Finite Group Databasebeta

← #17064322

Subgroups — #17064322

Order 4352.

880 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic17C211yesmaximal, characteristic
2#14531186C211yesmaximal, characteristic, complemented
3#14530350C211yesmaximal, characteristic, complemented
4#14530315C211yesmaximal, characteristic, complemented
5#14530263C211yesmaximal, characteristic, complemented
6#14530221C211yesmaximal, characteristic, complemented
7#14530217C211yesmaximal, characteristic, complemented
8#14530190C211yesmaximal, characteristic, complemented
9#14529970C211yesmaximal, characteristic
10#106×Dic17C211yesmaximal, characteristic, complemented
11#14528672C211yesmaximal, characteristic, complemented
12#102×Dic17C211yesmaximal, characteristic, complemented
13#102×Dic17C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic17C211yesmaximal, characteristic, complemented
15C17×#2034C211yesmaximal, characteristic
16C2×D4×Dic17—21no—
17C2×D4×Dic17C2211yescharacteristic
18C2×#10587563—21no—
19C22×C4×Dic17C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28#11762712C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762702—21no—
31#11762689—21no—
32#11762678C2211yescharacteristic
33#11762678C2211yescharacteristic
34#11762673—21no—
35#11762669C2211yescharacteristic
36#11762669C2211yescharacteristic
37#11762667—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41#11762660—21no—
42C22⋊C4×Dic17—21no—
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#121C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×C22⋊C8C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C23×Dic17D411yescharacteristic
64C23×Dic17D411yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C17×#121—22no—
68C17×#121—22no—
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79D4×Dic17—22no—
80D4×Dic17—22no—
81D4×Dic17—44no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×C4×Dic17—22no—
92C2×C4×Dic17—22no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17D411yescharacteristic
97C2×C4×Dic17D411yescharacteristic
98C2×C4×Dic17C2311yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102#10587563—44no—
103C17×#106—22no—
104C17×#102—22no—
105C17×#102—22no—
106C17×C22⋊C8—22no—
107C17×#95—21no—
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—
117#10587488—22no—
118C8×Dic17—22no—
119C8×Dic17—22no—
120#10587479—22no—
121#10587479—22no—
122C24×C17—21no—
123C24×C17C2×D411yescharacteristic
124C2×D4×C17—24no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—42no—
131C2×D4×C17—42no—
132C2×D4×C17#4112yes—
133C22×C4×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×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—41no—
153C22×Dic17—42no—
154C22×Dic17—42no—
155C22×Dic17—42no—
156C22×Dic17SD1614yes—
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17#4112yes—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17#4112yes—
173C22⋊C4×C17—24no—
174C22⋊C4×C17—42no—
175C22⋊Dic17—42no—
176C22⋊Dic17—42no—
177C22⋊Dic17—42no—
178C22⋊Dic17—42no—
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—21no—
188C4⋊Dic17—42no—
189#63225—42no—
190#63225—42no—
191#63225—42no—
192#63225—42no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—22no—
196C4×Dic17—22no—
197C4×Dic17—42no—
198C4×Dic17—42no—
199C4×Dic17—42no—
200C4×Dic17#4112yes—
201C4×Dic17#4112yes—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622#4112yes—
208#23755—171nomaximal, Sylow
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—22no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17C2×SD1612yes—
223C23×C17C22×D411yescharacteristic
224D4×C17—22no—
225D4×C17—22no—
226D4×C17—48no—
227D4×C17—41no—
228D4×C17—41no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—21no—
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×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—28no—
258C2×Dic17—44no—
259C2×Dic17—41no—
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#12312yes—
273C2×Dic17#12312yes—
274C2×Dic17C2×SD1612yes—
275C2×Dic17C2×SD1612yes—
276C2×Dic17#13512yes—
277C2×Dic17AGL(1,ℤ/8)12yes—
278C8×C17—22no—
279C8×C17—41no—
280#3622—22no—
281#3622—41no—
282C4×#121—171no—
283#2285—171no—
284#2100—171no—
285#2094—171no—
286#2084—171no—
287#2077—171no—
288#2077—171no—
289#2071—171no—
290#2034—171no—
291#2034D1711yescharacteristic
292C4×#106—171no—
293C4×#106—171no—
294C4×#102—171no—
295C4×#102—171no—
296C4×C22⋊C8—171no—
297C22×C17—22no—
298C22×C17—22no—
299C22×C17—22no—
300C22×C17—41no—
301C22×C17—41no—
302C22×C17—41no—
303C22×C17—41no—
304C22×C17—42no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—81no—
308C22×C17—81no—
309C22×C17—81no—
310C22×C17—81no—
311C22×C17C2×#12311yescharacteristic
312C22×C17#52511yescharacteristic
313C22×C17#53211yescharacteristic
314C22×C17C22×SD1611yescharacteristic
315C22×C17C2×#13511yescharacteristic
316C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
317C22×C17#57511yescharacteristic
318C4×C17—42no—
319C4×C17—41no—
320C4×C17—81no—
321C4×C17—81no—
322C4×C17#52911yescharacteristic
323C4×C17#52911yescharacteristic
324C4×C17#52911yescharacteristic
325C4×C17#52911yescharacteristic
326Dic17—22no—
327Dic17—22no—
328Dic17—22no—
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332Dic17—81no—
333Dic17#48114yes—
334C2×#121C2×D1711yescharacteristic
335C2×C4×D4—171no—
336C2×C4×D4—341no—
337C2×#117—341no—
338C22×C42—171no—
339C2×#106—171no—
340C2×#106—171no—
341C2×#106—171no—
342C2×#106C2×D1711yescharacteristic
343C2×#102—171no—
344C2×#102—171no—
345C2×#102C2×D1711yescharacteristic
346C2×#102C2×D1711yescharacteristic
347C2×C22⋊C8—171no—
348C2×C22⋊C8C2×D1711yescharacteristic
349C2×C4×C8—171no—
350C2×C4×C8—171no—
351#390—171no—
352#389—171no—
353#388—341no—
354#386—341no—
355#384—171no—
356#384—171no—
357#382—341no—
358#380—171no—
359#380—171no—
360#380—341no—
361#380C2×D1711yescharacteristic
362C4×C4⋊C4—171no—
363C4×C4⋊C4—171no—
364C4×C4⋊C4—171no—
365C4×C4⋊C4—341no—
366C4×C22⋊C4—341no—
367#340—171no—
368#340—171no—
369#340—171no—
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340C2×D1711yescharacteristic
375#336—171no—
376#336—171no—
377C2×C17—42no—
378C2×C17—82no—
379C2×C17#239511yescharacteristic
380C2×C17C2×#48111yescharacteristic
381C2×C17#309111yescharacteristic
382C2×C17#310111yescharacteristic
383C2×C17#310611yescharacteristic
384C2×C17#316711yescharacteristic
385C2×C17#317111yescharacteristic
386C22×D4—21no—
387C22×D4C22×D1711yescharacteristic
388C23×C4—171no—
389C23×C4—171no—
390C23×C4—341no—
391C23×C4C22×D1711yescharacteristic
392C22×C8—171no—
393C22×C8—171no—
394C22×C8C22×D1711yescharacteristic
395C22×C8C22×D1711yescharacteristic
396#121—22no—
397#121—22no—
398C4×D4—342no—
399C4×D4—342no—
400C4×D4—684no—
401#117—684no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—171no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—341no—
409C2×C4⋊C4—341no—
410C2×C4⋊C4—341no—
411C2×C4⋊C4C22×D1711yescharacteristic
412C2×C4⋊C4C22×D1711yescharacteristic
413C2×C4⋊C4C22×D1711yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—341no—
417C2×C22⋊C4—341no—
418C2×C22⋊C4—341no—
419C2×C22⋊C4—341no—
420C2×C22⋊C4—341no—
421C2×C42—171no—
422C2×C42—171no—
423C2×C42—171no—
424C2×C42—171no—
425C2×C42—342no—
426C2×C42—342no—
427C2×C42—341no—
428C2×C42—341no—
429C2×C42—341no—
430#106—22no—
431#106—342no—
432#106—342no—
433#106—342no—
434#102—22no—
435#102—22no—
436#102—342no—
437#102—342no—
438C22⋊C8—22no—
439C22⋊C8—342no—
440C4×C8—342no—
441C4×C8—342no—
442#95—21no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447#95—341no—
448#95—341no—
449C17#2375511yescharacteristic, complemented, Sylow
450C24—21no—
451C24D4×D1711yescharacteristic
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4#6325412yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—171no—
471C22×C4—171no—
472C22×C4—171no—
473C22×C4—171no—
474C22×C4—174no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—682no—
484C22×C4—682no—
485C22×C4—682no—
486C22×C4—681no—
487C22×C4D4×D1711yescharacteristic
488C22×C4C23×D1711yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—172no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8#6325412yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—341no—
513C4⋊C4—341no—
514C4⋊C4—682no—
515C4⋊C4—682no—
516C4⋊C4—682no—
517C4⋊C4—682no—
518C4⋊C4—682no—
519C4⋊C4#6325412yes—
520C22⋊C4—24no—
521C22⋊C4—42no—
522C22⋊C4—682no—
523C22⋊C4—682no—
524C22⋊C4—682no—
525C22⋊C4—682no—
526C22⋊C4—682no—
527C42—172no—
528C42—172no—
529C42—342no—
530C42—342no—
531C42—341no—
532C42—341no—
533C42—682no—
534C42—682no—
535C42—682no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23SD16×D1712yes—
550C23C2×D4×D1711yescharacteristic
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—172no—
573C2×C4—172no—
574C2×C4—172no—
575C2×C4—172no—
576C2×C4—172no—
577C2×C4—172no—
578C2×C4—348no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—684no—
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—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—1364no—
599C2×C4C2×#6325411yescharacteristic
600C2×C4C2×#6325411yescharacteristic
601C2×C4C2×#6325411yescharacteristic
602C2×C4#41×D1711yescharacteristic
603C2×C4#41×D1711yescharacteristic
604C2×C4#41×D1711yescharacteristic
605C8—22no—
606C8—41no—
607C8—342no—
608C8—681no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#123×D1711yescharacteristic
624C22#1176331511yescharacteristic
625C22#1176335411yescharacteristic
626C22C2×SD16×D1711yescharacteristic
627C22#135×D1711yescharacteristic
628C22AGL(1,ℤ/8)×D1711yescharacteristic
629C22#1176360511yescharacteristic
630C4—42no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—174no—
635C4—342no—
636C4—342no—
637C4—342no—
638C4—1361no—
639C4—1361no—
640C4—1361no—
641C4—1361no—
642C4#1176333611yescharacteristic
643C4#1176333811yescharacteristic
644C4#1176334011yescharacteristic
645C4#1176334011yescharacteristic
646C2—42no—
647C2—82no—
648C2#1453204011yescharacteristic
649C2D17×#48111yescharacteristic
650C2#1453869611yescharacteristic
651C2#1453878711yescharacteristic
652C2#1453883711yescharacteristic
653C2#1453933811yescharacteristic
654C2#1453937411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2034C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2034D17
hypocenterC17#23755
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
2. #14531186C2semidirect
3. #14530350C2semidirect
4. #14530315C2semidirect
5. #14530263C2semidirect
6. #14530221C2semidirect
7. #14530217C2semidirect
8. #14530190C2semidirect
10. #106×Dic17C2semidirect
11. #14528672C2semidirect
12. #102×Dic17C2semidirect
13. #102×Dic17C2semidirect
14. C22⋊C8×Dic17C2semidirect
449. C17#23755semidirect

Maximal subgroups

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

subgroupindex
#121×Dic172
#145311862
#145303502
#145303152
#145302632
#145302212
#145302172
#145301902
#145299702
#106×Dic172
#145286722
#102×Dic172
C22⋊C8×Dic172
C17×#20342
#2375517