Finite Group Databasebeta

← #17066482

Subgroups — #17066482

Order 4352.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762947C211yesmaximal, characteristic, complemented
2C22×#10587481C211yesmaximal, characteristic, complemented
3#14532149C211yesmaximal, characteristic, complemented
4#14529922C212yesmaximal, complemented
5#14529787C212yesmaximal, complemented
6#14529213C211yesmaximal, characteristic, complemented
7#14529118C212yesmaximal, complemented
8#14529091C212yesmaximal, complemented
9#14529077C211yesmaximal, characteristic, complemented
10#14528870C211yesmaximal, characteristic, complemented
11C2×C17×#547C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C22×#63229C2212yes—
14C23×#3622C2211yescharacteristic
15C2×#10587610C2212yes—
16#11762947—24no—
17C2×#10587481—24no—
18C2×#10587481C2212yes—
19C2×#10587481C2212yes—
20C2×#10587481C2212yes—
21C2×#10587503C2212yes—
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2212yes—
24#11762729—21no—
25#11762721—21no—
26#11762715C2212yes—
27#11762689—21no—
28#11762681—21no—
29#11762669C2211yescharacteristic
30#11762571C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C22×#41×C17—21no—
35C17×#547—24no—
36C17×#547—24no—
37C2×C17×C4⋊Q8C2211yescharacteristic
38C2×C17×#123—21no—
39C2×C17×#123—21no—
40C2×C17×C22⋊Q8—21no—
41C2×C17×#121C2211yescharacteristic
42C2×C17×#121C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C2×C4×D4×C17C2211yescharacteristic
45C22×C4⋊C4×C17C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic
47C2×#41×C17—28no—
48C2×#41×C17—44no—
49C2×#41×C17—44no—
50C22×Q8×C17—21no—
51C22×D4×C17—21no—
52C22×D4×C17—21no—
53C22×D4×C17D411yescharacteristic
54C22×D4×C17C2311yescharacteristic
55C23×C4×C17—21no—
56C23×C4×C17—21no—
57C23×C4×C17—21no—
58C23×C4×C17C2311yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C22×Dic34—22no—
62C22×Dic34—24no—
63C22×Dic34—22no—
64C22×Dic34—21no—
65C22×Dic34—42no—
66C22×Dic34C2312yes—
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8—22no—
69C17×#123—28no—
70C17×#123—28no—
71C17×C22⋊Q8—28no—
72C17×#121—22no—
73C17×#121—22no—
74C17×#121—22no—
75C17×#121D414yes—
76C4×D4×C17—22no—
77C4×D4×C17—22no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—22no—
83C2×C4⋊C4×C17—21no—
84C2×C4⋊C4×C17—22no—
85C2×C4⋊C4×C17—22no—
86C2×C4⋊C4×C17D411yescharacteristic
87C2×C4⋊C4×C17D414yes—
88C2×C4⋊C4×C17C2311yescharacteristic
89C2×C4⋊C4×C17C2311yescharacteristic
90C2×C4⋊C4×C17C2311yescharacteristic
91C2×C22⋊C4×C17—21no—
92C2×C22⋊C4×C17—21no—
93C2×C22⋊C4×C17—21no—
94C2×C22⋊C4×C17D411yescharacteristic
95C2×C22⋊C4×C17D411yescharacteristic
96C2×C42×C17C2311yescharacteristic
97C2×C22⋊Dic17—22no—
98C2×C22⋊Dic17—21no—
99C2×#63229—24no—
100C2×#63229—24no—
101C2×#63229—42no—
102C2×C4⋊Dic17C2312yes—
103#10587610—24no—
104#10587610—42no—
105C2×#63225—22no—
106C22×#3622—22no—
107C22×#3622—42no—
108C22×#3622C2312yes—
109C22×#3622C2312yes—
110#10587503—42no—
111#10587502—22no—
112#10587502—22no—
113#10587502—21no—
114C22⋊#3622—22no—
115C22⋊#3622—42no—
116#10587481—24no—
117#10587481—24no—
118#10587481—24no—
119C24×C17—21no—
120C24×C17C2×D411yescharacteristic
121C24×C17C2×D411yescharacteristic
122#41×C17—432no—
123C2×Q8×C17—24no—
124C2×Q8×C17—42no—
125C2×Q8×C17—42no—
126C2×D4×C17—22no—
127C2×D4×C17—22no—
128C2×D4×C17—24no—
129C2×D4×C17—24no—
130C2×D4×C17—21no—
131C2×D4×C17—22no—
132C2×D4×C17—22no—
133C2×D4×C17—42no—
134C2×D4×C17—42no—
135C2×D4×C17—42no—
136C2×D4×C17—42no—
137C2×D4×C17SD1614yescomplemented
138C2×D4×C17C2×D412yes—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—24no—
142C22×C4×C17—24no—
143C22×C4×C17—24no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—22no—
150C22×C4×C17—22no—
151C22×C4×C17—42no—
152C22×C4×C17—42no—
153C22×C4×C17—42no—
154C22×C4×C17—42no—
155C22×C4×C17—42no—
156C22×C4×C17—42no—
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D412yes—
161C22×C4×C17C2×D412yes—
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2411yescharacteristic
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—44no—
167C22×Dic17—42no—
168C2×Dic34—28no—
169C2×Dic34—22no—
170C2×Dic34—22no—
171C2×Dic34—22no—
172C2×Dic34—22no—
173C2×Dic34—22no—
174C2×Dic34—48no—
175C2×Dic34—44no—
176C2×Dic34—44no—
177C2×Dic34—42no—
178C2×Dic34#4114yescomplemented
179C4⋊C4×C17—24no—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—22no—
182C4⋊C4×C17—24no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17C2×D412yes—
187C4⋊C4×C17C2×D412yes—
188C4⋊C4×C17C2×D412yes—
189C22⋊C4×C17—24no—
190C22⋊C4×C17—24no—
191C22⋊C4×C17—22no—
192C22⋊C4×C17—24no—
193C22⋊C4×C17—22no—
194C42×C17—21no—
195C42×C17—21no—
196C22⋊Dic17—44no—
197C22⋊Dic17—42no—
198#63229—48no—
199C4⋊Dic17—22no—
200C4⋊Dic17—22no—
201#63225—44no—
202C2×#3622—22no—
203C2×#3622—22no—
204C2×#3622—22no—
205C2×#3622—22no—
206C2×#3622—22no—
207C2×#3622—22no—
208C2×#3622—44no—
209#23961—171nomaximal, Sylow
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—21no—
213C23×C17—22no—
214C23×C17—21no—
215C23×C17—22no—
216C23×C17—22no—
217C23×C17—42no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—42no—
221C23×C17#12012yes—
222C23×C17C2×SD1612yes—
223C23×C17C2×SD1612yes—
224C23×C17AGL(1,ℤ/8)12yes—
225C23×C17C22×D411yescharacteristic
226Q8×C17—48no—
227D4×C17—28no—
228D4×C17—24no—
229D4×C17—48no—
230D4×C17—48no—
231D4×C17—41no—
232D4×C17—41no—
233C2×C4×C17—21no—
234C2×C4×C17—22no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—24no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—48no—
247C2×C4×C17—48no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—48no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—42no—
258C2×C4×C17#12012yes—
259C2×C4×C17C2×SD1612yes—
260C2×C4×C17C22×D411yescharacteristic
261C2×C4×C17C22×D411yescharacteristic
262C2×C4×C17C2×#4112yes—
263C2×C4×C17H(5,𝔽2)12yes—
264C2×Dic17—24no—
265C2×Dic17—48no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271Dic34—24no—
272Dic34—24no—
273Dic34—48no—
274Dic34—42no—
275Dic34—42no—
276#3622—42no—
277#3622—42no—
278C2×#547D1711yescharacteristic
279C2×#448—171no—
280C22×#103—171no—
281#2409—171no—
282#2024—172no—
283#2000—172no—
284#1893—171no—
285#1877—172no—
286#1872—172no—
287#1869—171no—
288#1823—171no—
289C22×C17—22no—
290C22×C17—22no—
291C22×C17—24no—
292C22×C17—24no—
293C22×C17—22no—
294C22×C17—24no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—82no—
302C22×C17—82no—
303C22×C17#45014yes—
304C22×C17C2×#12011yescharacteristic
305C22×C17#54612yes—
306C22×C17C22×SD1611yescharacteristic
307C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C17#57812yes—
309C4×C17—22no—
310C4×C17—22no—
311C4×C17—41no—
312C4×C17—42no—
313C4×C17—42no—
314C4×C17—42no—
315C4×C17—41no—
316C4×C17#52511yescharacteristic
317C4×C17#53511yescharacteristic
318C4×C17#54612yes—
319Dic17—44no—
320Dic17—82no—
321C22×#41—21no—
322C23×Q8—171no—
323C22×SD16—172no—
324C23×C8—171no—
325#547—24no—
326#547—24no—
327C2×C4⋊Q8C2×D1711yescharacteristic
328C2×#123—21no—
329C2×#123—21no—
330C2×C22⋊Q8—21no—
331C2×C22⋊Q8—172no—
332C2×#121C2×D1711yescharacteristic
333C2×#121C2×D1711yescharacteristic
334C2×C4×D4C2×D1711yescharacteristic
335C2×C4×D4C2×D1711yescharacteristic
336C22×C4⋊C4C2×D1711yescharacteristic
337C22×C4⋊C4C2×D1711yescharacteristic
338#448—344no—
339C2×#103—172no—
340C2×#103—172no—
341C2×#103—172no—
342C2×#103—344no—
343C2×#102—172no—
344C2×C22⋊C8—171no—
345C2×C22⋊C8—172no—
346#395—341no—
347#393—341no—
348#391—172no—
349#386—341no—
350#385—341no—
351#380—171no—
352#340—172no—
353#340—172no—
354#340—172no—
355#336—172no—
356C2×C17—24no—
357C2×C17—42no—
358C2×C17—82no—
359C2×C17#240411yescharacteristic
360C2×C17C2×#45011yescharacteristic
361C2×C17#301111yescharacteristic
362C2×C17#329812yes—
363C2×C17#330812yes—
364C2×#41—28no—
365C2×#41—44no—
366C2×#41—44no—
367C22×Q8—21no—
368C22×Q8—172no—
369C22×Q8—344no—
370C22×Q8—341no—
371C22×Q8—342no—
372C22×Q8—342no—
373C22×Q8—682no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4C22⋊D1711yescharacteristic
377C22×D4C22×D1711yescharacteristic
378C23×C4—21no—
379C23×C4—21no—
380C23×C4—21no—
381C23×C4—341no—
382C23×C4C22×D1711yescharacteristic
383C23×C4C22×D1711yescharacteristic
384C2×SD16—344no—
385C2×SD16—344no—
386C2×SD16—682no—
387C22×C8—172no—
388C22×C8—172no—
389C22×C8—342no—
390C22×C8—682no—
391C4⋊Q8—22no—
392C4⋊Q8—22no—
393#123—28no—
394#123—28no—
395C22⋊Q8—28no—
396C22⋊Q8—344no—
397C22⋊Q8—682no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121C22⋊D1714yes—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—172no—
413C2×C4⋊C4—342no—
414C2×C4⋊C4C22⋊D1711yescharacteristic
415C2×C4⋊C4C22⋊D1714yes—
416C2×C4⋊C4C22×D1711yescharacteristic
417C2×C4⋊C4C22×D1711yescharacteristic
418C2×C4⋊C4C22×D1711yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—342no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4C22⋊D1711yescharacteristic
425C2×C22⋊C4C22⋊D1711yescharacteristic
426C2×C42C22×D1711yescharacteristic
427#103—344no—
428#103—344no—
429#103—344no—
430#102—682no—
431C22⋊C8—342no—
432C22⋊C8—682no—
433#95—341no—
434#95—342no—
435#95—342no—
436C17#2396111yescharacteristic, complemented, Sylow
437C24—21no—
438C24C2×C22⋊D1711yescharacteristic
439C24C2×C22⋊D1711yescharacteristic
440#41—432no—
441C2×Q8—24no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—174no—
445C2×Q8—348no—
446C2×Q8—342no—
447C2×Q8—342no—
448C2×Q8—342no—
449C2×Q8—342no—
450C2×Q8—342no—
451C2×Q8—684no—
452C2×Q8—688no—
453C2×Q8—684no—
454C2×Q8—682no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4#6322914yescomplemented
467C2×D4C2×C22⋊D1712yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—684no—
489C22×C4—682no—
490C22×C4C2×C22⋊D1711yescharacteristic
491C22×C4C2×C22⋊D1711yescharacteristic
492C22×C4C2×C22⋊D1712yes—
493C22×C4C2×C22⋊D1712yes—
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C23×D1711yescharacteristic
497SD16—688no—
498C2×C8—342no—
499C2×C8—342no—
500C2×C8—342no—
501C2×C8—342no—
502C2×C8—342no—
503C2×C8—342no—
504C2×C8—684no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—342no—
513C4⋊C4—342no—
514C4⋊C4—684no—
515C4⋊C4C2×C22⋊D1712yes—
516C4⋊C4C2×C22⋊D1712yes—
517C4⋊C4C2×C22⋊D1712yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—684no—
524C22⋊C4—682no—
525C42—21no—
526C42—21no—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23#1058761812yes—
539C23C2×#6322912yes—
540C23C2×#6322912yes—
541C23#1058763912yes—
542C23C22×C22⋊D1711yescharacteristic
543Q8—48no—
544Q8—344no—
545Q8—344no—
546Q8—688no—
547Q8—682no—
548Q8—682no—
549D4—28no—
550D4—24no—
551D4—48no—
552D4—48no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—344no—
581C2×C4—688no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4C2×#6322912yes—
588C2×C4#1058763912yes—
589C2×C4#41×D1712yes—
590C2×C4#1058769612yes—
591C2×C4C22×C22⋊D1711yescharacteristic
592C2×C4C22×C22⋊D1711yescharacteristic
593C8—682no—
594C8—682no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22—82no—
609C22#1176296014yes—
610C22C2×#1058763911yescharacteristic
611C22#1176344712yes—
612C22C22×#6322911yescharacteristic
613C22C2×#1058761811yescharacteristic
614C22#1176362412yes—
615C4—22no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—684no—
623C4—1362no—
624C4#1176331611yescharacteristic
625C4#1176337311yescharacteristic
626C4#1176344712yes—
627C2—24no—
628C2—42no—
629C2—82no—
630C2#1453209911yescharacteristic
631C2C2×#1176296011yescharacteristic
632C2#1453807911yescharacteristic
633C2#1454064312yes—
634C2#1454076912yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#547C2
socleC23×C17C22×D4
hypercenterC2×#547D17
hypocenterC17#23961
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#11762947C2semidirect
2. C22×#10587481C2semidirect
3. #14532149C2semidirect
4. #14529922C2semidirect
5. #14529787C2semidirect
6. #14529213C2semidirect
7. #14529118C2semidirect
8. #14529091C2semidirect
9. #14529077C2semidirect
10. #14528870C2semidirect
137. C2×D4×C17SD16semidirect
178. C2×Dic34#41semidirect
436. C17#23961semidirect
466. C2×D4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117629472
C22×#105874812
#145321492
#145299222
#145297872
#145292132
#145291182
#145290912
#145290772
#145288702
C2×C17×#5472
#2396117