Finite Group Databasebeta

← #17067296

Subgroups — #17067296

Order 4352.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763120C211yesmaximal, characteristic, complemented
2C22×#10587506C211yesmaximal, characteristic, complemented
3#14533092C211yesmaximal, characteristic, complemented
4#14530020C212yesmaximal, complemented
5#14529984C212yesmaximal, complemented
6#14529213C211yesmaximal, characteristic, complemented
7#14529077C211yesmaximal, characteristic, complemented
8#14529038C212yesmaximal, complemented
9#14528724C212yesmaximal, complemented
10#14528719C211yesmaximal, characteristic, complemented
11C2×C17×#548C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C2×#10587621—21no—
14C22×C4⋊Dic17C2211yescharacteristic
15#11763120—24no—
16C2×#10587478C2212yes—
17C2×#10587506—24no—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×#10587503C2212yes—
23C2×#10587503C2212yes—
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2212yes—
26#11762739—21no—
27#11762734—21no—
28#11762707C2212yes—
29#11762670—21no—
30#11762669C2211yescharacteristic
31#11762669C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C22×#41×C17C2211yescharacteristic
35C23×Q8×C17C2211yescharacteristic
36C17×#548—24no—
37C17×#548—24no—
38C2×C17×#124—21no—
39C2×C17×#124C2211yescharacteristic
40C2×C17×C22⋊Q8—21no—
41C2×C17×C22⋊Q8C2211yescharacteristic
42C2×C17×#121—21no—
43C2×C17×#121C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17—21no—
46C2×C4×D4×C17C2211yescharacteristic
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—24no—
50C2×#41×C17—42no—
51C22×Q8×C17—21no—
52C22×Q8×C17—22no—
53C22×Q8×C17—24no—
54C22×Q8×C17—42no—
55C22×Q8×C17D414yes—
56C22×Q8×C17C2311yescharacteristic
57C22×Q8×C17C2311yescharacteristic
58C22×D4×C17—21no—
59C22×D4×C17—21no—
60C22×D4×C17D411yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C23×C4×C17C2311yescharacteristic
65C23×C4×C17C2311yescharacteristic
66C23×Dic17—21no—
67C17×#124—28no—
68C17×#124—22no—
69C17×#124—22no—
70C17×C22⋊Q8—28no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—22no—
73C17×#121—28no—
74C17×#121—22no—
75C17×#121D414yes—
76C4×Q8×C17—22no—
77C4×Q8×C17—22no—
78C4×D4×C17—28no—
79C4×D4×C17—22no—
80C4×D4×C17—22no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17D411yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17D411yescharacteristic
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C42×C17—21no—
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—22no—
94C2×C22⋊Dic17—21no—
95#10587621—44no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—22no—
98C2×C4⋊Dic17—22no—
99C2×C4⋊Dic17C2312yes—
100C2×C4⋊Dic17C2312yes—
101C2×#63225—22no—
102C22×#3622—22no—
103C22×#3622—42no—
104C22×#3622C2312yes—
105C22×#3622C2312yes—
106#10587506—24no—
107#10587506—24no—
108#10587506—24no—
109#10587506—42no—
110#10587503—24no—
111#10587503—42no—
112#10587502—22no—
113#10587502—21no—
114#10587502—22no—
115C22⋊#3622—22no—
116C22⋊#3622—42no—
117#10587478—24no—
118C24×C17—21no—
119C24×C17C2×D411yescharacteristic
120C24×C17C2×D411yescharacteristic
121#41×C17—48no—
122#41×C17—44no—
123#41×C17—44no—
124C2×Q8×C17—24no—
125C2×Q8×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—22no—
128C2×Q8×C17—28no—
129C2×Q8×C17—22no—
130C2×Q8×C17—22no—
131C2×Q8×C17—44no—
132C2×Q8×C17—44no—
133C2×Q8×C17—48no—
134C2×Q8×C17C2×D412yes—
135C2×Q8×C17C2×D412yes—
136C2×Q8×C17#4114yes—
137C2×D4×C17—24no—
138C2×D4×C17—24no—
139C2×D4×C17—22no—
140C2×D4×C17—22no—
141C2×D4×C17—22no—
142C2×D4×C17—21no—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—44no—
146C2×D4×C17—42no—
147C2×D4×C17—42no—
148C2×D4×C17C2×D412yes—
149C22×C4×C17—21no—
150C22×C4×C17—22no—
151C22×C4×C17—22no—
152C22×C4×C17—24no—
153C22×C4×C17—24no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—22no—
159C22×C4×C17—22no—
160C22×C4×C17—44no—
161C22×C4×C17—44no—
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D412yes—
167C22×C4×C17C2×D412yes—
168C22×C4×C17C2411yescharacteristic
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—42no—
172C22×Dic17—42no—
173C22×Dic17—42no—
174C4⋊C4×C17—24no—
175C4⋊C4×C17—24no—
176C4⋊C4×C17—22no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17C2×D412yes—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—24no—
181C22⋊C4×C17—24no—
182C22⋊C4×C17—22no—
183C22⋊C4×C17—24no—
184C22⋊C4×C17—22no—
185C42×C17—24no—
186C42×C17—21no—
187C42×C17—21no—
188C22⋊Dic17—44no—
189C22⋊Dic17—42no—
190C4⋊Dic17—22no—
191C4⋊Dic17—22no—
192C4⋊Dic17—22no—
193C4⋊Dic17—22no—
194C4⋊Dic17—42no—
195#63225—44no—
196C2×#3622—22no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—22no—
201C2×#3622—22no—
202C2×#3622—44no—
203#24035—171nomaximal, Sylow
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—21no—
207C23×C17—22no—
208C23×C17—22no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—22no—
212C23×C17—42no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—42no—
216C23×C17#12012yes—
217C23×C17#13512yes—
218C23×C17AGL(1,ℤ/8)12yes—
219C23×C17C22×D411yescharacteristic
220Q8×C17—22no—
221Q8×C17—24no—
222Q8×C17—22no—
223Q8×C17—22no—
224Q8×C17—48no—
225Q8×C17—42no—
226Q8×C17#12314yescomplemented
227D4×C17—24no—
228D4×C17—48no—
229D4×C17—44no—
230D4×C17—48no—
231D4×C17—41no—
232D4×C17—41no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
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—21no—
244C2×C4×C17—22no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—22no—
248C2×C4×C17—48no—
249C2×C4×C17—48no—
250C2×C4×C17—42no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—42no—
256C2×C4×C17#12012yes—
257C2×C4×C17C22×D411yescharacteristic
258C2×C4×C17C22×D411yescharacteristic
259C2×C4×C17C2×#4112yes—
260C2×C4×C17H(5,𝔽2)12yes—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×Dic17—84no—
268#3622—42no—
269#3622—42no—
270C2×#548D1711yescharacteristic
271C2×#480—171no—
272C22×#103—171no—
273#2486—171no—
274#2042—172no—
275#2036—172no—
276#1893—171no—
277#1869—171no—
278#1863—172no—
279#1788—172no—
280#1787—171no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—24no—
284C22×C17—24no—
285C22×C17—42no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17—42no—
289C22×C17—41no—
290C22×C17—42no—
291C22×C17—42no—
292C22×C17—42no—
293C22×C17—82no—
294C22×C17—82no—
295C22×C17#44914yes—
296C22×C17C2×#12011yescharacteristic
297C22×C17#54612yes—
298C22×C17C2×#13511yescharacteristic
299C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C17#57812yes—
301C4×C17—22no—
302C4×C17—24no—
303C4×C17—41no—
304C4×C17—42no—
305C4×C17—41no—
306C4×C17—42no—
307C4×C17—41no—
308C4×C17C2×#12311yescharacteristic
309C4×C17#53411yescharacteristic
310C4×C17#54612yes—
311Dic17—82no—
312Dic17—82no—
313C22×#41C2×D1711yescharacteristic
314C23×Q8C2×D1711yescharacteristic
315C23×C8—171no—
316#548—24no—
317#548—24no—
318C2×#124—21no—
319C2×#124C2×D1711yescharacteristic
320C2×#123—341no—
321C2×C22⋊Q8—21no—
322C2×C22⋊Q8C2×D1711yescharacteristic
323C2×#121—21no—
324C2×#121C2×D1711yescharacteristic
325C2×C4×Q8C2×D1711yescharacteristic
326C2×C4×D4—21no—
327C2×C4×D4C2×D1711yescharacteristic
328C22×C4⋊C4—171no—
329#480—344no—
330C2×#107—172no—
331C2×#103—172no—
332C2×#103—172no—
333C2×#103—172no—
334C2×#103—172no—
335C2×#103—344no—
336C2×#102—172no—
337C2×#102—172no—
338C2×C22⋊C8—171no—
339C2×C22⋊C8—172no—
340#398—341no—
341#397—341no—
342#389—172no—
343#381—341no—
344#380—171no—
345#380—172no—
346#340—172no—
347#336—172no—
348C2×C17—24no—
349C2×C17—81no—
350C2×C17—82no—
351C2×C17#240511yescharacteristic
352C2×C17C2×#44911yescharacteristic
353C2×C17#301111yescharacteristic
354C2×C17#330012yes—
355C2×C17#330812yes—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—24no—
359C2×#41—42no—
360C22×Q8—21no—
361C22×Q8—22no—
362C22×Q8—24no—
363C22×Q8—42no—
364C22×Q8C22⋊D1714yes—
365C22×Q8C22×D1711yescharacteristic
366C22×Q8C22×D1711yescharacteristic
367C22×D4—21no—
368C22×D4—21no—
369C22×D4C22⋊D1711yescharacteristic
370C22×D4C22×D1711yescharacteristic
371C23×C4—21no—
372C23×C4—21no—
373C23×C4—341no—
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1711yescharacteristic
376C22×C8—172no—
377C22×C8—172no—
378C22×C8—342no—
379C22×C8—682no—
380#124—28no—
381#124—22no—
382#124—22no—
383#123—684no—
384C22⋊Q8—28no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387#121—28no—
388#121—22no—
389#121C22⋊D1714yes—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×D4—28no—
393C4×D4—22no—
394C4×D4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—172no—
398C2×C4⋊C4—172no—
399C2×C4⋊C4—342no—
400C2×C4⋊C4—342no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—342no—
403C2×C4⋊C4C22⋊D1711yescharacteristic
404C2×C4⋊C4C22×D1711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—341no—
410C2×C22⋊C4—342no—
411C2×C22⋊C4C22⋊D1711yescharacteristic
412C2×C22⋊C4C22⋊D1711yescharacteristic
413C2×C42—21no—
414C2×C42C22×D1711yescharacteristic
415#107—344no—
416#103—344no—
417#103—344no—
418#103—344no—
419#103—682no—
420#102—344no—
421#102—682no—
422C22⋊C8—342no—
423C22⋊C8—682no—
424#95—342no—
425#95—341no—
426#95—342no—
427C17#2403511yescharacteristic, complemented, Sylow
428C24—21no—
429C24C2×C22⋊D1711yescharacteristic
430C24C2×C22⋊D1711yescharacteristic
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—24no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—48no—
444C2×Q8#6325414yes—
445C2×Q8C2×C22⋊D1712yes—
446C2×Q8C2×C22⋊D1712yes—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×C22⋊D1712yes—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—682no—
475C22×C4—682no—
476C22×C4—682no—
477C22×C4C2×C22⋊D1711yescharacteristic
478C22×C4C2×C22⋊D1711yescharacteristic
479C22×C4C2×C22⋊D1711yescharacteristic
480C22×C4C2×C22⋊D1712yes—
481C22×C4C2×C22⋊D1712yes—
482C22×C4C2×C22⋊D1711yescharacteristic
483C22×C4C23×D1711yescharacteristic
484C2×C8—342no—
485C2×C8—342no—
486C2×C8—342no—
487C2×C8—342no—
488C2×C8—342no—
489C2×C8—342no—
490C2×C8—684no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—342no—
496C4⋊C4—342no—
497C4⋊C4—342no—
498C4⋊C4—342no—
499C4⋊C4—684no—
500C4⋊C4—682no—
501C4⋊C4C2×C22⋊D1712yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—684no—
509C22⋊C4—682no—
510C42—24no—
511C42—21no—
512C42—21no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23#1058763512yes—
526C23#1058763612yes—
527C23#1058763912yes—
528C23C22×C22⋊D1711yescharacteristic
529Q8—22no—
530Q8—24no—
531Q8—22no—
532Q8—22no—
533Q8—48no—
534Q8—42no—
535Q8#1058762114yescomplemented
536D4—24no—
537D4—48no—
538D4—44no—
539D4—48no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—682no—
566C2×C4—682no—
567C2×C4—682no—
568C2×C4—682no—
569C2×C4—682no—
570C2×C4—682no—
571C2×C4—1364no—
572C2×C4#1058763912yes—
573C2×C4C2×#6325412yes—
574C2×C4#1058769112yes—
575C2×C4C22×C22⋊D1711yescharacteristic
576C2×C4C22×C22⋊D1711yescharacteristic
577C8—682no—
578C8—682no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—82no—
592C22—82no—
593C22#1176295414yes—
594C22C2×#1058763911yescharacteristic
595C22#1176345312yes—
596C22C2×#1058763611yescharacteristic
597C22C2×#1058763511yescharacteristic
598C22#1176362412yes—
599C4—22no—
600C4—24no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—1362no—
607C4—1362no—
608C4C2×#1058762111yescharacteristic
609C4#1176336811yescharacteristic
610C4#1176345312yes—
611C2—24no—
612C2—81no—
613C2—82no—
614C2#1453210711yescharacteristic
615C2C2×#1176295411yescharacteristic
616C2#1453807911yescharacteristic
617C2#1454067112yes—
618C2#1454077512yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#548C2
socleC23×C17C22×D4
hypercenterC2×#548D17
hypocenterC17#24035
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#11763120C2semidirect
2. C22×#10587506C2semidirect
3. #14533092C2semidirect
4. #14530020C2semidirect
5. #14529984C2semidirect
6. #14529213C2semidirect
7. #14529077C2semidirect
8. #14529038C2semidirect
9. #14528724C2semidirect
10. #14528719C2semidirect
226. Q8×C17#123semidirect
427. C17#24035semidirect
535. Q8#10587621semidirect

Maximal subgroups

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

subgroupindex
C2×#117631202
C22×#105875062
#145330922
#145300202
#145299842
#145292132
#145290772
#145290382
#145287242
#145287192
C2×C17×#5482
#2403517