Finite Group Databasebeta

← #17051633

Subgroups — #17051633

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762811C211yesmaximal, characteristic
2#14531651C211yesmaximal, characteristic, complemented
3#14531624C211yesmaximal, characteristic
4#14529481C211yesmaximal, characteristic, complemented
5#14529467C211yesmaximal, characteristic, complemented
6#14529412C211yesmaximal, characteristic, complemented
7#14529405C211yesmaximal, characteristic, complemented
8#14529055C211yesmaximal, characteristic, complemented
9#14529041C211yesmaximal, characteristic, complemented
10#14528875C211yesmaximal, characteristic, complemented
11#14528817C211yesmaximal, characteristic, complemented
12#14528725C211yesmaximal, characteristic
13#14528615C211yesmaximal, characteristic, complemented
14C4×#10587487C211yesmaximal, characteristic, complemented
15C17×#1753C211yesmaximal, characteristic, complemented
16C22×#63252C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
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#11762711C2211yescharacteristic
33#11762711C2211yescharacteristic
34#11762709—21no—
35#11762709—21no—
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762694—21no—
39#11762694—21no—
40#11762670—21no—
41#11762669C2211yescharacteristic
42#11762669C2211yescharacteristic
43#11762668—21no—
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46#11762656—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×Dic34C2311yescharacteristic
72C2×M16×C17—41no—
73C2×M16×C17—41no—
74C2×M16×C17C2×C414yes—
75C22×C8×C17C2311yescharacteristic
76C22×C8×C17C2311yescharacteristic
77C22×C8×C17C2311yescharacteristic
78C22×C8×C17C2311yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85C2×#63227—21no—
86C2×#63227—21no—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—22no—
91C2×C4⋊Dic17C2×C414yes—
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×#63225—21no—
101C2×#63225—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106#10587563—44no—
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—41no—
110C4×Dic34—41no—
111C17×C22⋊C8—22no—
112C17×#97—41no—
113C4×C8×C17—41no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587492—41no—
123#10587492—41no—
124#10587492C2×C414yes—
125#10587492C2×C414yes—
126#10587489—41no—
127#10587488—41no—
128#10587487—22no—
129#10587487—22no—
130#10587487C2×C414yes—
131#10587487C2×C414yes—
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—21no—
150C22×Dic17—24no—
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×D68—21no—
167C2×D68—21no—
168C2×D68C22×C412yes—
169C2×C4×D17—24no—
170C2×C4×D17—42no—
171C2×C4×D17—42no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34C22×C412yes—
176C2×Dic34#4112yes—
177C2×Dic34#4112yes—
178M16×C17—44no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17—21no—
186C2×C8×C17—21no—
187C2×C8×C17—21no—
188C2×C8×C17—21no—
189C2×C8×C17C22×C412yes—
190C2×C8×C17C22×C412yes—
191C22⋊C4×C17—22no—
192C22⋊C4×C17—41no—
193C42×C17—21no—
194C42×C17—21no—
195C42×C17—21no—
196C42×C17—21no—
197C22⋊Dic17—42no—
198C22⋊Dic17—81no—
199#63227—42no—
200#63227—42no—
201#63227—81no—
202#63227—81no—
203C4⋊Dic17—24no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—21no—
214C4⋊Dic17C22×C412yes—
215C4⋊Dic17C22×C412yes—
216#63225—42no—
217#63225—42no—
218#63225—42no—
219#63225—42no—
220C4×Dic17—42no—
221C4×Dic17—42no—
222C4×Dic17—42no—
223C4×Dic17—42no—
224#22583—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—41no—
235C22×D17—41no—
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—22no—
270C2×Dic17—22no—
271C2×Dic17—48no—
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—41no—
291D68—41no—
292C4×D17—48no—
293Dic34—22no—
294Dic34—22no—
295Dic34—22no—
296Dic34#11714yescomplemented
297C8×C17—22no—
298C8×C17—22no—
299C8×C17—41no—
300C8×C17—41no—
301C2×#417—171no—
302#2344—171no—
303#2341—171no—
304#1936—171no—
305#1934—171no—
306#1926—171no—
307#1925—171no—
308#1865—171no—
309#1863—171no—
310#1824—171no—
311#1809—171no—
312#1788—171no—
313#1763—171no—
314C4×#103—171no—
315#1753D1711yescharacteristic, complemented
316C22×C17—22no—
317C22×C17—22no—
318C22×C17—22no—
319C22×C17—41no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17C2×C4×D411yescharacteristic
324C22×C17#54611yescharacteristic
325C22×C17#54611yescharacteristic
326C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
327C22×C17#57511yescharacteristic
328C22×C17#57711yescharacteristic
329C22×C17#57811yescharacteristic
330C2×D17—42no—
331C2×D17—42no—
332C2×D17—42no—
333C2×D17—81no—
334C2×D17—81no—
335C2×D17—81no—
336C2×D17—81no—
337C4×C17—22no—
338C4×C17—22no—
339C4×C17—41no—
340C4×C17—41no—
341C4×C17—41no—
342C4×C17C2×#11711yescharacteristic
343C4×C17#51811yescharacteristic
344C4×C17#55111yescharacteristic
345C4×C17#55111yescharacteristic
346Dic17—24no—
347Dic17—42no—
348Dic17—42no—
349Dic17—81no—
350Dic17—81no—
351Dic17—81no—
352Dic17—81no—
353C22×#41—171no—
354C22×M16C2×D1711yescharacteristic
355C2×C4×Q8—171no—
356C2×C4×Q8—171no—
357C2×#117—341no—
358C22×C4⋊C4—171no—
359C2×#107—171no—
360C2×#106—171no—
361#417—178no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#102—171no—
367C2×#102—171no—
368C2×#102—171no—
369C2×#102—171no—
370C2×C22⋊C8C2×D1711yescharacteristic
371C2×#97C2×D1711yescharacteristic
372C2×C4×C8C2×D1711yescharacteristic
373#390—171no—
374#390—171no—
375#390—341no—
376#390—341no—
377#389—171no—
378#389—171no—
379#387—341no—
380#387—341no—
381#381—341no—
382#380—171no—
383#380—171no—
384#380—341no—
385C4×C4⋊C4—171no—
386C4×C4⋊C4—171no—
387C4×C22⋊C4—341no—
388C4×C22⋊C4C2×D1711yescharacteristic
389#340—171no—
390#340—171no—
391#340—171no—
392#340—171no—
393#340—171no—
394#340—171no—
395#336C2×D1711yescharacteristic
396#336C2×D1711yescharacteristic
397C2×C17—42no—
398C2×C17#234211yescharacteristic
399C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
400C2×C17#294711yescharacteristic
401C2×C17#330811yescharacteristic
402C2×C17#331711yescharacteristic
403C2×C17#332711yescharacteristic
404C2×C17#333211yescharacteristic
405D17—82no—
406C2×#41—174no—
407C2×#41—342no—
408C2×#41—342no—
409C2×#41—342no—
410C2×#41—681no—
411C2×#41—681no—
412C22×Q8—171no—
413C22×D4—171no—
414C22×D4—341no—
415C23×C4—341no—
416C23×C4—341no—
417C23×C4C22×D1711yescharacteristic
418C2×M16—41no—
419C2×M16—41no—
420C2×M16C4×D1714yes—
421C22×C8C22×D1711yescharacteristic
422C22×C8C22×D1711yescharacteristic
423C22×C8C22×D1711yescharacteristic
424C22×C8C22×D1711yescharacteristic
425C4×Q8—342no—
426C4×Q8—342no—
427C4×Q8—681no—
428C4×Q8—681no—
429#117—684no—
430C2×C4⋊C4—174no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—171no—
434C2×C4⋊C4—171no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4—171no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—342no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4—341no—
442C2×C4⋊C4—341no—
443C2×C22⋊C4—341no—
444C2×C22⋊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×C42—341no—
454C2×C42—341no—
455C2×C42C22×D1711yescharacteristic
456C2×C42C22×D1711yescharacteristic
457#107—681no—
458#106—681no—
459#103—174no—
460#103—174no—
461#103—342no—
462#103—342no—
463#102—174no—
464#102—174no—
465#102—681no—
466#102—681no—
467C22⋊C8—22no—
468#97—41no—
469C4×C8—41no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95D6811yescharacteristic
477#95D6811yescharacteristic
478C17#2258311yescharacteristic, complemented, Sylow
479C24—341no—
480C24C2×D6811yescharacteristic
481#41—348no—
482#41—684no—
483#41—684no—
484#41—684no—
485C2×Q8—172no—
486C2×Q8—172no—
487C2×Q8—172no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×D4—172no—
492C2×D4—344no—
493C2×D4—341no—
494C2×D4—341no—
495C2×D4—341no—
496C2×D4—341no—
497C2×D4—341no—
498C2×D4—682no—
499C2×D4—682no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—344no—
508C22×C4—344no—
509C22×C4—341no—
510C22×C4—341no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4—682no—
514C22×C4—682no—
515C22×C4—682no—
516C22×C4—682no—
517C22×C4C2×C4×D1712yes—
518C22×C4C2×D6811yescharacteristic
519C22×C4C2×D6811yescharacteristic
520C22×C4C2×D6811yescharacteristic
521C22×C4C2×D6811yescharacteristic
522C22×C4C2×D6811yescharacteristic
523C22×C4#6325212yes—
524C22×C4C23×D1711yescharacteristic
525M16—44no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8C2×C4×D1712yes—
537C2×C8C2×C4×D1712yes—
538C4⋊C4—172no—
539C4⋊C4—172no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—344no—
543C4⋊C4—341no—
544C4⋊C4—341no—
545C4⋊C4—341no—
546C4⋊C4—341no—
547C4⋊C4—341no—
548C4⋊C4—341no—
549C4⋊C4—341no—
550C4⋊C4—341no—
551C4⋊C4—682no—
552C4⋊C4—682no—
553C4⋊C4—682no—
554C4⋊C4—682no—
555C22⋊C4—22no—
556C22⋊C4—41no—
557C22⋊C4—682no—
558C22⋊C4—682no—
559C22⋊C4—682no—
560C22⋊C4—1361no—
561C22⋊C4—1361no—
562C22⋊C4—1361no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—682no—
568C42—682no—
569C42—682no—
570C42—682no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—342no—
578C23—681no—
579C23—681no—
580C23—681no—
581C23—681no—
582C23—681no—
583C23—681no—
584C23C4×D6812yes—
585C23C22×D6811yescharacteristic
586Q8—174no—
587Q8—342no—
588Q8—342no—
589Q8—342no—
590D4—342no—
591D4—342no—
592D4—688no—
593D4—681no—
594D4—681no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—342no—
613C2×C4—342no—
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×C4C4×D6812yes—
636C2×C4#1058759412yes—
637C2×C4C22×C4×D1711yescharacteristic
638C2×C4C2×#6325211yescharacteristic
639C2×C4#1058769111yescharacteristic
640C2×C4#1058769111yescharacteristic
641C2×C4#41×D1711yescharacteristic
642C2×C4#41×D1711yescharacteristic
643C8—22no—
644C8—22no—
645C8—41no—
646C8—41no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—682no—
655C22—682no—
656C22—682no—
657C22—1361no—
658C22—1361no—
659C22—1361no—
660C22—1361no—
661C22C2×C4×D6811yescharacteristic
662C22#1176345411yescharacteristic
663C22#1176345411yescharacteristic
664C22C2×#1058759411yescharacteristic
665C22#1176360911yescharacteristic
666C22#1176362111yescharacteristic
667C22#1176362611yescharacteristic
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—344no—
674C4—682no—
675C4—682no—
676C4—1361no—
677C4—1361no—
678C4—1361no—
679C4—1361no—
680C4#117×D1711yescharacteristic
681C4#1176328211yescharacteristic
682C4#1176348811yescharacteristic
683C4#1176348811yescharacteristic
684C2—42no—
685C2—1362no—
686C2#1453163611yescharacteristic
687C2C4×#1058759411yescharacteristic
688C2#1453762911yescharacteristic
689C2#1454077711yescharacteristic
690C2#1454091211yescharacteristic
691C2#1454105411yescharacteristic
692C2#1454111211yescharacteristic
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#22583
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. #14529481C2semidirect
5. #14529467C2semidirect
6. #14529412C2semidirect
7. #14529405C2semidirect
8. #14529055C2semidirect
9. #14529041C2semidirect
10. #14528875C2semidirect
11. #14528817C2semidirect
13. #14528615C2semidirect
14. C4×#10587487C2semidirect
15. C17×#1753C2semidirect
23. #11762811C4semidirect
296. Dic34#117semidirect
315. #1753D17semidirect
478. C17#22583semidirect

Maximal subgroups

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

subgroupindex
C2×#117628112
#145316512
#145316242
#145294812
#145294672
#145294122
#145294052
#145290552
#145290412
#145288752
#145288172
#145287252
#145286152
C4×#105874872
C17×#17532
#2258317