Finite Group Databasebeta

← #17054056

Subgroups — #17054056

Order 4352.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762900C211yesmaximal, characteristic
2#14531756C211yesmaximal, characteristic
3#14531745C211yesmaximal, characteristic, complemented
4#14529639C211yesmaximal, characteristic
5#14529637C211yesmaximal, characteristic
6#14529426C211yesmaximal, characteristic, complemented
7#14529424C211yesmaximal, characteristic, complemented
8#14529406C211yesmaximal, characteristic
9#14528947C211yesmaximal, characteristic, complemented
10#14528798C211yesmaximal, characteristic, complemented
11#14528792C211yesmaximal, characteristic
12C4×#10587479C211yesmaximal, characteristic, complemented
13#14528672C211yesmaximal, characteristic, complemented
14#14528645C211yesmaximal, characteristic, complemented
15C17×#1925C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×#10587573C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19#11762900C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762714C2211yescharacteristic
34#11762712C2211yescharacteristic
35#11762703—21no—
36#11762697—21no—
37#11762693—21no—
38#11762691—21no—
39#11762676—21no—
40#11762674C2211yescharacteristic
41#11762674C2211yescharacteristic
42#11762670—21no—
43C4⋊C4×Dic17C2211yescharacteristic
44C4⋊C4×Dic17C2211yescharacteristic
45C4×#63225C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47C22⋊C4×Dic17—21no—
48#11762655—21no—
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C22×Dic34C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C22⋊C4×C17—21no—
69C2×C22⋊C4×C17—21no—
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75D4×Dic17—22no—
76D4×Dic17C2×C414yes—
77C2×#63229—22no—
78C2×#63229—22no—
79C2×#63229C2×C414yes—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225D411yescharacteristic
85C2×#63225D411yescharacteristic
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94#10587573—22no—
95#10587573C2×C414yes—
96C17×#107—41no—
97C17×#102—41no—
98C17×#102C2×C414yes—
99C17×#95—21no—
100#10587503—41no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502D411yescharacteristic
113#10587502D411yescharacteristic
114#10587489—41no—
115#10587487—41no—
116#10587487C2×C414yes—
117#10587485—22no—
118C8×Dic17—22no—
119C8×Dic17C2×C414yes—
120#10587481—41no—
121#10587479—22no—
122#10587479—22no—
123#10587479C2×C414yes—
124C4×#3622—22no—
125C24×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17C22×C412yes—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2411yescharacteristic
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—24no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—81no—
144C22×Dic17—81no—
145C22×Dic17C2×D411yescharacteristic
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34C22×C412yes—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17C22×C412yes—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17C22×C412yes—
167C22⋊C4×C17—42no—
168C22⋊C4×C17—81no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—42no—
172C22⋊Dic17—24no—
173C22⋊Dic17—42no—
174C22⋊Dic17—81no—
175#63229—28no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17C22×C412yes—
182#63225—22no—
183#63225—24no—
184#63225—22no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—24no—
188C4×Dic17—42no—
189C4×Dic17—81no—
190C4×Dic17—81no—
191C4×Dic17C22×C412yes—
192C4×Dic17#4112yes—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622C22×C412yes—
197C2×#3622#4112yes—
198C2×#3622#4112yes—
199#22806—171nomaximal, Sylow
200C23×C17—22no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17C22×D411yescharacteristic
208D4×C17—22no—
209D4×C17—22no—
210D4×C17—41no—
211D4×C17—41no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—22no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17#13512yes—
227C2×C4×C17C23×C411yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17H(5,𝔽2)11yescharacteristic
232C2×C4×C172−1+411yescharacteristic
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—84no—
255C2×Dic17C4×D412yes—
256C2×Dic17C4×D412yes—
257C2×Dic17#13512yes—
258Dic34—22no—
259Dic34—22no—
260Dic34—41no—
261Dic34—41no—
262C8×C17—22no—
263C8×C17—41no—
264#3622—41no—
265#3622#11714yes—
266C2×C4×SD16—171no—
267#2357—171no—
268#2356—171no—
269#1968—171no—
270#1968—171no—
271#1928—171no—
272#1928—171no—
273#1925—171no—
274#1925D1711yescharacteristic
275#1846—171no—
276#1805—171no—
277#1804—171no—
278C4×#106—171no—
279C4×#106—171no—
280C8×C4⋊C4—171no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17—81no—
287C22×C17—81no—
288C22×C17C2×C4×D411yescharacteristic
289C22×C17#54711yescharacteristic
290C22×C17#55011yescharacteristic
291C22×C17C2×#13511yescharacteristic
292C22×C17C2×#13511yescharacteristic
293C22×C17#57711yescharacteristic
294C22×C17#57711yescharacteristic
295C4×C17—22no—
296C4×C17—41no—
297C4×C17—42no—
298C4×C17—81no—
299C4×C17—81no—
300C4×C17C2×#11711yescharacteristic
301C4×C17#52011yescharacteristic
302C4×C17#55511yescharacteristic
303C4×C17#55611yescharacteristic
304Dic17—22no—
305Dic17—22no—
306Dic17—22no—
307Dic17—41no—
308Dic17—42no—
309Dic17—42no—
310Dic17—81no—
311Dic17—81no—
312C22×SD16—171no—
313C2×C4×Q8—171no—
314C2×C4×D4—171no—
315C4×SD16—178no—
316C2×#107—171no—
317C2×#107C2×D1711yescharacteristic
318C2×#106—171no—
319C2×#106—171no—
320C2×#106—171no—
321C2×C4⋊C8—171no—
322C2×#103—171no—
323C2×#103—171no—
324C2×#103—171no—
325C2×#102—171no—
326C2×#102C2×D1711yescharacteristic
327C2×#102C2×D1711yescharacteristic
328C2×C4×C8—171no—
329C2×C4×C8—171no—
330C2×C4×C8—171no—
331#391—171no—
332#391—171no—
333#390—171no—
334#390C2×D1711yescharacteristic
335#388—341no—
336#388—341no—
337#387—341no—
338#387—341no—
339#383—341no—
340#382—171no—
341#382—171no—
342#381—341no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C4⋊C4C2×D1711yescharacteristic
348C4×C22⋊C4—341no—
349#376—341no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340C2×D1711yescharacteristic
354#336—171no—
355#336C2×D1711yescharacteristic
356C2×C17—82no—
357C2×C17#235411yescharacteristic
358C2×C17C4×#13511yescharacteristic
359C2×C17#294911yescharacteristic
360C2×C17#333711yescharacteristic
361C2×C17#333711yescharacteristic
362C2×C17#339411yescharacteristic
363C2×C17#339411yescharacteristic
364C22×Q8—171no—
365C22×D4C22×D1711yescharacteristic
366C23×C4—341no—
367C2×SD16—174no—
368C2×SD16—342no—
369C2×SD16—342no—
370C22×C8—171no—
371C22×C8—171no—
372C22×C8C22×D1711yescharacteristic
373C22×C8C22×D1711yescharacteristic
374C4×Q8—174no—
375C4×Q8—342no—
376C4×D4—174no—
377C4×D4—342no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C42—21no—
393C2×C42—171no—
394C2×C42—171no—
395C2×C42—341no—
396C2×C42—341no—
397C2×C42—341no—
398C2×C42—341no—
399C2×C42C22×D1711yescharacteristic
400#107—41no—
401#107—681no—
402#106—174no—
403#106—342no—
404#106—342no—
405C4⋊C8—342no—
406#103—174no—
407#103—681no—
408#103—681no—
409#102—41no—
410#102—681no—
411#102C4×D1714yes—
412C4×C8—174no—
413C4×C8—342no—
414C4×C8—342no—
415#95—21no—
416#95—171no—
417#95—171no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429C17#2280611yescharacteristic, complemented, Sylow
430C24—21no—
431C2×Q8—172no—
432C2×Q8—341no—
433C2×Q8—341no—
434C2×Q8—341no—
435C2×Q8—341no—
436C2×Q8—341no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D1712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—344no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—1361no—
456C22×C4—1361no—
457C22×C4D4×D1711yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460SD16—348no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—172no—
467C2×C8—172no—
468C2×C8—172no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8C2×C4×D1712yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—172no—
479C4⋊C4—342no—
480C4⋊C4—344no—
481C4⋊C4—341no—
482C4⋊C4—342no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4C2×C4×D1712yes—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C22⋊C4—344no—
491C22⋊C4—682no—
492C22⋊C4—1361no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—172no—
497C42—172no—
498C42—344no—
499C42—341no—
500C42—341no—
501C42—682no—
502C42—1361no—
503C42—1361no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D1711yescharacteristic
512Q8—342no—
513Q8—342no—
514Q8—681no—
515Q8—681no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—172no—
535C2×C4—172no—
536C2×C4—172no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—342no—
545C2×C4—342no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—1364no—
559C2×C4#1058759812yes—
560C2×C4C22×C4×D1711yescharacteristic
561C2×C4#1058769111yescharacteristic
562C2×C4#41×D1711yescharacteristic
563C2×C4#41×D1711yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C2×C4#1058769711yescharacteristic
566C8—22no—
567C8—41no—
568C8—174no—
569C8—681no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D1711yescharacteristic
578C22#1176345711yescharacteristic
579C22#1176347711yescharacteristic
580C22#135×D1711yescharacteristic
581C22C2×#1058759811yescharacteristic
582C22#1176361811yescharacteristic
583C22#1176361911yescharacteristic
584C4—22no—
585C4—42no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—342no—
590C4—342no—
591C4—342no—
592C4—681no—
593C4—682no—
594C4—682no—
595C4—1361no—
596C4—1361no—
597C4#117×D1711yescharacteristic
598C4#1176328811yescharacteristic
599C4#1176352111yescharacteristic
600C4#1176352611yescharacteristic
601C2—82no—
602C2#1453173011yescharacteristic
603C2C4×#1058759811yescharacteristic
604C2#1453764011yescharacteristic
605C2#1454115611yescharacteristic
606C2#1454116711yescharacteristic
607C2#1454177511yescharacteristic
608C2#1454178411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1925C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1925D17
hypocenterC17#22806
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
3. #14531745C2semidirect
6. #14529426C2semidirect
7. #14529424C2semidirect
9. #14528947C2semidirect
10. #14528798C2semidirect
12. C4×#10587479C2semidirect
13. #14528672C2semidirect
14. #14528645C2semidirect
19. #11762900C4semidirect
429. C17#22806semidirect

Maximal subgroups

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

subgroupindex
C2×#117629002
#145317562
#145317452
#145296392
#145296372
#145294262
#145294242
#145294062
#145289472
#145287982
#145287922
C4×#105874792
#145286722
#145286452
C17×#19252
#2280617