Finite Group Databasebeta

← #17035565

Subgroups — #17035565

Order 4352.

1934 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587583C211yesmaximal, characteristic
2C2×#117×D17C211yesmaximal, characteristic
3C2×#11762819C211yesmaximal, characteristic, complemented
4C2×#11762817C211yesmaximal, characteristic, complemented
5C2×#11762772C211yesmaximal, characteristic
6#14529095C214yesmaximal, complemented
7#14529090C214yesmaximal, complemented
8#14528503C211yesmaximal, characteristic, complemented
9C17×#1736C211yesmaximal, characteristic, complemented
10C23×Dic34C2211yescharacteristic
11C23×C4×D17C2211yescharacteristic
12C22×#63223C2211yescharacteristic
13C2×#10587583—21no—
14C2×#10587583—21no—
15C2×#10587583—24no—
16C2×#10587583C418yescomplemented
17C2×#10587583C2212yes—
18C2×#10587583C2212yes—
19C2×#10587612C2211yescharacteristic
20C2×#10587578C2212yes—
21C2×#10587558—21no—
22C2×#10587563—21no—
23#117×D17—24no—
24#117×D17—24no—
25C22×C4⋊Dic17—21no—
26C2×C4⋊C4×D17C2212yes—
27C22×#63227—21no—
28C2×C22⋊C4×D17—21no—
29C2×C42×D17—21no—
30#11762819—24no—
31#11762817—24no—
32C2×#10587487C2214yescomplemented
33C2×#10587481C2214yescomplemented
34#11762772—24no—
35C2×#10587491C2212yes—
36C2×#10587491C2212yes—
37#11762571C2214yescomplemented
38C22×M16×C17C2211yescharacteristic
39C2×C17×#117C2211yescharacteristic
40C22×C4⋊C4×C17C2211yescharacteristic
41C17×#340C2214yescomplemented
42C24×D17D411yescharacteristic
43C23×C4×C17—21no—
44C23×C4×C17C2311yescharacteristic
45C23×Dic17—21no—
46C23×Dic17D411yescharacteristic
47C22×C4×D17—22no—
48C22×C4×D17—22no—
49C22×C4×D17—22no—
50C22×C4×D17—22no—
51C22×C4×D17—22no—
52C22×C4×D17C2×C414yes—
53C22×C4×D17D411yescharacteristic
54C22×C4×D17D411yescharacteristic
55C22×C4×D17D414yes—
56C22×C4×D17C2311yescharacteristic
57C22×C4×D17C2311yescharacteristic
58C22×Dic34—22no—
59C22×Dic34—22no—
60C22×Dic34—22no—
61C22×Dic34—22no—
62C22×Dic34—21no—
63C22×Dic34—22no—
64C22×Dic34C2×C414yescomplemented
65C22×Dic34C2312yes—
66C2×M16×C17—22no—
67C2×M16×C17—22no—
68C2×M16×C17—22no—
69C22×C8×C17C2312yes—
70C17×#117—22no—
71C17×#117D414yes—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—22no—
74C2×C4⋊C4×C17C2×C414yescomplemented
75C2×C4⋊C4×C17C2312yes—
76C2×C4⋊C4×C17C2312yes—
77C2×C22⋊C4×C17—21no—
78C2×C42×C17—21no—
79C2×C22⋊Dic17—21no—
80C2×#63227—28no—
81C2×#63227—22no—
82C2×#63227—22no—
83C2×#63227—22no—
84#10587612—22no—
85#10587612D414yes—
86C2×C4⋊Dic17—28no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17—22no—
89C2×C4⋊Dic17C2312yes—
90C2×#63225—22no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—22no—
93C2×#63223—22no—
94C2×#63223—22no—
95C2×#63223—22no—
96C22×#3622C2312yes—
97#10587583—28no—
98#10587583—28no—
99#10587583—24no—
100#10587583—24no—
101#10587583C2×C418yes—
102#10587583C2×C418yescomplemented
103#10587578—24no—
104#10587578—24no—
105C4⋊C4×D17—24no—
106C4⋊C4×D17—24no—
107C22⋊C4×D17—28no—
108#10587563—28no—
109#10587558—28no—
110C42×D17—28no—
111#10587491—24no—
112#10587491—24no—
113#10587487—28no—
114#10587481—28no—
115C24×C17C2×D411yescharacteristic
116C23×D17—22no—
117C23×D17—22no—
118C23×D17—24no—
119C23×D17—42no—
120C23×D17C22⋊C414yes—
121C23×D17C2×D411yescharacteristic
122C23×D17C2×D411yescharacteristic
123C22×C4×C17—24no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—44no—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C2×D412yes—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17#4112yes—
133C22×C4×C17#4112yes—
134C22×C4×C17C2411yescharacteristic
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—24no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—42no—
141C22×Dic17—42no—
142C22×Dic17C22⋊C414yes—
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C2×C4×D17—28no—
146C2×C4×D17—28no—
147C2×C4×D17—22no—
148C2×C4×D17—22no—
149C2×C4×D17—28no—
150C2×C4×D17—28no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—21no—
154C2×C4×D17—21no—
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—28no—
162C2×C4×D17—44no—
163C2×C4×D17—44no—
164C2×C4×D17C22⋊C414yes—
165C2×C4×D17C22⋊C414yes—
166C2×C4×D17C22×C412yes—
167C2×C4×D17C22×C412yes—
168C2×C4×D17C2×D412yes—
169C2×C4×D17C2×D412yes—
170C2×Dic34—28no—
171C2×Dic34—28no—
172C2×Dic34—22no—
173C2×Dic34—22no—
174C2×Dic34—22no—
175C2×Dic34—22no—
176C2×Dic34—24no—
177C2×Dic34—22no—
178C2×Dic34—42no—
179C2×Dic34—44no—
180C2×Dic34—44no—
181C2×Dic34—42no—
182C2×Dic34C22×C414yes—
183M16×C17—44no—
184C2×C8×C17—22no—
185C2×C8×C17—22no—
186C2×C8×C17—22no—
187C4⋊C4×C17—24no—
188C4⋊C4×C17—22no—
189C4⋊C4×C17—22no—
190C4⋊C4×C17C22×C414yes—
191C4⋊C4×C17C2×D414yes—
192C22⋊C4×C17—24no—
193C42×C17—24no—
194C22⋊Dic17—24no—
195#63227—28no—
196#63227—28no—
197#63227—28no—
198C4⋊Dic17—22no—
199C4⋊Dic17—28no—
200C4⋊Dic17—28no—
201C4⋊Dic17C2×D414yes—
202#63225—28no—
203C4×Dic17—28no—
204C4×Dic17—24no—
205#63223—44no—
206C2×#3622—22no—
207C2×#3622—22no—
208C2×#3622—22no—
209#20911—171nomaximal, Sylow
210C23×C17—22no—
211C23×C17—22no—
212C23×C17C2×C22⋊C412yes—
213C23×C17#12012yes—
214C23×C17#12312yes—
215C23×C17C22×D411yescharacteristic
216C22×D17—21no—
217C22×D17—21no—
218C22×D17—22no—
219C22×D17—21no—
220C22×D17—21no—
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—28no—
224C22×D17—28no—
225C22×D17—44no—
226C22×D17—48no—
227C22×D17—48no—
228C22×D17C2×C22⋊C412yes—
229C22×D17C2×C22⋊C412yes—
230C22×D17#13714yes—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—48no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—44no—
237C2×C4×C17C2×C22⋊C412yes—
238C2×C4×C17C4×D414yes—
239C2×C4×C17C4×D414yes—
240C2×C4×C17#12012yes—
241C2×C4×C17#12114yes—
242C2×C4×C17#12114yes—
243C2×C4×C17#12312yes—
244C2×C4×C17C23×C411yescharacteristic
245C2×C4×C17C22×D411yescharacteristic
246C2×C4×C17C22×D411yescharacteristic
247C2×C4×C17C22×D411yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—28no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—48no—
262C2×Dic17C2×C22⋊C412yes—
263C2×Dic17C2×C22⋊C412yes—
264C4×D17—22no—
265C4×D17—22no—
266C4×D17—22no—
267C4×D17—22no—
268C4×D17—22no—
269C4×D17—22no—
270C4×D17—22no—
271C4×D17—22no—
272C4×D17—28no—
273C4×D17—28no—
274C4×D17—416no—
275Dic34—24no—
276Dic34—24no—
277Dic34—42no—
278Dic34—42no—
279Dic34—48no—
280C8×C17—42no—
281#3622—42no—
282C22×C22⋊Q8—171no—
283C22×#117—171no—
284C2×#419—171no—
285C2×#419—171no—
286C2×#407—171no—
287#1873—174no—
288#1872—174no—
289#1736—171no—
290#1736D1711yescharacteristic, complemented
291C22×C17—24no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—24no—
295C22×C17#38614yes—
296C22×C17C22×C22⋊C411yescharacteristic
297C22×C17C2×#12011yescharacteristic
298C22×C17C2×#12311yescharacteristic
299C22×C17C2×#13712yes—
300C22×C17#57512yes—
301C2×D17—24no—
302C2×D17—22no—
303C2×D17—24no—
304C2×D17—22no—
305C2×D17—22no—
306C2×D17—22no—
307C2×D17—22no—
308C2×D17—416no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17#41914yescomplemented
312C4×C17—44no—
313C4×C17—44no—
314C4×C17#38614yes—
315C4×C17C2×C4×D411yescharacteristic
316C4×C17C2×C4×D411yescharacteristic
317C4×C17C2×#12111yescharacteristic
318C4×C17C2×#12111yescharacteristic
319Dic17—22no—
320Dic17—22no—
321Dic17—44no—
322Dic17—44no—
323C23×Q8—171no—
324C24×C4—171no—
325C22×M16—171no—
326C22×M16C2×D1711yescharacteristic
327C2×C22⋊Q8—178no—
328C2×C22⋊Q8—172no—
329C2×C22⋊Q8—172no—
330C2×C22⋊Q8—341no—
331C2×C22⋊Q8—344no—
332C2×C22⋊Q8—341no—
333C2×#117—171no—
334C2×#117—172no—
335C2×#117—341no—
336C2×#117—341no—
337C2×#117—344no—
338C2×#117—344no—
339C2×#117C2×D1711yescharacteristic
340C22×C4⋊C4—172no—
341C22×C4⋊C4—341no—
342C22×C4⋊C4C2×D1711yescharacteristic
343C22×C22⋊C4—341no—
344C22×C22⋊C4—341no—
345C22×C42—341no—
346#419—344no—
347#419—344no—
348C2×#103—174no—
349C2×#103—174no—
350#407—344no—
351C2×C22⋊C8—172no—
352C2×C22⋊C8—172no—
353#340—174no—
354#340C2×D1714yescomplemented
355C2×C17—24no—
356C2×C17C2×#38611yescharacteristic
357C2×C17C2×#41911yescharacteristic
358C2×C17#289611yescharacteristic
359C2×C17#300712yes—
360C2×C17#309512yes—
361D17—24no—
362D17—42no—
363C25—171no—
364C22×Q8—174no—
365C22×Q8—172no—
366C22×Q8—341no—
367C22×Q8—342no—
368C22×Q8—342no—
369C22×Q8—342no—
370C22×Q8—342no—
371C22×Q8—342no—
372C23×C4—21no—
373C23×C4—171no—
374C23×C4—174no—
375C23×C4—171no—
376C23×C4—174no—
377C23×C4—171no—
378C23×C4—171no—
379C23×C4—171no—
380C23×C4—342no—
381C23×C4—342no—
382C23×C4—342no—
383C23×C4—342no—
384C23×C4—342no—
385C23×C4—341no—
386C23×C4C22×D1711yescharacteristic
387C2×M16—22no—
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—342no—
391C2×M16—342no—
392C2×M16—342no—
393C22×C8—172no—
394C22×C8C22×D1712yes—
395C22⋊Q8—178no—
396C22⋊Q8—178no—
397C22⋊Q8—344no—
398C22⋊Q8—344no—
399C22⋊Q8—348no—
400C22⋊Q8—348no—
401#117—22no—
402#117—174no—
403#117—348no—
404#117—348no—
405#117—344no—
406#117—342no—
407#117—344no—
408#117D6814yes—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—172no—
412C2×C4⋊C4—344no—
413C2×C4⋊C4—342no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4—342no—
416C2×C4⋊C4—348no—
417C2×C4⋊C4—344no—
418C2×C4⋊C4C4×D1714yescomplemented
419C2×C4⋊C4C22×D1712yes—
420C2×C4⋊C4C22×D1712yes—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—348no—
423C2×C22⋊C4—342no—
424C2×C22⋊C4—342no—
425C2×C22⋊C4—342no—
426C2×C22⋊C4—348no—
427C2×C22⋊C4—341no—
428C2×C42—21no—
429C2×C42—348no—
430C2×C42—342no—
431C2×C42—341no—
432#103—348no—
433#103—348no—
434C22⋊C8—344no—
435C22⋊C8—344no—
436C17#2091111yescharacteristic, complemented, Sylow
437C24—171no—
438C24—171no—
439C24—174no—
440C24—344no—
441C24—342no—
442C24—342no—
443C24—682no—
444C24D4×D1711yescharacteristic
445C2×Q8—174no—
446C2×Q8—342no—
447C2×Q8—344no—
448C2×Q8—342no—
449C2×Q8—342no—
450C2×Q8—342no—
451C2×Q8—342no—
452C2×Q8—348no—
453C2×Q8—348no—
454C2×Q8—682no—
455C2×Q8—684no—
456C2×Q8—682no—
457C2×Q8—684no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—44no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—174no—
465C22×C4—172no—
466C22×C4—172no—
467C22×C4—172no—
468C22×C4—172no—
469C22×C4—174no—
470C22×C4—174no—
471C22×C4—348no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—348no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—348no—
486C22×C4—348no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—348no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—344no—
493C22×C4—684no—
494C22×C4—684no—
495C22×C4—682no—
496C22×C4—682no—
497C22×C4C2×C4×D1712yes—
498C22×C4C2×D6812yes—
499C22×C4#6325212yes—
500C22×C4D4×D1712yes—
501C22×C4D4×D1712yes—
502C22×C4D4×D1711yescharacteristic
503C22×C4#6325612yes—
504C22×C4C23×D1711yescharacteristic
505M16—44no—
506M16—684no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—342no—
511C2×C8—342no—
512C2×C8—342no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—174no—
517C4⋊C4—348no—
518C4⋊C4—348no—
519C4⋊C4—348no—
520C4⋊C4—342no—
521C4⋊C4C2×C4×D1714yes—
522C4⋊C4C2×D6814yes—
523C22⋊C4—24no—
524C22⋊C4—348no—
525C22⋊C4—348no—
526C22⋊C4—348no—
527C22⋊C4—344no—
528C42—24no—
529C42—348no—
530C42—344no—
531C23—22no—
532C23—22no—
533C23—172no—
534C23—174no—
535C23—172no—
536C23—348no—
537C23—341no—
538C23—341no—
539C23—342no—
540C23—341no—
541C23—341no—
542C23—342no—
543C23—342no—
544C23—348no—
545C23—684no—
546C23—688no—
547C23—688no—
548C23C22⋊C4×D1712yes—
549C23#1058756812yes—
550C23#1058758012yes—
551C23C2×D4×D1711yescharacteristic
552Q8—344no—
553Q8—344no—
554Q8—682no—
555Q8—682no—
556Q8—688no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—172no—
564C2×C4—172no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—348no—
570C2×C4—342no—
571C2×C4—342no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—342no—
575C2×C4—342no—
576C2×C4—342no—
577C2×C4—342no—
578C2×C4—348no—
579C2×C4—348no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—6816no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—688no—
588C2×C4C4×D6814yes—
589C2×C4C22⋊C4×D1712yes—
590C2×C4#1058756812yes—
591C2×C4#1058757014yes—
592C2×C4#1058757914yes—
593C2×C4#1058758012yes—
594C2×C4#1058758114yes—
595C2×C4C22×C4×D1711yescharacteristic
596C2×C4C22×D6811yescharacteristic
597C2×C4C2×#6325211yescharacteristic
598C2×C4C2×D4×D1711yescharacteristic
599C2×C4C2×D4×D1711yescharacteristic
600C2×C4C2×#6325611yescharacteristic
601C8—42no—
602C8—682no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—24no—
607C22—174no—
608C22—344no—
609C22—342no—
610C22—342no—
611C22—344no—
612C22—342no—
613C22—342no—
614C22—342no—
615C22—6816no—
616C22—682no—
617C22—682no—
618C22#1176268714yes—
619C22C2×C22⋊C4×D1711yescharacteristic
620C22C2×#1058756811yescharacteristic
621C22C2×#1058758011yescharacteristic
622C22#137×D1712yes—
623C22#1176360512yes—
624C4—44no—
625C4—44no—
626C4—342no—
627C4—342no—
628C4—684no—
629C4—684no—
630C4#1176268714yes—
631C4C2×#1058757911yescharacteristic
632C4C2×C4×D6811yescharacteristic
633C4C2×#1058757011yescharacteristic
634C4C2×#1058758111yescharacteristic
635C2—24no—
636C2—344no—
637C2—682no—
638C2C2×#1176268711yescharacteristic
639C2D17×#41911yescharacteristic
640C2#1453729811yescharacteristic
641C2#1453804512yes—
642C2#1453873412yes—
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1736C2
socleC23×C17C22×D4
prime radicalC24×D17D4
cosocleC22×C4×C17C24
hypercenter#1736D17
hypocenterC17#20911
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. C2×#11762819C2semidirect
4. C2×#11762817C2semidirect
6. #14529095C2semidirect
7. #14529090C2semidirect
8. #14528503C2semidirect
9. C17×#1736C2semidirect
16. C2×#10587583C4semidirect
32. C2×#10587487C22semidirect
33. C2×#10587481C22semidirect
37. #11762571C22semidirect
41. C17×#340C22semidirect
64. C22×Dic34C2×C4semidirect
74. C2×C4⋊C4×C17C2×C4semidirect
102. #10587583C2×C4semidirect
290. #1736D17semidirect
311. C2×D17#419semidirect
354. #340C2×D17semidirect
418. C2×C4⋊C4C4×D17semidirect
436. C17#20911semidirect

Maximal subgroups

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

subgroupindex
C22×#105875832
C2×#117×D172
C2×#117628192
C2×#117628172
C2×#117627722
#145290952
#145290902
#145285032
C17×#17362
#2091117