330011yescharacteristic
372C2×C17#331211yescharacteristic
373C2×C17#333111yescharacteristic
374C22×Q8C22×D1711yescharacteristic
375C22×Q8C22×D1711yescharacteristic
376C23×C4—341no—
377C23×C4C22×D1711yescharacteristic
378C2×Q16—22no—
379C2×Q16—41no—
380C2×Q16—41no—
381C22×C8—171no—
382C22×C8—171no—
383C22×C8C22×D1711yescharacteristic
384C22×C8C22×D1711yescharacteristic
385#125—342no—
386#125—681no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—41no—
391C4×Q8—342no—
392C4×Q8—681no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—342no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—681no—
406C2×C4⋊C4C22×D1711yescharacteristic
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—171no—
411C2×C22⋊C4—171no—
412C2×C22⋊C4—341no—
413C2×C42—171no—
414C2×C42—171no—
415C2×C42—341no—
416C2×C42—341no—
417C2×C42—341no—
418#106—342no—
419C4⋊C8—681no—
420#103—22no—
421#103—22no—
422#103—342no—
423#103—342no—
424#103—681no—
425#103—681no—
426C22⋊C8—41no—
427C22⋊C8—681no—
428C4×C8—681no—
429#95—171no—
430#95—171no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441C17#2672511yescharacteristic, complemented, Sylow
442C24D4×D1711yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8#6325412yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—171no—
468C22×C4—171no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4—682no—
474C22×C4—682no—
475C22×C4—1361no—
476C22×C4D4×D1711yescharacteristic
477C22×C4C23×D1711yescharacteristic
478Q16—44no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—172no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—682no—
504C4⋊C4—682no—
505C4⋊C4—682no—
506C4⋊C4—682no—
507C4⋊C4—1361no—
508C4⋊C4#6325412yes—
509C4⋊C4#6325412yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—342no—
513C22⋊C4—342no—
514C22⋊C4—682no—
515C42—341no—
516C42—341no—
517C42—341no—
518C42—341no—
519C42—682no—
520C42—682no—
521C42—1361no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1058760512yes—
529C23C2×D4×D1711yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—172no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—342no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—1364no—
582C2×C4C2×#6325411yescharacteristic
583C2×C4C2×#6325411yescharacteristic
584C2×C4C2×#6325411yescharacteristic
585C2×C4#1058769111yescharacteristic
586C2×C4#1058769111yescharacteristic
587C2×C4#41×D1711yescharacteristic
588C8—41no—
589C8—41no—
590C8—681no—
591C8—681no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1176331511yescharacteristic
600C22#1176344611yescharacteristic
601C22#1176344811yescharacteristic
602C22C2×#1058760511yescharacteristic
603C22#137×D1711yescharacteristic
604C22#1176361611yescharacteristic
605C22#1176362511yescharacteristic
606C4—42no—
607C4—41no—
608C4—42no—
609C4—42no—
610C4—81no—
611C4—342no—
612C4—681no—
613C4—681no—
614C4—682no—
615C4—681no—
616C4—1361no—
617C4—1361no—
618C4—1361no—
619C4#1176334011yescharacteristic
620C4#1176339411yescharacteristic
621C4#1176352311yescharacteristic
622C4#1176354511yescharacteristic
623C2—42no—
624C2#1453271611yescharacteristic
625C2#1453886511yescharacteristic
626C2#1453899111yescharacteristic
627C2#1454062511yescharacteristic
628C2#1454067511yescharacteristic
629C2#1454083011yescharacteristic
630C2#1454109311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2001C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2001D17
hypocenterC17#26725
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
3. #14530104C2semidirect
5. #14529933C2semidirect
6. #14529738C2semidirect
7. #14529462C2semidirect
8. #14529421C2semidirect
10. #14529372C2semidirect
11. #14529127C2semidirect
14. #103×Dic17C2semidirect
441. C17#26725semidirect

Maximal subgroups

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

subgroupindex
#145331272
#145317932
#145301042
#145299772
#145299332
#145297382
#145294622
#145294212
#145293872
#145293722
#145291272
#145290922
#145288712
#103×Dic172
C17×#20012
#2672517