Finite Group Databasebeta

← #17040888

Subgroups — #17040888

Order 4352.

1014 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763049C211yesmaximal, characteristic
2C2×#11762914C211yesmaximal, characteristic, complemented
3C2×#97×D17C211yesmaximal, characteristic
4C4×#10587583C211yesmaximal, characteristic
5C4×#10587632C211yesmaximal, characteristic
6#14529652C211yesmaximal, characteristic, complemented
7#14529521C211yesmaximal, characteristic, complemented
8#14529476C211yesmaximal, characteristic, complemented
9#14529472C211yesmaximal, characteristic, complemented
10#14529206C211yesmaximal, characteristic
11#14529204C211yesmaximal, characteristic
12#14528808C211yesmaximal, characteristic
13#14528682C211yesmaximal, characteristic, complemented
14#14528615C211yesmaximal, characteristic, complemented
15C17×#1763C211yesmaximal, characteristic, complemented
16C22×#63231C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C22×C8×D17C2211yescharacteristic
19C2×#10587583C2211yescharacteristic
20C2×#10587632C2211yescharacteristic
21C2×C4×Dic34C2211yescharacteristic
22C2×C42×D17C2211yescharacteristic
23#11763049C418yescomplemented
24#11762914—24no—
25C2×#10587478C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C2×#10587486C2211yescharacteristic
34C2×#10587486C2211yescharacteristic
35#97×D17—24no—
36#11762716C2211yescharacteristic
37#11762714C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762705C2211yescharacteristic
40#11762694—21no—
41#11762690—21no—
42#11762683—21no—
43#11762680—21no—
44#11762666C2211yescharacteristic
45#11762666C2211yescharacteristic
46C4×#63225—21no—
47C4×C4⋊Dic17—21no—
48C4×C4⋊Dic17C2211yescharacteristic
49C4×#63227—21no—
50C4×#63227—21no—
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C17×#97C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×C4×D17D411yescharacteristic
64C22×C4×D17D411yescharacteristic
65C22×C4×D17C2311yescharacteristic
66C22×Dic34C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C4×Q8×C17—22no—
70C4×Q8×C17—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C42×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17C2311yescharacteristic
78#10587632—22no—
79#10587632C2×C414yes—
80C2×#63231—22no—
81C2×#63231—22no—
82C2×#63231C2×C414yescomplemented
83C2×#63227—21no—
84C2×#63227—21no—
85C2×#63227—21no—
86C2×#63227—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17D411yescharacteristic
98C2×C4×Dic17D411yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C2×C8×D17—22no—
103C2×C8×D17—22no—
104C2×C8×D17—22no—
105C2×C8×D17—22no—
106C2×C8×D17—22no—
107C2×C8×D17C2×C414yes—
108#10587583—22no—
109#10587583C2×C414yes—
110C42×D17—22no—
111C42×D17—22no—
112C4×Dic34—22no—
113C4×Dic34—22no—
114C17×#103—22no—
115C17×#103C2×C414yescomplemented
116C17×#97—22no—
117C17×#95—21no—
118#10587506—22no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587487—22no—
129#10587487C2×C414yescomplemented
130#10587486—22no—
131#10587486—22no—
132#10587481—22no—
133#10587479—22no—
134#10587479—22no—
135#10587478C2×C414yescomplemented
136#10587474D414yes—
137C23×D17C2×D411yescharacteristic
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17C22×C412yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17C2×D411yescharacteristic
156C22×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—22no—
163C2×C4×D17—22no—
164C2×C4×D17—22no—
165C2×C4×D17—22no—
166C2×C4×D17—41no—
167C2×C4×D17—41no—
168C2×C4×D17C22×C412yes—
169C2×C4×D17#4112yes—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34C22×C412yes—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17C22×C412yes—
181C2×C8×C17#4112yes—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—42no—
188C4⋊C4×C17C22×C412yes—
189C42×C17—42no—
190C42×C17—42no—
191C42×C17C2×D412yes—
192C42×C17#4112yes—
193#63231—28no—
194#63227—24no—
195#63227—24no—
196#63227—42no—
197#63227—42no—
198C4⋊Dic17—21no—
199C4⋊Dic17—24no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—42no—
205C4⋊Dic17C22×C412yes—
206#63225—24no—
207#63225—42no—
208#63225—42no—
209C4×Dic17—22no—
210C4×Dic17—22no—
211C4×Dic17—21no—
212C4×Dic17—21no—
213C4×Dic17—42no—
214C4×Dic17—42no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622C22×C412yes—
218C2×#3622C2×D412yes—
219C2×#3622C2×D412yes—
220C2×#3622#4112yes—
221C8×D17—28no—
222C8×D17—28no—
223#21527—171nomaximal, Sylow
224C23×C17C22×D411yescharacteristic
225C22×D17—21no—
226C22×D17—21no—
227C22×D17—21no—
228C22×D17—21no—
229C22×D17C4×D412yes—
230C22×D17#13712yes—
231C22×D17#13712yes—
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17—41no—
235Q8×C17—41no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17#12112yes—
253C2×C4×C17#12112yes—
254C2×C4×C17C23×C411yescharacteristic
255C2×C4×C17C22×D411yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
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×Dic17C4×D412yes—
281C4×D17—22no—
282C4×D17—22no—
283C4×D17—22no—
284C4×D17—22no—
285C4×D17—44no—
286C4×D17—44no—
287Dic34—22no—
288Dic34—22no—
289Dic34—41no—
290Dic34—41no—
291C8×C17—22no—
292C8×C17—22no—
293#3622—22no—
294#3622C4×D414yes—
295C2×#467—171no—
296C2×#441—171no—
297C22×#97—171no—
298C4×C22⋊Q8—171no—
299C4×C22⋊Q8—171no—
300#1971—171no—
301#1943—171no—
302#1935—171no—
303#1935—171no—
304#1892—171no—
305#1892—171no—
306#1807—171no—
307#1776—171no—
308#1763—171no—
309#1763D1711yescharacteristic, complemented
310C22×C17C2×C4×D411yescharacteristic
311C22×C17C2×#12111yescharacteristic
312C22×C17#53211yescharacteristic
313C22×C17C2×#13711yescharacteristic
314C22×C17C2×#13711yescharacteristic
315C22×C17#57511yescharacteristic
316C22×C17#57511yescharacteristic
317C2×D17—22no—
318C2×D17—22no—
319C2×D17—22no—
320C2×D17—22no—
321C2×D17—22no—
322C2×D17—22no—
323C2×D17#44114yescomplemented
324C4×C17—22no—
325C4×C17—22no—
326C4×C17—42no—
327C4×C17—42no—
328C4×C17—81no—
329C4×C17—81no—
330C4×C17C2×C4×D411yescharacteristic
331C4×C17C4×#4111yescharacteristic
332C4×C17#52511yescharacteristic
333C4×C17#52911yescharacteristic
334Dic17—22no—
335Dic17—42no—
336Dic17—41no—
337Dic17—41no—
338Dic17—42no—
339Dic17—81no—
340Dic17—81no—
341C22×Q16—171no—
342C23×C8—171no—
343C23×C8—171no—
344C2×C22⋊Q8—171no—
345C2×C22⋊Q8—171no—
346C2×C4×Q8—171no—
347C2×C4×Q8C2×D1711yescharacteristic
348C22×C42—171no—
349#467—178no—
350#441—344no—
351C2×#107—171no—
352C2×#106—171no—
353C2×#106—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103C2×D1711yescharacteristic
359C2×#103C2×D1711yescharacteristic
360C2×#97—171no—
361C2×#97—171no—
362C2×#97—171no—
363C2×#97—344no—
364C2×#97C2×D1711yescharacteristic
365#391—171no—
366#391—171no—
367#389—171no—
368#389—171no—
369#387—341no—
370#387—341no—
371#385—341no—
372#385—341no—
373#380—171no—
374#380—171no—
375C4×C4⋊C4—171no—
376C4×C4⋊C4—341no—
377C4×C4⋊C4—341no—
378C4×C4⋊C4C2×D1711yescharacteristic
379C4×C22⋊C4—341no—
380C4×C22⋊C4—341no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340C2×D1711yescharacteristic
386#340C2×D1711yescharacteristic
387C2×C17C4×#12111yescharacteristic
388C2×C17C2×#44111yescharacteristic
389C2×C17#294311yescharacteristic
390C2×C17#305611yescharacteristic
391C2×C17#305611yescharacteristic
392C2×C17#318111yescharacteristic
393C2×C17#318111yescharacteristic
394D17—24no—
395C22×Q8—171no—
396C22×Q8C22×D1711yescharacteristic
397C23×C4—171no—
398C23×C4—171no—
399C23×C4—171no—
400C2×Q16—174no—
401C2×Q16—342no—
402C2×Q16—342no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—174no—
406C22×C8—342no—
407C22×C8—342no—
408C22×C8—342no—
409C22×C8—342no—
410C22×C8—342no—
411C22×C8C22×D1711yescharacteristic
412C22×C8C22×D1711yescharacteristic
413C22⋊Q8—174no—
414C22⋊Q8—174no—
415C22⋊Q8—342no—
416C22⋊Q8—342no—
417C4×Q8—22no—
418C4×Q8—22no—
419C4×Q8—342no—
420C4×Q8—342no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C22⋊C4—341no—
434C2×C22⋊C4—341no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—171no—
440C2×C42—171no—
441C2×C42—171no—
442C2×C42—342no—
443C2×C42—342no—
444C2×C42—341no—
445C2×C42—341no—
446C2×C42C22×D1711yescharacteristic
447#107—174no—
448#106—342no—
449#106—342no—
450#103—22no—
451#103—174no—
452#103—342no—
453#103—342no—
454#103—342no—
455#103C4×D1714yescomplemented
456#97—22no—
457#97—174no—
458#97—342no—
459#97—342no—
460#95—21no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469#95—341no—
470C17#2152711yescharacteristic, complemented, Sylow
471C24—171no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—172no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8—341no—
481C2×Q8—341no—
482C2×Q8—341no—
483C2×Q8C2×C4×D1712yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—171no—
489C22×C4—172no—
490C22×C4—172no—
491C22×C4—171no—
492C22×C4—171no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—681no—
506C22×C4—681no—
507C22×C4D4×D1711yescharacteristic
508C22×C4D4×D1711yescharacteristic
509C22×C4C23×D1711yescharacteristic
510Q16—348no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—172no—
516C2×C8—172no—
517C2×C8—172no—
518C2×C8—172no—
519C2×C8—348no—
520C2×C8—348no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8C2×C4×D1712yes—
524C2×C8#6325212yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—172no—
532C4⋊C4—344no—
533C4⋊C4—344no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—341no—
539C4⋊C4—682no—
540C4⋊C4—682no—
541C4⋊C4—682no—
542C4⋊C4C2×C4×D1712yes—
543C22⋊C4—344no—
544C22⋊C4—344no—
545C22⋊C4—682no—
546C22⋊C4—682no—
547C42—42no—
548C42—42no—
549C42—341no—
550C42—341no—
551C42—342no—
552C42—342no—
553C42—682no—
554C42—682no—
555C42#6325212yes—
556C42D4×D1712yes—
557C23—172no—
558C23—172no—
559C23—172no—
560C23—341no—
561C23—341no—
562C23—341no—
563C23—341no—
564C23C2×D4×D1711yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—342no—
570Q8—342no—
571Q8—681no—
572Q8—681no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—172no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—342no—
593C2×C4—342no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—342no—
597C2×C4—342no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—684no—
605C2×C4—684no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4#1058757012yes—
617C2×C4#1058757012yes—
618C2×C4C22×C4×D1711yescharacteristic
619C2×C4C2×#6325211yescharacteristic
620C2×C4C2×#6325211yescharacteristic
621C2×C4C2×D4×D1711yescharacteristic
622C2×C4#41×D1711yescharacteristic
623C2×C4#41×D1711yescharacteristic
624C8—22no—
625C8—22no—
626C8—174no—
627C8—342no—
628C22—174no—
629C22—342no—
630C22—342no—
631C22—342no—
632C22—342no—
633C22—342no—
634C22—342no—
635C22C4×D4×D1711yescharacteristic
636C22C2×#1058757011yescharacteristic
637C22#1176335611yescharacteristic
638C22#137×D1711yescharacteristic
639C22#137×D1711yescharacteristic
640C22#1176360511yescharacteristic
641C22#1176360811yescharacteristic
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—342no—
649C4—682no—
650C4—681no—
651C4—681no—
652C4—682no—
653C4—1361no—
654C4—1361no—
655C4C4×D4×D1711yescharacteristic
656C4C4×#6325211yescharacteristic
657C4#1176331811yescharacteristic
658C4#1176333711yescharacteristic
659C2—344no—
660C2C4×#1058757011yescharacteristic
661C2D17×#44111yescharacteristic
662C2#1453757911yescharacteristic
663C2#1453841211yescharacteristic
664C2#1453841711yescharacteristic
665C2#1453948911yescharacteristic
666C2#1453949111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1763C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1763D17
hypocenterC17#21527
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#11762914C2semidirect
6. #14529652C2semidirect
7. #14529521C2semidirect
8. #14529476C2semidirect
9. #14529472C2semidirect
13. #14528682C2semidirect
14. #14528615C2semidirect
15. C17×#1763C2semidirect
23. #11763049C4semidirect
82. C2×#63231C2×C4semidirect
115. C17×#103C2×C4semidirect
129. #10587487C2×C4semidirect
135. #10587478C2×C4semidirect
309. #1763D17semidirect
323. C2×D17#441semidirect
455. #103C4×D17semidirect
470. C17#21527semidirect

Maximal subgroups

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

subgroupindex
C2×#117630492
C2×#117629142
C2×#97×D172
C4×#105875832
C4×#105876322
#145296522
#145295212
#145294762
#145294722
#145292062
#145292042
#145288082
#145286822
#145286152
C17×#17632
#2152717