Finite Group Databasebeta

← #17045534

Subgroups — #17045534

Order 4352.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762942C211yesmaximal, characteristic
2#121×Dic17C211yesmaximal, characteristic, complemented
3C2×#11762710C211yesmaximal, characteristic, complemented
4#14529657C212yesmaximal, complemented
5#14529482C212yesmaximal, complemented
6#14529409C212yesmaximal, complemented
7#14529130C212yesmaximal, complemented
8#14528977C211yesmaximal, characteristic, complemented
9#14528716C211yesmaximal, characteristic, complemented
10#14528548C211yesmaximal, characteristic, complemented
11C17×#1874C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C22×#63228C2212yes—
14C2×D4×Dic17—21no—
15C2×D4×Dic17C2211yescharacteristic, complemented
16C22×#63227—21no—
17C22×#63227—21no—
18C22×C4×Dic17C2211yescharacteristic
19#11762942C418yescomplemented
20C2×#10587479C2212yescomplemented
21C2×#10587492C2212yes—
22C2×#10587492C2212yes—
23C2×#10587480C2212yes—
24C2×C22⋊#3622C2211yescharacteristic
25C2×#10587486C2212yescomplemented
26#11762712C2211yescharacteristic, complemented
27#11762710—21no—
28#11762710—21no—
29#11762710C2212yes—
30#11762710C2212yes—
31#11762708C2211yescharacteristic, complemented
32#11762702—21no—
33#11762689—21no—
34#11762669C2211yescharacteristic
35C4⋊C4×Dic17C2211yescharacteristic, complemented
36C22⋊C4×Dic17—21no—
37#11762571C2212yescomplemented
38#11762563C2212yescomplemented
39C2×C17×#121C2211yescharacteristic
40C22×C4⋊C4×C17C2211yescharacteristic
41C2×C17×#102C2212yescomplemented
42C2×C17×C22⋊C8C2211yescharacteristic
43C17×#340C2212yescomplemented
44C24×D17—21no—
45C22×D4×C17—21no—
46C22×D4×C17C2311yescharacteristic
47C23×C4×C17—21no—
48C23×C4×C17C2311yescharacteristic
49C23×Dic17—21no—
50C23×Dic17D411yescharacteristic
51C23×Dic17D411yescharacteristic
52C22×D68—21no—
53C22×D68—22no—
54C22×D68—42no—
55C22×D68—42no—
56C22×D68C2×C414yescomplemented
57C22×D68C2312yes—
58C22×C8×C17C2312yes—
59C17×#121—22no—
60C17×#121C2×C414yes—
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17—42no—
63C2×C4⋊C4×C17C2312yes—
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C22⋊C4×C17—21no—
66C2×C22⋊Dic17—21no—
67C2×C22⋊Dic17—21no—
68C2×C22⋊Dic17—21no—
69C2×C22⋊Dic17—21no—
70D4×Dic17—44no—
71D4×Dic17—42no—
72C2×#63228—42no—
73C2×#63228—42no—
74C2×#63228C2×C418yescomplemented
75C2×#63227—22no—
76C2×#63227—22no—
77C2×#63227—22no—
78C2×#63227—22no—
79C2×#63227—44no—
80C2×#63227—44no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4×Dic17—21no—
85C2×C4×Dic17—42no—
86C2×C4×Dic17D411yescharacteristic
87C2×C4×Dic17D411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2312yes—
91C17×#102—42no—
92C17×C22⋊C8—22no—
93#10587502—21no—
94#10587502—21no—
95#10587502—21no—
96C22⋊#3622C2×C414yescomplemented
97#10587492—42no—
98#10587492—42no—
99#10587486—42no—
100#10587480C2×C418yescomplemented
101#10587479—42no—
102C24×C17—21no—
103C24×C17C2×D411yescharacteristic
104C23×D17—24no—
105C23×D17—22no—
106C23×D17—42no—
107C23×D17—42no—
108C23×D17—42no—
109C23×D17—42no—
110C23×D17—42no—
111C23×D17—42no—
112C2×D4×C17—24no—
113C2×D4×C17—21no—
114C2×D4×C17—22no—
115C2×D4×C17—22no—
116C2×D4×C17—42no—
117C2×D4×C17—42no—
118C2×D4×C17C22×C412yes—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—22no—
123C22×C4×C17—42no—
124C22×C4×C17—42no—
125C22×C4×C17—42no—
126C22×C4×C17C22×C412yes—
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17#4112yes—
129C22×C4×C17C2411yescharacteristic
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—41no—
136C22×Dic17—41no—
137C22×Dic17—41no—
138C22×Dic17—42no—
139C22×Dic17—42no—
140C22×Dic17—42no—
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C2×D68—22no—
146C2×D68—22no—
147C2×D68—22no—
148C2×D68—22no—
149C2×D68—24no—
150C2×D68—22no—
151C2×D68—48no—
152C2×D68—42no—
153C2×D68—42no—
154C2×D68—48no—
155C2×D68C22×C414yes—
156C2×C8×C17—22no—
157C2×C8×C17—22no—
158C2×C8×C17—22no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17C22×C412yes—
164C22⋊C4×C17—24no—
165C22⋊Dic17—42no—
166C22⋊Dic17—42no—
167C22⋊Dic17—42no—
168C22⋊Dic17—42no—
169#63228—48no—
170#63227—44no—
171#63227—44no—
172#63227—44no—
173#63227—44no—
174C4⋊Dic17—22no—
175C4⋊Dic17—22no—
176C4⋊Dic17—42no—
177C4×Dic17—22no—
178C4×Dic17—22no—
179C4×Dic17—41no—
180C4×Dic17—41no—
181C4×Dic17—42no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622C22×C414yes—
185#22046—171nomaximal, Sylow
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—22no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17C4×D412yes—
198C23×C17AGL(1,ℤ/8)12yes—
199C23×C17AGL(1,ℤ/8)12yes—
200C23×C17C22×D411yescharacteristic
201C22×D17—24no—
202C22×D17—48no—
203C22×D17—48no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—48no—
211C22×D17—84no—
212C22×D17—84no—
213C22×D17—84no—
214C22×D17—84no—
215D4×C17—24no—
216D4×C17—48no—
217D4×C17—42no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—84no—
229C2×C4×C17C4×D412yes—
230C2×C4×C17C23×C411yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17H(5,𝔽2)12yes—
233C2×C4×C17H(5,𝔽2)12yes—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—44no—
246C2×Dic17—44no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—84no—
254C2×Dic17AGL(1,ℤ/8)12yes—
255D68—24no—
256D68—24no—
257D68—48no—
258D68—42no—
259D68—42no—
260C8×C17—42no—
261#3622—24no—
262C2×#447—171no—
263C4×#121—171no—
264C2×#390—171no—
265#1972—172no—
266#1936—172no—
267#1926—172no—
268#1879—172no—
269#1874D1711yescharacteristic, complemented
270#1853—171no—
271#1786—171no—
272#1749—171no—
273C22×C17—22no—
274C22×C17—22no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—42no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—81no—
283C22×C17—81no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17#44214yes—
287C22×C17C2×C4×D411yescharacteristic
288C22×C17#53411yescharacteristic
289C22×C17#53511yescharacteristic
290C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C17#57511yescharacteristic
294C2×D17—44no—
295C2×D17—44no—
296C2×D17—44no—
297C2×D17—816no—
298C2×D17—82no—
299C2×D17—82no—
300C2×D17—82no—
301C2×D17—82no—
302C4×C17—22no—
303C4×C17—42no—
304C4×C17—82no—
305C4×C17#51812yes—
306C4×C17#53812yes—
307Dic17—22no—
308Dic17—41no—
309Dic17—41no—
310Dic17—41no—
311Dic17—81no—
312Dic17—81no—
313C23×D4—171no—
314C22×D8—172no—
315C2×#121C2×D1711yescharacteristic
316C2×C4×D4—171no—
317C2×C4×D4—341no—
318C22×C4⋊C4C2×D1711yescharacteristic
319C22×C22⋊C4—341no—
320C22×C22⋊C4—341no—
321C22×C42—171no—
322#447—178no—
323C2×#106—172no—
324C2×#102—172no—
325C2×#102—172no—
326C2×#102—172no—
327C2×#102C2×D1712yescomplemented
328C2×C22⋊C8—171no—
329C2×C22⋊C8C2×D1711yescharacteristic
330C2×#97—172no—
331#390—171no—
332#390—172no—
333#390—172no—
334#390—341no—
335#390—341no—
336#389—171no—
337#388—341no—
338#386—341no—
339#380—171no—
340C4×C4⋊C4—171no—
341C4×C22⋊C4—341no—
342#340—172no—
343#340C2×D1712yescomplemented
344#336—172no—
345C2×C17—24no—
346C2×C17—82no—
347C2×C17#232111yescharacteristic
348C2×C17C2×#44211yescharacteristic
349C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
350C2×C17#319211yescharacteristic
351C2×C17#319411yescharacteristic
352C2×C17#321311yescharacteristic
353C2×C17#321711yescharacteristic
354D17—84no—
355C25—341no—
356C22×D4—21no—
357C22×D4—174no—
358C22×D4—172no—
359C22×D4—341no—
360C22×D4—342no—
361C22×D4—682no—
362C22×D4—682no—
363C22×D4C22×D1711yescharacteristic
364C23×C4—21no—
365C23×C4—171no—
366C23×C4—171no—
367C23×C4—341no—
368C23×C4C22×D1711yescharacteristic
369C2×D8—178no—
370C2×D8—682no—
371C2×D8—682no—
372C22×C8—172no—
373C22×C8C22×D1712yes—
374#121—22no—
375#121C4×D1714yes—
376C4×D4—682no—
377C4×D4—684no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4C22×D1712yes—
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—342no—
387C2×C22⋊C4—342no—
388C2×C22⋊C4—342no—
389C2×C22⋊C4—342no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—684no—
395C2×C22⋊C4—684no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—341no—
401C2×C42—682no—
402#106—682no—
403#102—42no—
404#102—178no—
405#102—682no—
406#102—682no—
407C22⋊C8—22no—
408C22⋊C8—174no—
409#97—682no—
410#95—341no—
411#95—341no—
412#95—341no—
413C17#2204611yescharacteristic, complemented, Sylow
414C24—21no—
415C24—342no—
416C24—344no—
417C24—682no—
418C24—682no—
419C24—682no—
420C24—682no—
421C24—682no—
422C24—682no—
423C24D4×D1711yescharacteristic
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—174no—
431C2×D4—342no—
432C2×D4—342no—
433C2×D4—342no—
434C2×D4—342no—
435C2×D4—342no—
436C2×D4—344no—
437C2×D4—688no—
438C2×D4—682no—
439C2×D4—682no—
440C2×D4—688no—
441C2×D4C2×C4×D1712yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—171no—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—342no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—682no—
459C22×C4—681no—
460C22×C4—681no—
461C22×C4—681no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4C2×C4×D1712yes—
465C22×C4D4×D1711yescharacteristic
466C22×C4#6325412yes—
467C22×C4C23×D1711yescharacteristic
468D8—688no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—174no—
473C2×C8—342no—
474C2×C8—342no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—342no—
480C4⋊C4—342no—
481C4⋊C4—682no—
482C4⋊C4C2×C4×D1712yes—
483C22⋊C4—24no—
484C22⋊C4—684no—
485C22⋊C4—684no—
486C22⋊C4—684no—
487C22⋊C4—684no—
488C22⋊C4—682no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C42—342no—
493C42—342no—
494C42—681no—
495C42—681no—
496C42—682no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—344no—
509C23—688no—
510C23—688no—
511C23—682no—
512C23—682no—
513C23—682no—
514C23—682no—
515C23—682no—
516C23—682no—
517C23—688no—
518C23—1364no—
519C23—1364no—
520C23—1364no—
521C23—1364no—
522C23#1058756712yes—
523C23#1058759712yes—
524C23#1058760012yes—
525C23C2×D4×D1711yescharacteristic
526D4—24no—
527D4—48no—
528D4—42no—
529D4—344no—
530D4—344no—
531D4—688no—
532D4—682no—
533D4—682no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—84no—
545C2×C4—172no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—684no—
558C2×C4—684no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—1364no—
566C2×C4#1058756712yes—
567C2×C4C22×C4×D1711yescharacteristic
568C2×C4C2×#6325411yescharacteristic
569C2×C4#1058769112yes—
570C2×C4#1058769612yes—
571C8—42no—
572C8—344no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—684no—
587C22—684no—
588C22—684no—
589C22—13616no—
590C22—1362no—
591C22—1362no—
592C22—1362no—
593C22—1362no—
594C22#1176291814yes—
595C22C2×#1058756711yescharacteristic
596C22#1176337011yescharacteristic
597C22#1176337411yescharacteristic
598C22AGL(1,ℤ/8)×D1711yescharacteristic
599C22C2×#1058759711yescharacteristic
600C22C2×#1058760011yescharacteristic
601C22#1176360811yescharacteristic
602C4—22no—
603C4—42no—
604C4—82no—
605C4—342no—
606C4—681no—
607C4—681no—
608C4—681no—
609C4—1361no—
610C4—1361no—
611C4#1176328112yes—
612C4#1176339612yes—
613C2—24no—
614C2—82no—
615C2—1364no—
616C2#1453144711yescharacteristic
617C2C2×#1176291811yescharacteristic
618C2#1453759711yescharacteristic
619C2#1453958311yescharacteristic
620C2#1453959711yescharacteristic
621C2#1453977011yescharacteristic
622C2#1453980011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1874C2
socleC23×C17C22×D4
prime radicalC2×#11762942C2
cosocleC22×C4×C17C24
hypercenter#1874D17
hypocenterC17#22046
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. #121×Dic17C2semidirect
3. C2×#11762710C2semidirect
4. #14529657C2semidirect
5. #14529482C2semidirect
6. #14529409C2semidirect
7. #14529130C2semidirect
8. #14528977C2semidirect
9. #14528716C2semidirect
10. #14528548C2semidirect
11. C17×#1874C2semidirect
15. C2×D4×Dic17C22semidirect
19. #11762942C4semidirect
20. C2×#10587479C22semidirect
25. C2×#10587486C22semidirect
26. #11762712C22semidirect
31. #11762708C22semidirect
35. C4⋊C4×Dic17C22semidirect
37. #11762571C22semidirect
38. #11762563C22semidirect
41. C2×C17×#102C22semidirect
43. C17×#340C22semidirect
56. C22×D68C2×C4semidirect
74. C2×#63228C2×C4semidirect
96. C22⋊#3622C2×C4semidirect
100. #10587480C2×C4semidirect
269. #1874D17semidirect
327. C2×#102C2×D17semidirect
343. #340C2×D17semidirect
413. C17#22046semidirect

Maximal subgroups

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

subgroupindex
C2×#117629422
#121×Dic172
C2×#117627102
#145296572
#145294822
#145294092
#145291302
#145289772
#145287162
#145285482
C17×#18742
#2204617