Finite Group Databasebeta

← #17091461

Subgroups — #17091461

Order 4352.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763095C211yesmaximal, characteristic, complemented
2C22×#10587487C211yesmaximal, characteristic
3#14532486C211yesmaximal, characteristic
4C22⋊C4×Dic34C211yesmaximal, characteristic
5#14530326C211yesmaximal, characteristic, complemented
6#14530164C211yesmaximal, characteristic, complemented
7#14530164C211yesmaximal, characteristic, complemented
8#14529926C211yesmaximal, characteristic, complemented
9#14529055C211yesmaximal, characteristic, complemented
10#14529055C211yesmaximal, characteristic, complemented
11#14528979C211yesmaximal, characteristic
12#14528809C211yesmaximal, characteristic
13#14528790C211yesmaximal, characteristic, complemented
14#14528720C211yesmaximal, characteristic
15C17×#1850C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587564—21no—
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×Dic34C2211yescharacteristic
21C22×C4⋊Dic17C2211yescharacteristic
22#11763095—24no—
23C2×#10587488C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26C2×#10587488C2211yescharacteristic
27C2×#10587487—24no—
28C2×#10587487C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33C2×#10587487C2211yescharacteristic
34C2×#10587487C2211yescharacteristic
35C2×#10587487C2211yescharacteristic
36#11762741—21no—
37#11762730—21no—
38#11762715C2211yescharacteristic
39#11762715C2211yescharacteristic
40#11762713—21no—
41#11762713—21no—
42#11762707C2211yescharacteristic
43#11762707C2211yescharacteristic
44#11762684—21no—
45#11762679C2211yescharacteristic
46#11762669C2211yescharacteristic
47#11762669C2211yescharacteristic
48#11762667—21no—
49C22⋊C4×Dic17—21no—
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54C23×C8×C17C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C23×Dic17—21no—
64C22×Dic34—22no—
65C22×Dic34—22no—
66C22×Dic34—21no—
67C22×Dic34—22no—
68C22×Dic34—21no—
69C22×Dic34—41no—
70C22×Dic34—41no—
71C22×Dic34Q814yescomplemented
72C22×Dic34C2311yescharacteristic
73C22×Dic34C2311yescharacteristic
74C22×C8×C17—22no—
75C22×C8×C17—22no—
76C22×C8×C17—41no—
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×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—22no—
92C2×C4⋊Dic17—41no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105#10587564—44no—
106#10587555—22no—
107#10587555—41no—
108C4×Dic34—22no—
109C4×Dic34—22no—
110C4×Dic34—41no—
111C4×Dic34—41no—
112C17×C4⋊C8—22no—
113C17×C4⋊C8—41no—
114C17×C22⋊C8—22no—
115C17×#95D411yescharacteristic
116C17×#95D411yescharacteristic
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587488—22no—
126#10587488—22no—
127#10587488—41no—
128#10587488—41no—
129#10587487—22no—
130#10587487—22no—
131#10587487—22no—
132#10587487—22no—
133#10587487—22no—
134#10587487—41no—
135#10587487—41no—
136#10587487D414yes—
137C24×C17C2×D411yescharacteristic
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×Q812yes—
148C22×C4×C17#4112yes—
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—24no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—42no—
160C22×Dic17—42no—
161C22×Dic17—42no—
162C22×Dic17—42no—
163C22×Dic17—81no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—28no—
172C2×Dic34—24no—
173C2×Dic34—21no—
174C2×Dic34—44no—
175C2×Dic34—44no—
176C2×Dic34—44no—
177C2×Dic34—42no—
178C2×Dic34—42no—
179C2×Dic34—42no—
180C2×Dic34—42no—
181C2×Dic34—81no—
182C2×Dic34C2×D412yes—
183C2×Dic34C2×Q812yes—
184C2×Dic34C2×Q812yes—
185C2×Dic34#4112yes—
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×C17—21no—
196C2×C8×C17—21no—
197C2×C8×C17—44no—
198C2×C8×C17C2×D412yes—
199C22⋊C4×C17—22no—
200C22⋊C4×C17SD1614yes—
201C42×C17—21no—
202C42×C17—21no—
203C42×C17—21no—
204C42×C17—21no—
205C22⋊Dic17—42no—
206C22⋊Dic17—42no—
207C4⋊Dic17—21no—
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⋊Dic17—42no—
219C4⋊Dic17—42no—
220C4⋊Dic17—42no—
221C4⋊Dic17—81no—
222C4⋊Dic17C2×D412yes—
223#63225—42no—
224#63225—42no—
225#63225—42no—
226#63225—42no—
227C4×Dic17—42no—
228C4×Dic17—42no—
229#25955—171nomaximal, Sylow
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17—21no—
234C23×C17C22⋊Q812yes—
235C23×C17C2×SD1612yes—
236C23×C17#13712yes—
237C23×C17C22×D411yescharacteristic
238C2×C4×C17—21no—
239C2×C4×C17—21no—
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×C17C22⋊Q812yes—
255C2×C4×C17C2×SD1612yes—
256C2×C4×C17C2×SD1612yes—
257C2×C4×C17C22×D411yescharacteristic
258C2×C4×C17C22×Q811yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17H(5,𝔽2)11yescharacteristic
262C2×C4×C17H(5,𝔽2)11yescharacteristic
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—48no—
269C2×Dic17—41no—
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—
288C2×Dic17—84no—
289Dic34—22no—
290Dic34—22no—
291Dic34—22no—
292Dic34—22no—
293Dic34—22no—
294Dic34—48no—
295Dic34—41no—
296Dic34—41no—
297Dic34—84no—
298Dic34C22⋊Q814yescomplemented
299C8×C17—22no—
300C8×C17—41no—
301C8×C17—41no—
302C8×C17—41no—
303C2×#475—171no—
304C22×#103—171no—
305#2440—171no—
306Q8×C22⋊C4—171no—
307#2096—171no—
308#2066—171no—
309#2066—171no—
310#2024—171no—
311#1865—171no—
312#1865—171no—
313#1853—171no—
314#1850D1711yescharacteristic
315#1807—171no—
316#1803—171no—
317#1787—171no—
318C22×C17—22no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—22no—
322C22×C17—22no—
323C22×C17—22no—
324C22×C17#47514yescomplemented
325C22×C17C2×C22⋊Q811yescharacteristic
326C22×C17D4211yescharacteristic
327C22×C17#54611yescharacteristic
328C22×C17C22×SD1611yescharacteristic
329C22×C17C2×#13711yescharacteristic
330C22×C17#57711yescharacteristic
331C22×C17#57811yescharacteristic
332C4×C17—22no—
333C4×C17—22no—
334C4×C17—22no—
335C4×C17—41no—
336C4×C17—41no—
337C4×C17C2×C22⋊Q811yescharacteristic
338C4×C17#54311yescharacteristic
339C4×C17#54611yescharacteristic
340C4×C17#55111yescharacteristic
341Dic17—24no—
342Dic17—42no—
343Dic17—42no—
344Dic17—81no—
345Dic17—81no—
346Dic17—81no—
347Dic17—81no—
348Dic17—81no—
349C23×Q8—171no—
350C23×C8C2×D1711yescharacteristic
351C2×C4⋊Q8—171no—
352C2×C22⋊Q8—341no—
353C2×C4×Q8—171no—
354C2×C4×Q8—171no—
355C22×C4⋊C4—171no—
356#475—344no—
357C2×#106—171no—
358C2×#106—171no—
359C2×#106—171no—
360C2×#106—171no—
361C2×C4⋊C8C2×D1711yescharacteristic
362C2×C4⋊C8C2×D1711yescharacteristic
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#103—171no—
370C2×#103—171no—
371C2×#103—344no—
372C2×C22⋊C8C2×D1711yescharacteristic
373#398—341no—
374#395—341no—
375#391—171no—
376#391—171no—
377#391—341no—
378#391—341no—
379#389—171no—
380#389—171no—
381#385—341no—
382#384—171no—
383#380—171no—
384#380—171no—
385#380—341no—
386C4×C22⋊C4—341no—
387C4×C22⋊C4C2×D1711yescharacteristic
388#340—171no—
389#340—171no—
390#340—171no—
391#340—171no—
392#336C2×D1711yescharacteristic
393#336C2×D1711yescharacteristic
394C2×C17—24no—
395C2×C17#243711yescharacteristic
396C2×C17C2×#47511yescharacteristic
397C2×C17#308211yescharacteristic
398C2×C17#327011yescharacteristic
399C2×C17D4×SD1611yescharacteristic
400C2×C17#329811yescharacteristic
401C2×C17#331611yescharacteristic
402C22×Q8—174no—
403C22×Q8—171no—
404C22×Q8—171no—
405C22×Q8—342no—
406C22×Q8—342no—
407C22×Q8—341no—
408C22×Q8—341no—
409C22×Q8—342no—
410C22×Q8—681no—
411C22×Q8—681no—
412C23×C4—341no—
413C23×C4—341no—
414C23×C4C22×D1711yescharacteristic
415C22×C8—22no—
416C22×C8—22no—
417C22×C8—41no—
418C22×C8C22×D1711yescharacteristic
419C22×C8C22×D1711yescharacteristic
420C22×C8C22×D1711yescharacteristic
421C22×C8C22×D1711yescharacteristic
422C4⋊Q8—342no—
423C4⋊Q8—681no—
424C22⋊Q8—684no—
425C4×Q8—342no—
426C4×Q8—342no—
427C4×Q8—681no—
428C4×Q8—681no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—171no—
434C2×C4⋊C4—171no—
435C2×C4⋊C4—342no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4—341no—
442C2×C4⋊C4—341no—
443C2×C4⋊C4—341no—
444C2×C4⋊C4—681no—
445C2×C22⋊C4—341no—
446C2×C22⋊C4—341no—
447C2×C22⋊C4D6811yescharacteristic
448C2×C22⋊C4D6811yescharacteristic
449C2×C42—341no—
450C2×C42—341no—
451C2×C42C22×D1711yescharacteristic
452C2×C42C22×D1711yescharacteristic
453#106—342no—
454#106—342no—
455#106—681no—
456#106—681no—
457C4⋊C8—22no—
458C4⋊C8—41no—
459#103—174no—
460#103—342no—
461#103—342no—
462#103—342no—
463#103—342no—
464#103—342no—
465#103—681no—
466#103—681no—
467C22⋊C8—22no—
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#95D6811yescharacteristic
477#95D6811yescharacteristic
478C17#2595511yescharacteristic, complemented, Sylow
479C24C2×D6811yescharacteristic
480C2×Q8—172no—
481C2×Q8—172no—
482C2×Q8—172no—
483C2×Q8—172no—
484C2×Q8—348no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—344no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×Q8—341no—
494C2×Q8—682no—
495C2×Q8—682no—
496C2×Q8—682no—
497C2×Q8—684no—
498C2×Q8—684no—
499C2×Q8—684no—
500C2×Q8—682no—
501C2×Q8—1361no—
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—341no—
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—1361no—
520C22×C4C2×D6811yescharacteristic
521C22×C4C2×D6811yescharacteristic
522C22×C4C2×D6811yescharacteristic
523C22×C4C2×D6811yescharacteristic
524C22×C4C2×D6811yescharacteristic
525C22×C4#6325412yes—
526C22×C4Q8×D1712yes—
527C22×C4C23×D1711yescharacteristic
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×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—44no—
540C2×C8D4×D1712yes—
541C4⋊C4—172no—
542C4⋊C4—341no—
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—341no—
552C4⋊C4—341no—
553C4⋊C4—682no—
554C4⋊C4—682no—
555C4⋊C4—682no—
556C4⋊C4—682no—
557C4⋊C4—682no—
558C4⋊C4—682no—
559C4⋊C4—682no—
560C4⋊C4—1361no—
561C22⋊C4—22no—
562C22⋊C4—682no—
563C22⋊C4—682no—
564C22⋊C4#6321914yes—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—682no—
570C42—682no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#1058758312yes—
576C23C2×#6321912yes—
577C23#1058759512yes—
578C23C22×D6811yescharacteristic
579Q8—174no—
580Q8—342no—
581Q8—342no—
582Q8—342no—
583Q8—342no—
584Q8—342no—
585Q8—688no—
586Q8—681no—
587Q8—681no—
588Q8—1364no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—342no—
606C2×C4—342no—
607C2×C4—342no—
608C2×C4—342no—
609C2×C4—342no—
610C2×C4—688no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
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—1364no—
631C2×C4#1058758312yes—
632C2×C4C2×#6321912yes—
633C2×C4C2×#6321912yes—
634C2×C4C2×D4×D1711yescharacteristic
635C2×C4C2×#6325411yescharacteristic
636C2×C4#1058769111yescharacteristic
637C2×C4C2×Q8×D1711yescharacteristic
638C2×C4#41×D1711yescharacteristic
639C2×C4#1058769611yescharacteristic
640C8—22no—
641C8—41no—
642C8—41no—
643C8—41no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1176309514yescomplemented
651C22C2×#1058758311yescharacteristic
652C22D4×D6811yescharacteristic
653C22#1176345411yescharacteristic
654C22C22×#6321911yescharacteristic
655C22C2×#1058759511yescharacteristic
656C22#1176362111yescharacteristic
657C22#1176362611yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—344no—
664C4—682no—
665C4—682no—
666C4—1361no—
667C4—1361no—
668C4—1361no—
669C4—1361no—
670C4—1361no—
671C4C22⋊Q8×D1711yescharacteristic
672C4#1176343411yescharacteristic
673C4#1176344611yescharacteristic
674C4#1176348211yescharacteristic
675C2—24no—
676C2#1453244811yescharacteristic
677C2C2×#1176309511yescharacteristic
678C2#1453862711yescharacteristic
679C2#1454035911yescharacteristic
680C2D4×#6321911yescharacteristic
681C2#1454065011yescharacteristic
682C2#1454089711yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1850C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1850D17
hypocenterC17#25955
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. C2×#11763095C2semidirect
5. #14530326C2semidirect
6. #14530164C2semidirect
7. #14530164C2semidirect
8. #14529926C2semidirect
9. #14529055C2semidirect
10. #14529055C2semidirect
13. #14528790C2semidirect
71. C22×Dic34Q8semidirect
298. Dic34C22⋊Q8semidirect
324. C22×C17#475semidirect
478. C17#25955semidirect
650. C22#11763095semidirect

Maximal subgroups

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

subgroupindex
C2×#117630952
C22×#105874872
#145324862
C22⋊C4×Dic342
#145303262
#145301642
#145299262
#145290552
#145289792
#145288092
#145287902
#145287202
C17×#18502
#2595517