Finite Group Databasebeta

← #16965534

Subgroups — #16965534

Order 4352.

6494 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
1C24×C4×D17C211yesmaximal, characteristic
2C2×#11762684C211yesmaximal, characteristic, complemented
3C2×#11762683C214yesmaximal, complemented
4C2×#11762682C211yesmaximal, characteristic, complemented
5C2×#11762664C211yesmaximal, characteristic
6C2×#11762664C211yesmaximal, characteristic
7#14527006C211yesmaximal, characteristic, complemented
8#14527005C214yesmaximal, complemented
9C17×C22⋊#95C211yesmaximal, characteristic, complemented
10C25×D17C2211yescharacteristic
11C24×Dic17C2211yescharacteristic
12C23×C4×D17—24no—
13C23×C4×D17—24no—
14C23×C4×D17—24no—
15C23×C4×D17—216no—
16C23×C4×D17C2214yes—
17C22×C4⋊Dic17C2212yescomplemented
18C22×#63225C2212yescomplemented
19C22×C22⋊Dic17C2211yescharacteristic
20C22×C22⋊Dic17C2211yescharacteristic
21C22×#63227C2212yes—
22C22×#63227C2212yes—
23C22×#63227C2214yes—
24#11762684—24no—
25#11762683—216no—
26#11762682—24no—
27#11762664—24no—
28#11762664C418yescomplemented
29C2×#10587502C2212yescomplemented
30C2×#10587502C2212yescomplemented
31C2×#10587502C2214yescomplemented
32C24×C4×C17C2211yescharacteristic
33C22×C22⋊C4×C17C2211yescharacteristic
34C22×C22⋊C4×C17C2211yescharacteristic
35C2×C17×#95C2214yescomplemented
36C25×C17C2311yescharacteristic
37C24×D17—22no—
38C24×D17—22no—
39C24×D17—22no—
40C24×D17—22no—
41C24×D17—28no—
42C24×D17—28no—
43C24×D17C2×C414yes—
44C24×D17D414yes—
45C24×D17Q814yes—
46C24×D17C2312yes—
47C23×C4×C17—22no—
48C23×C4×C17—22no—
49C23×C4×C17—28no—
50C23×C4×C17D414yes—
51C23×C4×C17C2312yes—
52C23×C4×C17C2312yes—
53C23×C4×C17C2312yes—
54C23×Dic17—22no—
55C23×Dic17—22no—
56C23×Dic17—28no—
57C23×Dic17D414yes—
58C23×Dic17C2311yescharacteristic
59C23×Dic17C2311yescharacteristic
60C23×Dic17C2312yes—
61C23×Dic17C2312yes—
62C22×C4×D17—232no—
63C22×C4×D17—264no—
64C22×C4×D17—264no—
65C22×C4×D17—264no—
66C22×C4×D17—28no—
67C22×C4×D17—28no—
68C22×C4×D17—28no—
69C22×C4×D17—28no—
70C22×C4×D17—28no—
71C22×C4×D17—28no—
72C22×C4×D17D4116yes—
73C2×C22⋊C4×C17—22no—
74C2×C22⋊C4×C17—22no—
75C2×C22⋊C4×C17—22no—
76C2×C22⋊C4×C17—22no—
77C2×C22⋊C4×C17—22no—
78C2×C22⋊C4×C17C2×C414yescomplemented
79C2×C22⋊Dic17—22no—
80C2×C22⋊Dic17—22no—
81C2×C22⋊Dic17—22no—
82C2×C22⋊Dic17—22no—
83C2×C22⋊Dic17—22no—
84C2×C22⋊Dic17C2×C414yescomplemented
85C2×#63227—28no—
86C2×#63227—24no—
87C2×#63227—24no—
88C2×#63227—28no—
89C2×#63227—28no—
90C2×#63227—28no—
91C2×#63227C2×C4116yescomplemented
92C2×C4⋊Dic17—28no—
93C2×C4⋊Dic17—24no—
94C2×#63225—24no—
95C2×#63225—28no—
96C17×#95—28no—
97#10587502—24no—
98#10587502—28no—
99#10587502—24no—
100C24×C17—24no—
101C24×C17—24no—
102C24×C17C22×C412yes—
103C24×C17C2×D412yes—
104C24×C17C2×D412yes—
105C24×C17C2×D412yes—
106C24×C17C2×Q812yes—
107C24×C17#4112yes—
108C24×C17#4112yes—
109C24×C17C2411yescharacteristic
110C23×D17—216no—
111C23×D17—216no—
112C23×D17—232no—
113C23×D17—232no—
114C23×D17—216no—
115C23×D17—216no—
116C23×D17—28no—
117C23×D17—28no—
118C23×D17—28no—
119C23×D17—28no—
120C23×D17—28no—
121C23×D17—28no—
122C23×D17—22no—
123C23×D17—24no—
124C23×D17—22no—
125C23×D17—22no—
126C23×D17—22no—
127C23×D17—24no—
128C23×D17—232no—
129C23×D17—216no—
130C23×D17—232no—
131C23×D17—216no—
132C23×D17C4⋊C4116yescomplemented
133C23×D17C22×C414yes—
134C23×D17C2×D414yes—
135C23×D17C2×D418yes—
136C23×D17C2×Q814yes—
137C23×D17#4118yes—
138C22×C4×C17—216no—
139C22×C4×C17—216no—
140C22×C4×C17—28no—
141C22×C4×C17—22no—
142C22×C4×C17—24no—
143C22×C4×C17—22no—
144C22×C4×C17—216no—
145C22×C4×C17—22no—
146C22×C4×C17—24no—
147C22×C4×C17—24no—
148C22×C4×C17—22no—
149C22×C4×C17—24no—
150C22×C4×C17—22no—
151C22×C4×C17—24no—
152C22×C4×C17—22no—
153C22×C4×C17—22no—
154C22×C4×C17C22×C414yescomplemented
155C22×C4×C17C2×D414yes—
156C22×C4×C17C2×D418yes—
157C22×Dic17—216no—
158C22×Dic17—216no—
159C22×Dic17—28no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—24no—
163C22×Dic17—216no—
164C22×Dic17—24no—
165C22×Dic17—22no—
166C22×Dic17—24no—
167C22×Dic17—22no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17—21no—
171C22×Dic17—24no—
172C22×Dic17—21no—
173C22×Dic17—21no—
174C22×Dic17—21no—
175C22×Dic17—24no—
176C22×Dic17C22×C414yescomplemented
177C22×Dic17C2×D414yes—
178C22×Dic17C2×D418yes—
179C2×C4×D17—2256no—
180C2×C4×D17—232no—
181C2×C4×D17—232no—
182C2×C4×D17—232no—
183C2×C4×D17—232no—
184C2×C4×D17—232no—
185C2×C4×D17—232no—
186C2×C4×D17—232no—
187C22⋊C4×C17—28no—
188C22⋊C4×C17—44no—
189C22⋊Dic17—28no—
190C22⋊Dic17—44no—
191#63227—232no—
192#63227—48no—
193#63227—48no—
194C4⋊Dic17—48no—
195#63225—48no—
196#14693—171nomaximal, Sylow
197C23×C17—28no—
198C23×C17—24no—
199C23×C17—24no—
200C23×C17—24no—
201C23×C17—28no—
202C23×C17—24no—
203C23×C17—24no—
204C23×C17—24no—
205C23×C17—28no—
206C23×C17—28no—
207C23×C17C2×C4⋊C414yes—
208C23×C17C4×D414yes—
209C23×C17C4×D414yes—
210C23×C17#12014yes—
211C23×C17C22⋊Q814yes—
212C23×C17C22⋊Q814yes—
213C23×C17#12314yes—
214C23×C17C23×C411yescharacteristic
215C23×C17C22×D411yescharacteristic
216C23×C17C22×D411yescharacteristic
217C23×C17C22×D414yes—
218C23×C17C22×D411yescharacteristic
219C23×C17C22×Q811yescharacteristic
220C23×C17C2×#4111yescharacteristic
221C23×C17C2×#4111yescharacteristic
222C23×C17C2×#4114yes—
223C22×D17—232no—
224C22×D17—264no—
225C22×D17—264no—
226C22×D17—264no—
227C22×D17—232no—
228C22×D17—264no—
229C22×D17—264no—
230C22×D17—28no—
231C22×D17—24no—
232C22×D17—24no—
233C22×D17—28no—
234C22×D17—24no—
235C22×D17—28no—
236C22×D17—24no—
237C22×D17—28no—
238C22×D17—24no—
239C22×D17—24no—
240C22×D17—24no—
241C22×D17—24no—
242C22×D17—28no—
243C22×D17—28no—
244C22×D17—28no—
245C22×D17—28no—
246C22×D17—28no—
247C22×D17—28no—
248C22×D17—232no—
249C22×D17—264no—
250C22×D17C2×C4⋊C418yes—
251C22×D17C4×D4116yes—
252C22×D17#12018yes—
253C22×D17C22⋊Q8116yes—
254C22×D17#12318yes—
255C2×C4×C17—232no—
256C2×C4×C17—28no—
257C2×C4×C17—28no—
258C2×C4×C17—24no—
259C2×C4×C17—24no—
260C2×C4×C17—24no—
261C2×C4×C17—28no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17—44no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17—42no—
268C2×C4×C17—44no—
269C2×C4×C17—44no—
270C2×C4×C17#12018yes—
271C2×Dic17—232no—
272C2×Dic17—24no—
273C2×Dic17—28no—
274C2×Dic17—28no—
275C2×Dic17—24no—
276C2×Dic17—24no—
277C2×Dic17—28no—
278C2×Dic17—42no—
279C2×Dic17—44no—
280C2×Dic17—42no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17#12018yes—
290C4×D17—2128no—
291C25×C4—171no—
292C2×#385—174no—
293C2×#385—171no—
294C2×#385—171no—
295C2×#379—171no—
296C2×#379—171no—
297C22⋊#95—174no—
298C22⋊#95—171no—
299C22⋊#95D1711yescharacteristic, complemented
300C22×C17—24no—
301C22×C17—28no—
302C22×C17—28no—
303C22×C17—24no—
304C22×C17—28no—
305C22×C17—28no—
306C22×C17—28no—
307C22×C17—28no—
308C22×C17#38518yescomplemented
309C22×C17C22×C4⋊C411yescharacteristic
310C22×C17C2×C4×D414yes—
311C22×C17C2×C4×D411yescharacteristic
312C22×C17C2×C4×D411yescharacteristic
313C22×C17C2×#12012yes—
314C22×C17C2×#12012yes—
315C22×C17C2×#12012yes—
316C22×C17C2×#12011yescharacteristic
317C22×C17C2×C22⋊Q814yes—
318C22×C17C2×C22⋊Q811yescharacteristic
319C22×C17C2×C22⋊Q811yescharacteristic
320C22×C17C2×#12312yes—
321C22×C17C2×#12311yescharacteristic
322C22×C17#52512yes—
323C22×C17#52514yes—
324C22×C17#52514yes—
325C22×C17#52512yes—
326C2×D17—2128no—
327C2×D17—28no—
328C2×D17—216no—
329C2×D17—216no—
330C2×D17—28no—
331C2×D17—28no—
332C2×D17—28no—
333C2×D17—28no—
334C2×D17—28no—
335C2×D17—216no—
336C2×D17—216no—
337C2×D17#385116yescomplemented
338C4×C17—28no—
339C4×C17—44no—
340C4×C17—82no—
341Dic17—28no—
342Dic17—44no—
343Dic17—81no—
344Dic17—81no—
345C26—171no—
346C24×C4—171no—
347C24×C4—174no—
348C24×C4—344no—
349C24×C4—3416no—
350C24×C4—344no—
351C24×C4—344no—
352C24×C4C2×D1711yescharacteristic
353C22×C4⋊C4—172no—
354C22×C4⋊C4—172no—
355C22×C22⋊C4—171no—
356C22×C22⋊C4—174no—
357C22×C22⋊C4—171no—
358C22×C22⋊C4—172no—
359C22×C22⋊C4—172no—
360C22×C22⋊C4C2×D1711yescharacteristic
361C22×C22⋊C4C2×D1711yescharacteristic
362#385—3416no—
363#385—344no—
364#385—344no—
365#379—178no—
366#379—344no—
367C2×#95—174no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95C2×D1714yescomplemented
371C2×C17—28no—
372C2×C17C2×#38512yes—
373C2×C17C2×#38511yescharacteristic
374C2×C17C4×#12012yes—
375C2×C17C4×#12012yes—
376C2×C17#284611yescharacteristic
377C2×C17#284812yes—
378C2×C17#284812yes—
379C2×C17#284912yes—
380C2×C17#284911yescharacteristic
381D17—216no—
382C25—172no—
383C25—174no—
384C25—174no—
385C25—174no—
386C25—348no—
387C25—348no—
388C25—342no—
389C25—342no—
390C25—342no—
391C25—342no—
392C25C22×D1711yescharacteristic
393C23×C4—22no—
394C23×C4—22no—
395C23×C4—28no—
396C23×C4—172no—
397C23×C4—174no—
398C23×C4—172no—
399C23×C4—1716no—
400C23×C4—171no—
401C23×C4—171no—
402C23×C4—348no—
403C23×C4—348no—
404C23×C4—348no—
405C23×C4—348no—
406C23×C4—3464no—
407C23×C4—3464no—
408C23×C4—3432no—
409C23×C4—3464no—
410C23×C4—348no—
411C23×C4—342no—
412C23×C4—342no—
413C23×C4—348no—
414C23×C4—348no—
415C23×C4D6814yes—
416C23×C4C22×D1712yes—
417C23×C4C22×D1712yes—
418C23×C4C22×D1712yes—
419C2×C4⋊C4—348no—
420C2×C4⋊C4—344no—
421C2×C4⋊C4—344no—
422C2×C4⋊C4—348no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—174no—
429C2×C22⋊C4—1716no—
430C2×C22⋊C4—348no—
431C2×C22⋊C4—348no—
432C2×C22⋊C4—342no—
433C2×C22⋊C4—342no—
434C2×C22⋊C4—344no—
435C2×C22⋊C4—344no—
436C2×C22⋊C4—348no—
437C2×C22⋊C4—348no—
438C2×C22⋊C4—342no—
439C2×C22⋊C4—342no—
440C2×C22⋊C4—342no—
441C2×C22⋊C4C4×D1714yescomplemented
442#95—28no—
443#95—344no—
444#95—348no—
445#95—344no—
446C17#1469311yescharacteristic, complemented, Sylow
447C24—24no—
448C24—24no—
449C24—174no—
450C24—178no—
451C24—174no—
452C24—174no—
453C24—178no—
454C24—1716no—
455C24—3416no—
456C24—3432no—
457C24—3432no—
458C24—3416no—
459C24—3416no—
460C24—3416no—
461C24—3416no—
462C24—3432no—
463C24—3432no—
464C24—3416no—
465C24—342no—
466C24—342no—
467C24—344no—
468C24—342no—
469C24—342no—
470C24—344no—
471C24—348no—
472C24—348no—
473C24—348no—
474C24—348no—
475C24—348no—
476C24—348no—
477C24C2×C4×D1712yes—
478C24C2×D6812yes—
479C24#6325212yes—
480C24D4×D1712yes—
481C24D4×D1712yes—
482C24#6325412yes—
483C24Q8×D1712yes—
484C24C23×D1711yescharacteristic
485C22×C4—216no—
486C22×C4—216no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—28no—
491C22×C4—216no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—178no—
502C22×C4—174no—
503C22×C4—174no—
504C22×C4—3432no—
505C22×C4—3432no—
506C22×C4—34256no—
507C22×C4—3432no—
508C22×C4—3432no—
509C22×C4—3432no—
510C22×C4—3432no—
511C22×C4—3416no—
512C22×C4—3416no—
513C22×C4—3416no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—344no—
517C22×C4—348no—
518C22×C4—3432no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—344no—
522C22×C4—344no—
523C22×C4—341no—
524C22×C4—341no—
525C22×C4—341no—
526C22×C4—341no—
527C22×C4—341no—
528C22×C4—344no—
529C22×C4—341no—
530C22×C4—344no—
531C22×C4C2×C4×D1714yescomplemented
532C22×C4C2×D6814yes—
533C22×C4D4×D1718yes—
534C4⋊C4—688no—
535C4⋊C4—688no—
536C22⋊C4—28no—
537C22⋊C4—44no—
538C22⋊C4—3432no—
539C22⋊C4—348no—
540C22⋊C4—688no—
541C22⋊C4—688no—
542C22⋊C4—684no—
543C23—28no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—24no—
553C23—178no—
554C23—1716no—
555C23—178no—
556C23—1716no—
557C23—178no—
558C23—3464no—
559C23—3432no—
560C23—3464no—
561C23—3464no—
562C23—3464no—
563C23—3432no—
564C23—3464no—
565C23—3464no—
566C23—348no—
567C23—344no—
568C23—344no—
569C23—348no—
570C23—344no—
571C23—348no—
572C23—344no—
573C23—348no—
574C23—344no—
575C23—344no—
576C23—344no—
577C23—344no—
578C23—348no—
579C23—348no—
580C23—348no—
581C23—348no—
582C23—348no—
583C23—348no—
584C23—3432no—
585C23C4×D6814yes—
586C23#1058756714yes—
587C23#1058756814yes—
588C23#1058756914yes—
589C23C4⋊C4×D1714yes—
590C23#1058758214yes—
591C23#1058758314yes—
592C23C22×C4×D1711yescharacteristic
593C23C22×D6811yescharacteristic
594C23C2×#6325211yescharacteristic
595C23C2×D4×D1711yescharacteristic
596C23C2×D4×D1714yes—
597C23C2×D4×D1711yescharacteristic
598C23C2×#6325411yescharacteristic
599C23C2×Q8×D1711yescharacteristic
600C23#41×D1714yes—
601C2×C4—232no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—178no—
617C2×C4—34128no—
618C2×C4—3432no—
619C2×C4—344no—
620C2×C4—348no—
621C2×C4—348no—
622C2×C4—344no—
623C2×C4—348no—
624C2×C4—344no—
625C2×C4—682no—
626C2×C4—684no—
627C2×C4—682no—
628C2×C4—682no—
629C2×C4—682no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—682no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4—682no—
636C2×C4#1058756818yes—
637C22—24no—
638C22—28no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—1716no—
646C22—34128no—
647C22—348no—
648C22—3416no—
649C22—3416no—
650C22—348no—
651C22—348no—
652C22—348no—
653C22—348no—
654C22—348no—
655C22—3416no—
656C22—3416no—
657C22#1176268318yescomplemented
658C22C2×C4⋊C4×D1711yescharacteristic
659C22C4×D4×D1714yes—
660C22C2×#1058756711yescharacteristic
661C22C2×C4×D6811yescharacteristic
662C22#120×D1712yes—
663C22#120×D1712yes—
664C22C2×#1058756812yes—
665C22C2×#1058756811yescharacteristic
666C22C22⋊Q8×D1714yes—
667C22C2×#1058758211yescharacteristic
668C22C2×#1058758311yescharacteristic
669C22#123×D1712yes—
670C22C2×#1058756911yescharacteristic
671C22#1176331712yes—
672C22#1176331814yes—
673C22#1176331914yes—
674C22#1176332112yes—
675C4—28no—
676C4—44no—
677C4—82no—
678C4—348no—
679C4—684no—
680C4—1361no—
681C4—1361no—
682C2—28no—
683C2—3416no—
684C2D17×#38512yes—
685C2C2×#1176268311yescharacteristic
686C2C4×#1058756812yes—
687C2#1453128912yes—
688C2#1453706211yescharacteristic
689C2#1453707012yes—
690C2#1453707212yes—
691C2#1453707512yes—
692C2#1453707711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×C22⋊#95C2
socle, cosocleC24×C17C24
prime radicalC25×D17C22
hypercenterC22⋊#95D17
hypocenterC17#14693
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#11762684C2semidirect
3. C2×#11762683C2semidirect
4. C2×#11762682C2semidirect
7. #14527006C2semidirect
8. #14527005C2semidirect
9. C17×C22⋊#95C2semidirect
17. C22×C4⋊Dic17C22semidirect
18. C22×#63225C22semidirect
28. #11762664C4semidirect
29. C2×#10587502C22semidirect
30. C2×#10587502C22semidirect
31. C2×#10587502C22semidirect
35. C2×C17×#95C22semidirect
78. C2×C22⋊C4×C17C2×C4semidirect
84. C2×C22⋊Dic17C2×C4semidirect
91. C2×#63227C2×C4semidirect
132. C23×D17C4⋊C4semidirect
154. C22×C4×C17C22×C4semidirect
176. C22×Dic17C22×C4semidirect
299. C22⋊#95D17semidirect
308. C22×C17#385semidirect
337. C2×D17#385semidirect
370. C2×#95C2×D17semidirect
441. C2×C22⋊C4C4×D17semidirect
446. C17#14693semidirect
531. C22×C4C2×C4×D17semidirect
657. C22#11762683semidirect

Maximal subgroups

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

subgroupindex
C24×C4×D172
C2×#117626842
C2×#117626832
C2×#117626822
C2×#117626642
#145270062
#145270052
C17×C22⋊#952
#1469317