Finite Group Databasebeta

← #17051618

Subgroups — #17051618

Order 4352.

1038 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762811C211yesmaximal, characteristic
2#14531651C211yesmaximal, characteristic, complemented
3#14531615C211yesmaximal, characteristic
4#14529474C211yesmaximal, characteristic, complemented
5#14529460C211yesmaximal, characteristic, complemented
6#14529426C211yesmaximal, characteristic, complemented
7#14529419C211yesmaximal, characteristic, complemented
8#14529048C211yesmaximal, characteristic, complemented
9#14529034C211yesmaximal, characteristic, complemented
10#14528875C211yesmaximal, characteristic, complemented
11#14528817C211yesmaximal, characteristic, complemented
12#14528725C211yesmaximal, characteristic
13#14528610C211yesmaximal, characteristic, complemented
14C4×#10587492C211yesmaximal, characteristic, complemented
15C17×#1753C211yesmaximal, characteristic, complemented
16C22×#63252C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587563—21no—
20C22×C4⋊Dic17C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23#11762811C418yescomplemented
24C2×#10587487C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587492C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31C2×#10587492C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762715C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762694—21no—
37#11762694—21no—
38#11762685—21no—
39#11762685—21no—
40#11762671—21no—
41#11762670—21no—
42#11762669C2211yescharacteristic
43#11762669C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46#95×D17—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×C22⋊C8C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C4×C22⋊C4×C17C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C22×C22⋊D17—21no—
62C23×Dic17—21no—
63C2×#63252—22no—
64C2×#63252—22no—
65C2×#63252—22no—
66C2×#63252—41no—
67C2×#63252—41no—
68C2×#63252C2×C414yes—
69C22×D68C2311yescharacteristic
70C22×C4×D17—21no—
71C22×C4×D17—21no—
72C22×C4×D17—21no—
73C22×Dic34C2311yescharacteristic
74C2×M16×C17—41no—
75C2×M16×C17—41no—
76C2×M16×C17C2×C414yescomplemented
77C22×C8×C17C2311yescharacteristic
78C22×C8×C17C2311yescharacteristic
79C22×C8×C17C2311yescharacteristic
80C22×C8×C17C2311yescharacteristic
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C22⋊C4×C17D411yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—22no—
91C2×C4⋊Dic17C2×C414yescomplemented
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×#63225—21no—
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102#10587563—44no—
103C4×D68—22no—
104C4×D68—22no—
105C4×D68—41no—
106C4×D68—41no—
107C17×C22⋊C8—22no—
108C17×#97—41no—
109C4×C8×C17—41no—
110C17×#95D411yescharacteristic
111C17×#95D411yescharacteristic
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#10587502—21no—
121#10587502—21no—
122#10587492—22no—
123#10587492—22no—
124#10587492C2×C414yes—
125#10587492C2×C414yes—
126#10587489—41no—
127#10587488—41no—
128#10587487—41no—
129#10587487—41no—
130#10587487C2×C414yescomplemented
131#10587487C2×C414yescomplemented
132C24×C17C2×D411yescharacteristic
133C23×D17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17#4112yes—
145C22×C4×C17C2411yescharacteristic
146C2×C22⋊D17—24no—
147C2×C22⋊D17—42no—
148C2×C22⋊D17—42no—
149C22×Dic17—24no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159#63252—28no—
160#63252—44no—
161#63252—44no—
162#63252—44no—
163C2×D68—21no—
164C2×D68—21no—
165C2×D68—21no—
166C2×D68C22×C412yes—
167C2×D68#4112yes—
168C2×D68#4112yes—
169C2×C4×D17—24no—
170C2×C4×D17—42no—
171C2×C4×D17—42no—
172C2×C4×D17—42no—
173C2×C4×D17—42no—
174C2×C4×D17—42no—
175C2×C4×D17—42no—
176C2×C4×D17—81no—
177C2×C4×D17—81no—
178C2×Dic34—21no—
179C2×Dic34—21no—
180C2×Dic34—21no—
181C2×Dic34—21no—
182C2×Dic34—21no—
183C2×Dic34C22×C412yes—
184M16×C17—44no—
185C2×C8×C17—21no—
186C2×C8×C17—21no—
187C2×C8×C17—21no—
188C2×C8×C17—21no—
189C2×C8×C17—21no—
190C2×C8×C17—21no—
191C2×C8×C17—21no—
192C2×C8×C17—21no—
193C2×C8×C17—21no—
194C2×C8×C17—21no—
195C2×C8×C17C22×C412yes—
196C2×C8×C17C22×C412yes—
197C22⋊C4×C17—22no—
198C22⋊C4×C17—41no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17—21no—
202C42×C17—21no—
203C22⋊Dic17—42no—
204C22⋊Dic17—81no—
205#63227—42no—
206#63227—42no—
207C4⋊Dic17—24no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—21no—
214C4⋊Dic17—21no—
215C4⋊Dic17—21no—
216C4⋊Dic17—21no—
217C4⋊Dic17—21no—
218C4⋊Dic17C22×C412yes—
219C4⋊Dic17C22×C412yes—
220#63225—42no—
221#63225—42no—
222C4×Dic17—42no—
223C4×Dic17—42no—
224#22582—171nomaximal, Sylow
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17C4×D412yes—
232C23×C17C22×D411yescharacteristic
233C22×D17—22no—
234C22×D17—22no—
235C22×D17—22no—
236C22×D17—41no—
237C22×D17—41no—
238C22×D17—41no—
239C22×D17—41no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17C4×D412yes—
258C2×C4×C17AGL(1,ℤ/8)12yes—
259C2×C4×C17C23×C411yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17H(5,𝔽2)11yescharacteristic
264C2×C4×C17H(5,𝔽2)11yescharacteristic
265C22⋊D17—48no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—48no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288D68—22no—
289D68—22no—
290D68—22no—
291D68#11714yescomplemented
292C4×D17—48no—
293C4×D17—84no—
294C4×D17—84no—
295Dic34—22no—
296Dic34—22no—
297Dic34—41no—
298Dic34—41no—
299C8×C17—22no—
300C8×C17—22no—
301C8×C17—41no—
302C8×C17—41no—
303C2×#417—171no—
304#2344—171no—
305#2339—171no—
306#1935—171no—
307#1933—171no—
308#1928—171no—
309#1927—171no—
310#1864—171no—
311#1862—171no—
312#1824—171no—
313#1809—171no—
314#1788—171no—
315#1762—171no—
316C4×#102—171no—
317#1753D1711yescharacteristic, complemented
318C22×C17—22no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17—41no—
325C22×C17C2×C4×D411yescharacteristic
326C22×C17#54611yescharacteristic
327C22×C17#54611yescharacteristic
328C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
329C22×C17#57511yescharacteristic
330C22×C17#57711yescharacteristic
331C22×C17#57811yescharacteristic
332C2×D17—24no—
333C2×D17—42no—
334C2×D17—42no—
335C2×D17—42no—
336C2×D17—42no—
337C2×D17—42no—
338C2×D17—42no—
339C4×C17—22no—
340C4×C17—22no—
341C4×C17—41no—
342C4×C17—41no—
343C4×C17—41no—
344C4×C17C2×#11711yescharacteristic
345C4×C17#51811yescharacteristic
346C4×C17#55111yescharacteristic
347C4×C17#55111yescharacteristic
348Dic17—42no—
349Dic17—42no—
350Dic17—42no—
351Dic17—81no—
352Dic17—81no—
353Dic17—81no—
354Dic17—81no—
355C22×#41—171no—
356C22×M16C2×D1711yescharacteristic
357C2×C4×D4—171no—
358C2×C4×D4—171no—
359C2×#117—341no—
360C22×C4⋊C4—171no—
361C2×#107—171no—
362C2×#106—171no—
363#417—178no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#102—171no—
369C2×#102—171no—
370C2×#102—171no—
371C2×#102—171no—
372C2×C22⋊C8C2×D1711yescharacteristic
373C2×#97C2×D1711yescharacteristic
374C2×C4×C8C2×D1711yescharacteristic
375#391—171no—
376#391—171no—
377#389—171no—
378#389—171no—
379#387—341no—
380#387—341no—
381#386—341no—
382#386—341no—
383#381—341no—
384#381—341no—
385#380—171no—
386#380—171no—
387C4×C4⋊C4—171no—
388C4×C4⋊C4—171no—
389C4×C22⋊C4C2×D1711yescharacteristic
390C2×#95—341no—
391#340—171no—
392#340—171no—
393#340—171no—
394#340—171no—
395#340—171no—
396#340—171no—
397#336C2×D1711yescharacteristic
398#336C2×D1711yescharacteristic
399C2×C17—42no—
400C2×C17#234211yescharacteristic
401C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
402C2×C17#294711yescharacteristic
403C2×C17#330611yescharacteristic
404C2×C17#331511yescharacteristic
405C2×C17#332311yescharacteristic
406C2×C17#333011yescharacteristic
407D17—44no—
408C2×#41—174no—
409C2×#41—342no—
410C2×#41—342no—
411C2×#41—342no—
412C2×#41—681no—
413C2×#41—681no—
414C22×Q8—171no—
415C22×D4—171no—
416C22×D4—341no—
417C23×C4—341no—
418C23×C4—341no—
419C23×C4—341no—
420C23×C4—341no—
421C23×C4C22×D1711yescharacteristic
422C2×M16—41no—
423C2×M16—41no—
424C2×M16C4×D1714yescomplemented
425C22×C8C22×D1711yescharacteristic
426C22×C8C22×D1711yescharacteristic
427C22×C8C22×D1711yescharacteristic
428C22×C8C22×D1711yescharacteristic
429C4×D4—342no—
430C4×D4—342no—
431C4×D4—681no—
432C4×D4—681no—
433#117—684no—
434C2×C4⋊C4—174no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4—171no—
437C2×C4⋊C4—171no—
438C2×C4⋊C4—171no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4—171no—
441C2×C4⋊C4—341no—
442C2×C4⋊C4—342no—
443C2×C4⋊C4—341no—
444C2×C4⋊C4—341no—
445C2×C22⋊C4—341no—
446C2×C22⋊C4—341no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4D6811yescharacteristic
450C2×C22⋊C4D6811yescharacteristic
451C2×C42—341no—
452C2×C42—341no—
453C2×C42C22×D1711yescharacteristic
454C2×C42C22×D1711yescharacteristic
455#107—681no—
456#106—681no—
457#103—174no—
458#103—174no—
459#103—681no—
460#103—681no—
461#102—174no—
462#102—174no—
463#102—342no—
464#102—342no—
465C22⋊C8—22no—
466#97—41no—
467C4×C8—41no—
468#95—341no—
469#95—341no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478#95D6811yescharacteristic
479#95D6811yescharacteristic
480C17#2258211yescharacteristic, complemented, Sylow
481C24—341no—
482C24C2×D6811yescharacteristic
483#41—348no—
484#41—684no—
485#41—684no—
486#41—684no—
487C2×Q8—172no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×D4—172no—
494C2×D4—172no—
495C2×D4—172no—
496C2×D4—344no—
497C2×D4—341no—
498C2×D4—341no—
499C2×D4—341no—
500C2×D4—682no—
501C2×D4—682no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—344no—
510C22×C4—344no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—341no—
515C22×C4—682no—
516C22×C4—682no—
517C22×C4—682no—
518C22×C4—682no—
519C22×C4—682no—
520C22×C4—682no—
521C22×C4—682no—
522C22×C4—682no—
523C22×C4—1361no—
524C22×C4—1361no—
525C22×C4C2×C4×D1712yes—
526C22×C4C2×D6811yescharacteristic
527C22×C4C2×D6811yescharacteristic
528C22×C4C2×D6811yescharacteristic
529C22×C4C2×D6811yescharacteristic
530C22×C4C2×D6811yescharacteristic
531C22×C4#6325212yes—
532C22×C4C23×D1711yescharacteristic
533M16—44no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8C2×C4×D1712yes—
545C2×C8C2×C4×D1712yes—
546C4⋊C4—172no—
547C4⋊C4—172no—
548C4⋊C4—341no—
549C4⋊C4—341no—
550C4⋊C4—344no—
551C4⋊C4—341no—
552C4⋊C4—341no—
553C4⋊C4—341no—
554C4⋊C4—341no—
555C4⋊C4—341no—
556C4⋊C4—341no—
557C4⋊C4—341no—
558C4⋊C4—341no—
559C4⋊C4—682no—
560C4⋊C4—682no—
561C22⋊C4—22no—
562C22⋊C4—41no—
563C22⋊C4—682no—
564C22⋊C4—682no—
565C22⋊C4—682no—
566C22⋊C4—1361no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—682no—
572C42—682no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—342no—
580C23—342no—
581C23—342no—
582C23—681no—
583C23—681no—
584C23—681no—
585C23—681no—
586C23C4×D6812yes—
587C23C22×D6811yescharacteristic
588Q8—342no—
589Q8—342no—
590Q8—681no—
591Q8—681no—
592D4—174no—
593D4—342no—
594D4—342no—
595D4—342no—
596D4—688no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—688no—
618C2×C4—688no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4—681no—
635C2×C4—681no—
636C2×C4—681no—
637C2×C4—1364no—
638C2×C4—1364no—
639C2×C4C4×D6812yes—
640C2×C4#1058759412yes—
641C2×C4C22×C4×D1711yescharacteristic
642C2×C4C2×#6325211yescharacteristic
643C2×C4#1058769111yescharacteristic
644C2×C4#1058769111yescharacteristic
645C2×C4#41×D1711yescharacteristic
646C2×C4#41×D1711yescharacteristic
647C8—22no—
648C8—22no—
649C8—41no—
650C8—41no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—344no—
659C22—682no—
660C22—682no—
661C22—682no—
662C22—682no—
663C22—682no—
664C22—682no—
665C22C2×C4×D6811yescharacteristic
666C22#1176345411yescharacteristic
667C22#1176345411yescharacteristic
668C22C2×#1058759411yescharacteristic
669C22#1176360911yescharacteristic
670C22#1176362111yescharacteristic
671C22#1176362611yescharacteristic
672C4—22no—
673C4—22no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—682no—
678C4—682no—
679C4—682no—
680C4—1361no—
681C4—1361no—
682C4—1361no—
683C4—1361no—
684C4#117×D1711yescharacteristic
685C4#1176328211yescharacteristic
686C4#1176348811yescharacteristic
687C4#1176348811yescharacteristic
688C2—42no—
689C2—684no—
690C2#1453163611yescharacteristic
691C2C4×#1058759411yescharacteristic
692C2#1453762911yescharacteristic
693C2#1454074711yescharacteristic
694C2#1454088211yescharacteristic
695C2#1454100211yescharacteristic
696C2#1454109011yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1753C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#1753D17
hypocenterC17#22582
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
2. #14531651C2semidirect
4. #14529474C2semidirect
5. #14529460C2semidirect
6. #14529426C2semidirect
7. #14529419C2semidirect
8. #14529048C2semidirect
9. #14529034C2semidirect
10. #14528875C2semidirect
11. #14528817C2semidirect
13. #14528610C2semidirect
14. C4×#10587492C2semidirect
15. C17×#1753C2semidirect
23. #11762811C4semidirect
76. C2×M16×C17C2×C4semidirect
91. C2×C4⋊Dic17C2×C4semidirect
130. #10587487C2×C4semidirect
131. #10587487C2×C4semidirect
291. D68#117semidirect
317. #1753D17semidirect
424. C2×M16C4×D17semidirect
480. C17#22582semidirect

Maximal subgroups

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

subgroupindex
C2×#117628112
#145316512
#145316152
#145294742
#145294602
#145294262
#145294192
#145290482
#145290342
#145288752
#145288172
#145287252
#145286102
C4×#105874922
C17×#17532
#2258217