Finite Group Databasebeta

← #17064355

Subgroups — #17064355

Order 4352.

836 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
1#121×Dic17C211yesmaximal, characteristic
2#14531179C211yesmaximal, characteristic, complemented
3#14530355C211yesmaximal, characteristic, complemented
4#14530303C211yesmaximal, characteristic, complemented
5#14530268C211yesmaximal, characteristic, complemented
6#14530220C211yesmaximal, characteristic, complemented
7#14530212C211yesmaximal, characteristic, complemented
8#14530200C211yesmaximal, characteristic, complemented
9#14529970C211yesmaximal, characteristic
10#107×Dic17C211yesmaximal, characteristic, complemented
11#14528672C211yesmaximal, characteristic, complemented
12#102×Dic17C211yesmaximal, characteristic, complemented
13#102×Dic17C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic17C211yesmaximal, characteristic, complemented
15C17×#2033C211yesmaximal, characteristic
16C2×D4×Dic17—21no—
17C2×D4×Dic17C2211yescharacteristic
18C22×C4×Dic17C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C8×Dic17C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762712C2211yescharacteristic
28#11762708C2211yescharacteristic
29#11762702—21no—
30#11762689—21no—
31#11762676—21no—
32#11762674C2211yescharacteristic
33#11762674C2211yescharacteristic
34#11762670—21no—
35#11762669C2211yescharacteristic
36#11762669C2211yescharacteristic
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C22⋊C4×Dic17—21no—
41C22⋊C4×Dic17—21no—
42#11762654—21no—
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#121C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
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×C4×Dic17—22no—
88C2×C4×Dic17—22no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17D411yescharacteristic
93C2×C4×Dic17D411yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C17×#107—22no—
99C17×#102—22no—
100C17×#102—22no—
101C17×C22⋊C8—22no—
102C17×#95—21no—
103#10587503—22no—
104#10587503—22no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115C22⋊#3622—22no—
116#10587489—22no—
117C8×Dic17—22no—
118C8×Dic17—22no—
119#10587479—22no—
120#10587479—22no—
121C24×C17—21no—
122C24×C17C2×D411yescharacteristic
123C2×D4×C17—24no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—42no—
130C2×D4×C17—42no—
131C2×D4×C17#4112yes—
132C22×C4×C17—21no—
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×C17C2×D411yescharacteristic
142C22×C4×C17C2411yescharacteristic
143C22×Dic17—22no—
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×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17#4112yes—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17#4112yes—
172C22⋊C4×C17—24no—
173C22⋊C4×C17—42no—
174C22⋊Dic17—42no—
175C22⋊Dic17—42no—
176C22⋊Dic17—42no—
177C22⋊Dic17—42no—
178C22⋊Dic17—42no—
179C4⋊Dic17—21no—
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—21no—
189C4×Dic17—21no—
190C4×Dic17—22no—
191C4×Dic17—22no—
192C4×Dic17—42no—
193C4×Dic17—42no—
194C4×Dic17—42no—
195C4×Dic17#4112yes—
196C4×Dic17#4112yes—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622#4112yes—
203#23758—171nomaximal, Sylow
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—22no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17#13512yes—
218C23×C17C22×D411yescharacteristic
219D4×C17—22no—
220D4×C17—22no—
221D4×C17—48no—
222D4×C17—41no—
223D4×C17—41no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—22no—
227C2×C4×C17—21no—
228C2×C4×C17—41no—
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×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—44no—
255C2×Dic17—44no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
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—84no—
269C2×Dic17#12312yes—
270C2×Dic17#12312yes—
271C2×Dic17#13512yes—
272C2×Dic17AGL(1,ℤ/8)12yes—
273C8×C17—22no—
274C8×C17—41no—
275#3622—22no—
276#3622—41no—
277C4×#121—171no—
278#2283—171no—
279#2101—171no—
280#2092—171no—
281#2085—171no—
282#2077—171no—
283#2076—171no—
284#2073—171no—
285#2034—171no—
286#2033D1711yescharacteristic
287C4×#107—171no—
288C4×#106—171no—
289C4×#102—171no—
290C4×#102—171no—
291C4×C22⋊C8—171no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—22no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—41no—
298C22×C17—41no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—42no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—81no—
305C22×C17—81no—
306C22×C17C2×#12311yescharacteristic
307C22×C17#52511yescharacteristic
308C22×C17#53211yescharacteristic
309C22×C17C2×#13511yescharacteristic
310C22×C17C2×#13511yescharacteristic
311C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C17#57511yescharacteristic
313C4×C17—42no—
314C4×C17—41no—
315C4×C17—81no—
316C4×C17—81no—
317C4×C17#52911yescharacteristic
318C4×C17#52911yescharacteristic
319C4×C17#52911yescharacteristic
320C4×C17#52911yescharacteristic
321Dic17—22no—
322Dic17—41no—
323Dic17—41no—
324Dic17—81no—
325Dic17—81no—
326Dic17—81no—
327Dic17—81no—
328Dic17#48214yes—
329C2×#121C2×D1711yescharacteristic
330C2×C4×D4—171no—
331C2×C4×D4—341no—
332C22×C42—171no—
333C2×#107—171no—
334C2×#107C2×D1711yescharacteristic
335C2×#106—171no—
336C2×#106—171no—
337C2×#102—171no—
338C2×#102—171no—
339C2×#102C2×D1711yescharacteristic
340C2×#102C2×D1711yescharacteristic
341C2×C22⋊C8—171no—
342C2×C22⋊C8C2×D1711yescharacteristic
343C2×C4×C8—171no—
344C2×C4×C8—171no—
345#390—171no—
346#389—171no—
347#388—341no—
348#386—341no—
349#383—341no—
350#382—171no—
351#382—171no—
352#381—341no—
353#380—171no—
354#380—171no—
355#380C2×D1711yescharacteristic
356C4×C4⋊C4—171no—
357C4×C4⋊C4—171no—
358C4×C4⋊C4—171no—
359C4×C22⋊C4—341no—
360C4×C22⋊C4—341no—
361#376—341no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340—171no—
367#340—171no—
368#340—171no—
369#340C2×D1711yescharacteristic
370#336—171no—
371#336—171no—
372C2×C17—42no—
373C2×C17—82no—
374C2×C17#239511yescharacteristic
375C2×C17C2×#48211yescharacteristic
376C2×C17#309111yescharacteristic
377C2×C17#310211yescharacteristic
378C2×C17#310511yescharacteristic
379C2×C17#316811yescharacteristic
380C2×C17#317111yescharacteristic
381C22×D4—21no—
382C22×D4C22×D1711yescharacteristic
383C23×C4—171no—
384C23×C4—171no—
385C23×C4—341no—
386C23×C4C22×D1711yescharacteristic
387C22×C8—171no—
388C22×C8—171no—
389C22×C8C22×D1711yescharacteristic
390C22×C8C22×D1711yescharacteristic
391#121—22no—
392#121—22no—
393C4×D4—342no—
394C4×D4—342no—
395C4×D4—684no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—341no—
407C2×C22⋊C4—341no—
408C2×C22⋊C4—341no—
409C2×C22⋊C4—341no—
410C2×C22⋊C4—341no—
411C2×C42—171no—
412C2×C42—171no—
413C2×C42—171no—
414C2×C42—171no—
415C2×C42—342no—
416C2×C42—342no—
417C2×C42—341no—
418C2×C42—341no—
419C2×C42—341no—
420#107—22no—
421#107—342no—
422#106—342no—
423#106—342no—
424#102—22no—
425#102—22no—
426#102—342no—
427#102—342no—
428C22⋊C8—22no—
429C22⋊C8—342no—
430C4×C8—342no—
431C4×C8—342no—
432#95—21no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441#95—341no—
442#95—341no—
443C17#2375811yescharacteristic, complemented, Sylow
444C24—21no—
445C24D4×D1711yescharacteristic
446C2×D4—24no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4#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—171no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—171no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—682no—
478C22×C4—682no—
479C22×C4—682no—
480C22×C4—681no—
481C22×C4D4×D1711yescharacteristic
482C22×C4C23×D1711yescharacteristic
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—172no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8#6325412yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—682no—
509C4⋊C4#6325412yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—682no—
513C22⋊C4—682no—
514C22⋊C4—682no—
515C22⋊C4—682no—
516C22⋊C4—682no—
517C42—172no—
518C42—172no—
519C42—342no—
520C42—342no—
521C42—341no—
522C42—341no—
523C42—682no—
524C42—682no—
525C42—682no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23#1058759812yes—
540C23C2×D4×D1711yescharacteristic
541D4—22no—
542D4—22no—
543D4—48no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—172no—
563C2×C4—172no—
564C2×C4—172no—
565C2×C4—172no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—341no—
570C2×C4—341no—
571C2×C4—341no—
572C2×C4—341no—
573C2×C4—341no—
574C2×C4—684no—
575C2×C4—684no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—1364no—
589C2×C4C2×#6325411yescharacteristic
590C2×C4C2×#6325411yescharacteristic
591C2×C4C2×#6325411yescharacteristic
592C2×C4#41×D1711yescharacteristic
593C2×C4#41×D1711yescharacteristic
594C2×C4#41×D1711yescharacteristic
595C8—22no—
596C8—41no—
597C8—342no—
598C8—681no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#123×D1711yescharacteristic
614C22#1176331511yescharacteristic
615C22#1176335411yescharacteristic
616C22#135×D1711yescharacteristic
617C22C2×#1058759811yescharacteristic
618C22AGL(1,ℤ/8)×D1711yescharacteristic
619C22#1176360811yescharacteristic
620C4—42no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—174no—
625C4—342no—
626C4—681no—
627C4—681no—
628C4—1361no—
629C4—1361no—
630C4—1361no—
631C4—1361no—
632C4#1176333611yescharacteristic
633C4#1176333811yescharacteristic
634C4#1176334011yescharacteristic
635C4#1176334011yescharacteristic
636C2—42no—
637C2—82no—
638C2#1453204011yescharacteristic
639C2D17×#48211yescharacteristic
640C2#1453869211yescharacteristic
641C2#1453880311yescharacteristic
642C2#1453882611yescharacteristic
643C2#1453934911yescharacteristic
644C2#1453937211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2033C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2033D17
hypocenterC17#23758
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. #14531179C2semidirect
3. #14530355C2semidirect
4. #14530303C2semidirect
5. #14530268C2semidirect
6. #14530220C2semidirect
7. #14530212C2semidirect
8. #14530200C2semidirect
10. #107×Dic17C2semidirect
11. #14528672C2semidirect
12. #102×Dic17C2semidirect
13. #102×Dic17C2semidirect
14. C22⋊C8×Dic17C2semidirect
443. C17#23758semidirect

Maximal subgroups

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

subgroupindex
#121×Dic172
#145311792
#145303552
#145303032
#145302682
#145302202
#145302122
#145302002
#145299702
#107×Dic172
#145286722
#102×Dic172
C22⋊C8×Dic172
C17×#20332
#2375817