Finite Group Databasebeta

← #17061801

Subgroups — #17061801

Order 4352.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762845C211yesmaximal, characteristic, complemented
2C2×#11762805C211yesmaximal, characteristic, complemented
3C2×#11762805C211yesmaximal, characteristic, complemented
4C22×C22⋊#3622C211yesmaximal, characteristic, complemented
5#14531167C211yesmaximal, characteristic, complemented
6#14529984C212yesmaximal, complemented
7#14529977C212yesmaximal, complemented
8#14529970C212yesmaximal, complemented
9#14529963C212yesmaximal, complemented
10#14528500C211yesmaximal, characteristic, complemented
11C2×C17×#525C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C23×#3622C2211yescharacteristic
14C2×#10587612C2211yescharacteristic
15C2×#10587612C2211yescharacteristic
16#11762845—24no—
17C2×#10587478C2212yescomplemented
18C2×#10587479C2212yescomplemented
19#11762805—24no—
20#11762805—24no—
21C2×#10587506C2211yescharacteristic, complemented
22C2×#10587506C2212yescomplemented
23C2×#10587506C2211yescharacteristic, complemented
24C2×#10587503C2211yescharacteristic, complemented
25C2×#10587503C2212yescomplemented
26C2×#10587503C2211yescharacteristic, complemented
27C2×C22⋊#3622—24no—
28C2×C22⋊#3622C2212yes—
29C2×C22⋊#3622C2212yes—
30#11762672—21no—
31#11762669C2212yes—
32#11762669C2212yes—
33#11762665—21no—
34C4×C22⋊Dic17—22no—
35#11762571C2212yescomplemented
36#11762571C2212yescomplemented
37C22×#41×C17C2211yescharacteristic
38C24×C4×C17C2211yescharacteristic
39C17×#525—24no—
40C17×#525—24no—
41C2×C17×#123—21no—
42C2×C17×C22⋊Q8C2212yes—
43C2×C17×#121C2212yes—
44C2×C17×#120—21no—
45C2×C4×D4×C17—22no—
46C2×C17×#117C2211yescharacteristic
47C25×C17D411yescharacteristic
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17—22no—
51C2×#41×C17—22no—
52C2×#41×C17—22no—
53C2×#41×C17D414yes—
54C22×Q8×C17C2311yescharacteristic
55C22×D4×C17—22no—
56C22×D4×C17—21no—
57C22×D4×C17C2311yescharacteristic
58C23×C4×C17—21no—
59C23×C4×C17—22no—
60C23×C4×C17—22no—
61C23×C4×C17—22no—
62C23×C4×C17—22no—
63C23×C4×C17—22no—
64C23×C4×C17—22no—
65C23×C4×C17D412yes—
66C23×C4×C17D411yescharacteristic
67C23×C4×C17C2312yes—
68C23×C4×C17C2311yescharacteristic
69C17×#123—28no—
70C17×C22⋊Q8—24no—
71C17×C22⋊Q8—24no—
72C17×#121—24no—
73C17×#121—24no—
74C17×#120—28no—
75C4×D4×C17—216no—
76C17×#117—22no—
77C17×#117D414yes—
78C2×C4⋊C4×C17—22no—
79C2×C4⋊C4×C17C2312yes—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—22no—
82C2×C22⋊C4×C17—22no—
83C2×C42×C17—21no—
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—22no—
87C2×C22⋊Dic17—22no—
88#10587612—22no—
89#10587612—22no—
90#10587612—22no—
91#10587612—22no—
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2312yes—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C22×#3622—22no—
98C22×#3622—22no—
99C22×#3622—22no—
100C22×#3622—22no—
101C22×#3622—22no—
102C22×#3622D414yescomplemented
103C22×#3622C2312yes—
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106#10587506—22no—
107#10587506—22no—
108#10587506—24no—
109#10587503—22no—
110#10587503—22no—
111#10587503—24no—
112#10587502—22no—
113#10587502—22no—
114C22⋊#3622—24no—
115C22⋊#3622—24no—
116#10587479—24no—
117#10587478—24no—
118C24×C17—22no—
119C24×C17—24no—
120C24×C17—22no—
121C24×C17—22no—
122C24×C17—21no—
123C24×C17—42no—
124C24×C17C2×D412yes—
125C24×C17C2×D411yescharacteristic
126#41×C17—28no—
127#41×C17—28no—
128#41×C17—44no—
129#41×C17—44no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17C2×D412yes—
136C2×D4×C17—28no—
137C2×D4×C17—24no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17—44no—
144C2×D4×C17—42no—
145C2×D4×C17—42no—
146C2×D4×C17—44no—
147C2×D4×C17C2×D412yes—
148C22×C4×C17—22no—
149C22×C4×C17—22no—
150C22×C4×C17—22no—
151C22×C4×C17—21no—
152C22×C4×C17—22no—
153C22×C4×C17—24no—
154C22×C4×C17—24no—
155C22×C4×C17—24no—
156C22×C4×C17—24no—
157C22×C4×C17—22no—
158C22×C4×C17—22no—
159C22×C4×C17—22no—
160C22×C4×C17—22no—
161C22×C4×C17—22no—
162C22×C4×C17—22no—
163C22×C4×C17—44no—
164C22×C4×C17—44no—
165C22×C4×C17—42no—
166C22×C4×C17—44no—
167C22×C4×C17—42no—
168C22×C4×C17—44no—
169C22×C4×C17C2×D412yes—
170C22×C4×C17C2×D412yes—
171C22×C4×C17C2×D412yes—
172C22×C4×C17C2×D412yes—
173C22×C4×C17C2×D411yescharacteristic
174C22×C4×C17#4112yes—
175C22×C4×C17#4112yes—
176C22×C4×C17#4112yes—
177C22×C4×C17C2411yescharacteristic
178C22×Dic17—22no—
179C22×Dic17—21no—
180C22×Dic17—21no—
181C4⋊C4×C17—28no—
182C4⋊C4×C17—22no—
183C4⋊C4×C17C2×D414yes—
184C22⋊C4×C17—28no—
185C22⋊C4×C17—28no—
186C22⋊C4×C17—24no—
187C42×C17—24no—
188C22⋊Dic17—44no—
189C22⋊Dic17—42no—
190C22⋊Dic17—42no—
191C22⋊Dic17—44no—
192C4⋊Dic17—22no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—22no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4×Dic17—42no—
199C4×Dic17—42no—
200C2×#3622—28no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—22no—
206C2×#3622—22no—
207C2×#3622—22no—
208C2×#3622—44no—
209C2×#3622C2×D412yes—
210C2×#3622C2×D412yes—
211#23479—171nomaximal, Sylow
212C23×C17—22no—
213C23×C17—22no—
214C23×C17—28no—
215C23×C17—22no—
216C23×C17—22no—
217C23×C17—22no—
218C23×C17—22no—
219C23×C17—22no—
220C23×C17—22no—
221C23×C17—22no—
222C23×C17—48no—
223C23×C17—48no—
224C23×C17—44no—
225C23×C17—44no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—42no—
229C23×C17—42no—
230C23×C17#12012yes—
231C23×C17#12112yes—
232C23×C17#12312yes—
233C23×C17#13514yes—
234C23×C17C22×D411yescharacteristic
235Q8×C17—22no—
236Q8×C17—22no—
237Q8×C17—41no—
238Q8×C17—41no—
239D4×C17—22no—
240D4×C17—22no—
241D4×C17—416no—
242D4×C17—48no—
243D4×C17—41no—
244D4×C17—41no—
245C2×C4×C17—24no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—44no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—42no—
258C2×C4×C17—42no—
259C2×C4×C17—48no—
260C2×C4×C17—48no—
261C2×C4×C17#12012yes—
262C2×C4×C17#12114yes—
263C2×C4×C17#12114yes—
264C2×C4×C17#12112yes—
265C2×C4×C17#12314yes—
266C2×C4×C17#12312yes—
267C2×C4×C17#12414yes—
268C2×C4×C17C22×D411yescharacteristic
269C2×C4×C17C22×D411yescharacteristic
270C2×C4×C17C22×D411yescharacteristic
271C2×C4×C17C2×#4111yescharacteristic
272C2×C4×C17C2×#4111yescharacteristic
273C2×C4×C17C2×#4111yescharacteristic
274C2×Dic17—42no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281#3622—22no—
282#3622—22no—
283#3622—42no—
284C2×#525D1711yescharacteristic
285C2×#427—171no—
286C2×#416—171no—
287C2×#416—171no—
288C22×C22⋊C8—171no—
289#2280—171no—
290#2036—172no—
291#2035—172no—
292#2034—172no—
293#2033—172no—
294#1735—171no—
295C22×C17—24no—
296C22×C17—24no—
297C22×C17—22no—
298C22×C17—22no—
299C22×C17—24no—
300C22×C17—24no—
301C22×C17—24no—
302C22×C17—416no—
303C22×C17—42no—
304C22×C17—42no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17—44no—
309C22×C17—82no—
310C22×C17—81no—
311C22×C17—81no—
312C22×C17#39414yes—
313C22×C17C2×#12011yescharacteristic
314C22×C17C2×#12111yescharacteristic
315C22×C17C2×#12311yescharacteristic
316C22×C17C2×#13512yes—
317C22×C17#57512yes—
318C4×C17—42no—
319C4×C17—42no—
320C4×C17—44no—
321C4×C17#39414yes—
322C4×C17C2×#12111yescharacteristic
323C4×C17C2×#12111yescharacteristic
324C4×C17C2×#12311yescharacteristic
325C4×C17C2×#12411yescharacteristic
326Dic17—82no—
327Dic17—81no—
328Dic17—81no—
329C22×#41C2×D1711yescharacteristic
330C24×C4C2×D1711yescharacteristic
331C23×C8—171no—
332C23×C8—171no—
333#525—24no—
334#525—24no—
335C2×#123—21no—
336C2×C22⋊Q8C2×D1712yes—
337C2×#121C2×D1712yes—
338C2×#120—21no—
339C2×C4×D4—22no—
340C2×#117—171no—
341C2×#117—171no—
342C2×#117C2×D1711yescharacteristic
343#427—344no—
344C2×#107—172no—
345C2×#106—172no—
346#416—344no—
347#416—344no—
348C2×#103—171no—
349C2×#103—172no—
350C2×#103—171no—
351C2×#102—171no—
352C2×#102—172no—
353C2×#102—171no—
354C2×C22⋊C8—172no—
355C2×C22⋊C8—172no—
356C2×C22⋊C8—344no—
357#381—341no—
358#380—172no—
359#380—172no—
360#379—341no—
361C4×C22⋊C4—342no—
362#340—172no—
363#340—172no—
364C2×C17—24no—
365C2×C17—44no—
366C2×C17—82no—
367C2×C17C2×#39411yescharacteristic
368C2×C17#300212yes—
369C2×C17#305011yescharacteristic
370C2×C17#305411yescharacteristic
371C2×C17#309112yes—
372C25C22⋊D1711yescharacteristic
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41C22⋊D1714yes—
379C22×Q8C22×D1711yescharacteristic
380C22×D4—22no—
381C22×D4—21no—
382C22×D4C22×D1711yescharacteristic
383C23×C4—22no—
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—21no—
390C23×C4C22⋊D1711yescharacteristic
391C23×C4C22⋊D1712yes—
392C23×C4C22×D1712yes—
393C23×C4C22×D1711yescharacteristic
394C22×C8—174no—
395C22×C8—171no—
396C22×C8—171no—
397C22×C8—172no—
398C22×C8—342no—
399C22×C8—342no—
400C22×C8—342no—
401C22×C8—342no—
402C22×C8—342no—
403#123—28no—
404C22⋊Q8—24no—
405C22⋊Q8—24no—
406#121—24no—
407#121—24no—
408#120—28no—
409C4×D4—216no—
410#117—22no—
411#117—342no—
412#117—342no—
413#117—342no—
414#117—342no—
415#117C22⋊D1714yes—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—172no—
420C2×C4⋊C4C22×D1712yes—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—341no—
425C2×C22⋊C4—341no—
426C2×C22⋊C4—342no—
427C2×C22⋊C4—342no—
428C2×C42—21no—
429C2×C42—341no—
430C2×C42—341no—
431#107—344no—
432#106—344no—
433#103—342no—
434#103—342no—
435#103—344no—
436#102—342no—
437#102—342no—
438#102—344no—
439C22⋊C8—344no—
440C22⋊C8—344no—
441#95—342no—
442#95—342no—
443C17#2347911yescharacteristic, complemented, Sylow
444C24—22no—
445C24—24no—
446C24—21no—
447C24—22no—
448C24—22no—
449C24—42no—
450C24C2×C22⋊D1712yes—
451C24C2×C22⋊D1711yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×C22⋊D1712yes—
462C2×D4—28no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—44no—
473C2×D4C2×C22⋊D1712yes—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—342no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4D4×D1712yes—
499C22×C4#6325412yes—
500C22×C4#6325412yes—
501C22×C4#6325412yes—
502C22×C4C2×C22⋊D1712yes—
503C22×C4C2×C22⋊D1712yes—
504C22×C4C2×C22⋊D1712yes—
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C23×D1711yescharacteristic
507C2×C8—172no—
508C2×C8—172no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—348no—
514C2×C8—342no—
515C2×C8—342no—
516C2×C8—342no—
517C2×C8—684no—
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—342no—
525C4⋊C4—342no—
526C4⋊C4C2×C22⋊D1714yes—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—684no—
531C22⋊C4—682no—
532C22⋊C4—684no—
533C22⋊C4—682no—
534C42—24no—
535C42—682no—
536C42—682no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—48no—
548C23—48no—
549C23—44no—
550C23—44no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23#1058762112yes—
556C23#1058762512yes—
557C23#1058763614yes—
558C23#1058763912yes—
559C23C22×C22⋊D1711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—416no—
567D4—48no—
568D4—41no—
569D4—41no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—682no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4#1058762112yes—
594C2×C4#1058762114yes—
595C2×C4#1058762214yes—
596C2×C4#1058762414yes—
597C2×C4#1058762512yes—
598C2×C4#1058762514yes—
599C2×C4#1058763912yes—
600C2×C4C2×D4×D1711yescharacteristic
601C2×C4C2×#6325411yescharacteristic
602C2×C4C2×#6325411yescharacteristic
603C2×C4C2×#6325411yescharacteristic
604C2×C4C22×C22⋊D1711yescharacteristic
605C2×C4C22×C22⋊D1711yescharacteristic
606C8—342no—
607C8—342no—
608C8—682no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—82no—
626C22#1176272314yes—
627C22C2×#1058763911yescharacteristic
628C22C2×#1058762511yescharacteristic
629C22C2×#1058762111yescharacteristic
630C22C2×#1058763612yes—
631C22#1176360712yes—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—1362no—
636C4—1361no—
637C4—1361no—
638C4#1176272314yes—
639C4C2×#1058762411yescharacteristic
640C4C2×#1058762511yescharacteristic
641C4C2×#1058762111yescharacteristic
642C4C2×#1058762211yescharacteristic
643C2—24no—
644C2—44no—
645C2—82no—
646C2C2×#1176272311yescharacteristic
647C2#1453800212yes—
648C2#1453836211yescharacteristic
649C2#1453839011yescharacteristic
650C2#1453869312yes—
subgroupG/H
centerC22×C4C2×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#525C2
socleC23×C17C22×D4
hypercenterC2×#525D17
hypocenterC17#23479
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C2×#11762845C2semidirect
2. C2×#11762805C2semidirect
3. C2×#11762805C2semidirect
4. C22×C22⋊#3622C2semidirect
5. #14531167C2semidirect
6. #14529984C2semidirect
7. #14529977C2semidirect
8. #14529970C2semidirect
9. #14529963C2semidirect
10. #14528500C2semidirect
17. C2×#10587478C22semidirect
18. C2×#10587479C22semidirect
21. C2×#10587506C22semidirect
22. C2×#10587506C22semidirect
23. C2×#10587506C22semidirect
24. C2×#10587503C22semidirect
25. C2×#10587503C22semidirect
26. C2×#10587503C22semidirect
35. #11762571C22semidirect
36. #11762571C22semidirect
102. C22×#3622D4semidirect
443. C17#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#117628452
C2×#117628052
C22×C22⋊#36222
#145311672
#145299842
#145299772
#145299702
#145299632
#145285002
C2×C17×#5252
#2347917