Finite Group Databasebeta

← #17063256

Subgroups — #17063256

Order 4352.

892 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537347C211yesmaximal, characteristic, complemented
2C2×#11762825C211yesmaximal, characteristic
3#14537304C211yesmaximal, characteristic
4C2×#11762791C211yesmaximal, characteristic
5#14530240C218yesmaximal, complemented
6C2×#11762580C211yesmaximal, characteristic, complemented
7C2×#11762566C211yesmaximal, characteristic, complemented
8C17×#1738C211yesmaximal, characteristic, complemented
9C22×#63252C2211yescharacteristic
10C22×#63223C2211yescharacteristic
11C2×#10587612C2211yescharacteristic
12C2×#10587612C2211yescharacteristic
13#11762837—22no—
14#11762835C2214yescomplemented
15#11762832—22no—
16#11762825—22no—
17#11762825C2214yes—
18C2×#10587496—22no—
19#11762819—21no—
20#11762815—21no—
21#11762811—22no—
22#11762806C2214yes—
23#11762791—22no—
24#11762791C2214yes—
25C2×#10587495—21no—
26C2×#10587494—21no—
27#11762580C2214yescomplemented
28#11762566C2214yescomplemented
29C22×M16×C17C2211yescharacteristic
30C22×M16×C17C2211yescharacteristic
31C22×M16×C17C2211yescharacteristic
32C17×#341C2214yescomplemented
33C23×C4×C17C2311yescharacteristic
34C22×C22⋊D17—21no—
35C2×#63252—21no—
36C2×#63252—21no—
37C2×#63252—21no—
38C2×#63252—21no—
39C2×#63252—21no—
40C2×#63252—21no—
41C2×#63252—21no—
42C2×#63252—21no—
43C2×#63252—21no—
44C2×#63252—21no—
45C2×#63252D412yes—
46C2×#63252C2312yes—
47C22×D68D411yescharacteristic
48C22×C4×D17—21no—
49C22×Dic34D411yescharacteristic
50C2×M16×C17—21no—
51C2×M16×C17—21no—
52C2×M16×C17—21no—
53C2×M16×C17—21no—
54C2×M16×C17—21no—
55C2×M16×C17—21no—
56C2×M16×C17—21no—
57C2×M16×C17—21no—
58C2×M16×C17—21no—
59C2×M16×C17—21no—
60C2×M16×C17—21no—
61C2×M16×C17—21no—
62C2×M16×C17—21no—
63C2×M16×C17—21no—
64C2×M16×C17—21no—
65C2×M16×C17—21no—
66C2×M16×C17C2312yes—
67C2×M16×C17C2312yes—
68C22×C8×C17—21no—
69C22×C8×C17—21no—
70C22×C8×C17C2312yes—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73#10587612—21no—
74#10587612—21no—
75#10587612—21no—
76#10587612—21no—
77#10587612—21no—
78#10587612D412yes—
79#10587612C2312yes—
80#10587612C2312yes—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17D412yes—
85C2×#63223—21no—
86C2×#63223—21no—
87C2×#63223—21no—
88C2×#63223—21no—
89C2×#63223—21no—
90C2×#63223C2312yes—
91C22×#3622—21no—
92#10587496—28no—
93#10587495—24no—
94#10587494—24no—
95#10587492—24no—
96#10587487—24no—
97#10587485—28no—
98C24×C17#4111yescharacteristic
99C23×D17—21no—
100C22×C4×C17C2×D411yescharacteristic
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17C2×D411yescharacteristic
104C22×C4×C17#4111yescharacteristic
105C22×C4×C17#4111yescharacteristic
106C22×C4×C17#4111yescharacteristic
107C22×C4×C17#4111yescharacteristic
108C22×C4×C17#4111yescharacteristic
109C22×C4×C17#4111yescharacteristic
110C22×C4×C17#4111yescharacteristic
111C22×C4×C17#4111yescharacteristic
112C22×C4×C17#4111yescharacteristic
113C22×C4×C17C2411yescharacteristic
114C2×C22⋊D17—22no—
115C2×C22⋊D17—22no—
116C2×C22⋊D17—41no—
117C2×C22⋊D17—41no—
118C2×C22⋊D17—41no—
119C2×C22⋊D17—41no—
120C22×Dic17—21no—
121C22×Dic17—22no—
122C22×Dic17—41no—
123#63252—22no—
124#63252—22no—
125#63252—22no—
126#63252—22no—
127#63252—22no—
128#63252—22no—
129#63252—22no—
130#63252—22no—
131#63252—41no—
132#63252—41no—
133#63252—41no—
134#63252—41no—
135#63252—41no—
136#63252—41no—
137#63252—41no—
138#63252—41no—
139C2×D68—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68C2×D412yes—
145C2×C4×D17—22no—
146C2×C4×D17—22no—
147C2×C4×D17—41no—
148C2×C4×D17—41no—
149C2×C4×D17—41no—
150C2×C4×D17—41no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34C2×D412yes—
157M16×C17—22no—
158M16×C17—22no—
159M16×C17—22no—
160M16×C17—22no—
161M16×C17—41no—
162M16×C17—41no—
163M16×C17—41no—
164M16×C17—41no—
165M16×C17—41no—
166M16×C17—41no—
167M16×C17—41no—
168M16×C17—41no—
169C2×C8×C17—22no—
170C2×C8×C17—22no—
171C2×C8×C17—22no—
172C2×C8×C17—22no—
173C2×C8×C17—22no—
174C2×C8×C17—41no—
175C2×C8×C17—41no—
176C2×C8×C17—41no—
177C2×C8×C17—41no—
178C22⋊Dic17—22no—
179C22⋊Dic17—22no—
180C22⋊Dic17—22no—
181C22⋊Dic17—41no—
182C4⋊Dic17—22no—
183C4⋊Dic17—22no—
184C4⋊Dic17—22no—
185C4⋊Dic17—41no—
186C4×Dic17—22no—
187C4×Dic17—22no—
188C4×Dic17—41no—
189C4×Dic17C2×D414yes—
190#63223—22no—
191#63223—22no—
192#63223—41no—
193#63223—41no—
194C2×#3622—22no—
195C2×#3622—41no—
196C2×#3622—41no—
197#23649—171nomaximal, Sylow
198C23×C17—81no—
199C23×C17#12311yescharacteristic
200C23×C17#12311yescharacteristic
201C23×C17#12311yescharacteristic
202C23×C17#12311yescharacteristic
203C23×C17#12611yescharacteristic
204C23×C17#12611yescharacteristic
205C23×C17C2×#4111yescharacteristic
206C22×D17—22no—
207C22×D17—41no—
208C22×D17—41no—
209C22×D17—82no—
210C2×C4×C17#12111yescharacteristic
211C2×C4×C17#12111yescharacteristic
212C2×C4×C17#12111yescharacteristic
213C2×C4×C17#12111yescharacteristic
214C2×C4×C17#12311yescharacteristic
215C2×C4×C17#12311yescharacteristic
216C2×C4×C17#12311yescharacteristic
217C2×C4×C17#12311yescharacteristic
218C2×C4×C17#12311yescharacteristic
219C2×C4×C17#12311yescharacteristic
220C2×C4×C17#12311yescharacteristic
221C2×C4×C17#12311yescharacteristic
222C2×C4×C17#12411yescharacteristic
223C2×C4×C17#12411yescharacteristic
224C2×C4×C17#12411yescharacteristic
225C2×C4×C17#12411yescharacteristic
226C2×C4×C17#12611yescharacteristic
227C2×C4×C17#12611yescharacteristic
228C2×C4×C17#12611yescharacteristic
229C2×C4×C17#12611yescharacteristic
230C2×C4×C17#12611yescharacteristic
231C2×C4×C17#12611yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C22×D411yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C22⋊D17—42no—
239C22⋊D17—42no—
240C22⋊D17—42no—
241C22⋊D17—42no—
242C2×Dic17—24no—
243C2×Dic17—22no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—42no—
247C2×Dic17—82no—
248C2×Dic17—82no—
249D68—22no—
250D68—22no—
251D68—41no—
252D68—41no—
253C4×D17—42no—
254C4×D17—42no—
255C4×D17—42no—
256C4×D17—42no—
257Dic34—22no—
258Dic34—22no—
259Dic34—41no—
260Dic34—41no—
261C8×C17—42no—
262C8×C17—42no—
263C8×C17—42no—
264#3622—42no—
265#2905—171no—
266C2×#421—171no—
267#2896—171no—
268C2×#413—171no—
269#2080—178no—
270C2×#344—171no—
271#1738D1711yescharacteristic, complemented
272C2×H(3,ℤ/4)—171no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17#39711yescharacteristic
281C22×C17#39711yescharacteristic
282C22×C17#39711yescharacteristic
283C22×C17#39711yescharacteristic
284C22×C17C2×#12311yescharacteristic
285C22×C17C2×#12311yescharacteristic
286C22×C17C2×#12611yescharacteristic
287C2×D17—42no—
288C2×D17—82no—
289C2×D17—82no—
290C2×D17—82no—
291C4×C17#39711yescharacteristic
292C4×C17#39711yescharacteristic
293C4×C17#39711yescharacteristic
294C4×C17#39711yescharacteristic
295C4×C17C2×#12111yescharacteristic
296C4×C17C2×#12311yescharacteristic
297C4×C17C2×#12411yescharacteristic
298C4×C17C2×#12611yescharacteristic
299Dic17—44no—
300Dic17—42no—
301Dic17—82no—
302C22×#41—171no—
303C22×M16—171no—
304C22×M16C2×D1711yescharacteristic
305C22×M16C2×D1711yescharacteristic
306C22×M16C2×D1711yescharacteristic
307C2×#117—171no—
308C2×#117—171no—
309#424—174no—
310#424—342no—
311#423—342no—
312#421—174no—
313#421—342no—
314C2×#104—342no—
315#419—341no—
316#418—341no—
317#417—342no—
318#416—174no—
319#413—174no—
320#413—342no—
321C2×#101—341no—
322C2×#100—341no—
323#344—174no—
324#341C2×D1714yescomplemented
325H(3,ℤ/4)—174no—
326C2×C17—21no—
327C2×C17—21no—
328C2×C17—21no—
329C2×C17—21no—
330C2×C17—21no—
331C2×C17—21no—
332C2×C17—21no—
333C2×C17C2×#39711yescharacteristic
334D17—82no—
335C2×#41—172no—
336C2×#41—172no—
337C2×#41—341no—
338C2×#41—341no—
339C2×#41—341no—
340C2×#41—341no—
341C2×#41—341no—
342C2×#41—341no—
343C2×#41—341no—
344C2×#41—341no—
345C2×#41—341no—
346C2×#41—341no—
347C22×Q8—171no—
348C22×D4—171no—
349C22×D4—341no—
350C23×C4—341no—
351C23×C4C22×D1711yescharacteristic
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—172no—
369C2×M16—341no—
370C2×M16—341no—
371C2×M16—341no—
372C2×M16—341no—
373C2×M16—341no—
374C2×M16C22×D1712yes—
375C2×M16C22×D1712yes—
376C22×C8—21no—
377C22×C8—21no—
378C22×C8—341no—
379C22×C8C22×D1712yes—
380#117—172no—
381#117—172no—
382#117—172no—
383#117—341no—
384#117—341no—
385#117—341no—
386#117—341no—
387#117—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C42—172no—
393C2×C42—341no—
394C4⋊C8—348no—
395#104—348no—
396#103—344no—
397#102—344no—
398#101—344no—
399#100—344no—
400C17#2364911yescharacteristic, complemented, Sylow
401C24—341no—
402C24#6325611yescharacteristic
403#41—342no—
404#41—342no—
405#41—342no—
406#41—342no—
407#41—342no—
408#41—342no—
409#41—342no—
410#41—342no—
411#41—681no—
412#41—681no—
413#41—681no—
414#41—681no—
415#41—681no—
416#41—681no—
417#41—681no—
418#41—681no—
419C2×Q8—172no—
420C2×Q8—341no—
421C2×Q8—341no—
422C2×Q8—341no—
423C2×Q8—341no—
424C2×Q8—341no—
425C2×D4—172no—
426C2×D4—342no—
427C2×D4—342no—
428C2×D4—341no—
429C2×D4—341no—
430C2×D4—341no—
431C2×D4—341no—
432C2×D4—341no—
433C2×D4—681no—
434C2×D4—681no—
435C2×D4—681no—
436C2×D4—681no—
437C22×C4—341no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—681no—
442C22×C4—681no—
443C22×C4—681no—
444C22×C4—681no—
445C22×C4—681no—
446C22×C4#6325211yescharacteristic
447C22×C4#6325211yescharacteristic
448C22×C4#6325211yescharacteristic
449C22×C4#6325211yescharacteristic
450C22×C4#6325211yescharacteristic
451C22×C4#6325211yescharacteristic
452C22×C4D4×D1711yescharacteristic
453C22×C4D4×D1711yescharacteristic
454C22×C4D4×D1711yescharacteristic
455C22×C4D4×D1711yescharacteristic
456C22×C4#6325411yescharacteristic
457C22×C4#6325411yescharacteristic
458C22×C4#6325611yescharacteristic
459C22×C4C23×D1711yescharacteristic
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—342no—
473M16—342no—
474M16—681no—
475M16—681no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—342no—
486C2×C8—681no—
487C2×C8—681no—
488C4⋊C4—342no—
489C4⋊C4—342no—
490C4⋊C4—342no—
491C4⋊C4—681no—
492C22⋊C4—342no—
493C22⋊C4—342no—
494C22⋊C4—342no—
495C22⋊C4—681no—
496C42—174no—
497C42—342no—
498C42—342no—
499C42—681no—
500C23—81no—
501C23—342no—
502C23—681no—
503C23—681no—
504C23—1362no—
505C23#1058758011yescharacteristic
506C23#1058758011yescharacteristic
507C23#1058758011yescharacteristic
508C23#1058758011yescharacteristic
509C23#1058758411yescharacteristic
510C23#1058758411yescharacteristic
511C23C2×#6325611yescharacteristic
512Q8—342no—
513Q8—342no—
514Q8—681no—
515Q8—681no—
516D4—342no—
517D4—342no—
518D4—682no—
519D4—682no—
520D4—682no—
521D4—682no—
522D4—681no—
523D4—681no—
524C2×C4—344no—
525C2×C4—342no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—682no—
529C2×C4—682no—
530C2×C4—682no—
531C2×C4—682no—
532C2×C4—682no—
533C2×C4—1362no—
534C2×C4—1362no—
535C2×C4#1058756211yescharacteristic
536C2×C4#1058756211yescharacteristic
537C2×C4#1058756211yescharacteristic
538C2×C4#1058756211yescharacteristic
539C2×C4#1058756911yescharacteristic
540C2×C4#1058756911yescharacteristic
541C2×C4#1058756911yescharacteristic
542C2×C4#1058756911yescharacteristic
543C2×C4#1058757011yescharacteristic
544C2×C4#1058757011yescharacteristic
545C2×C4#1058757011yescharacteristic
546C2×C4#1058757011yescharacteristic
547C2×C4#1058757111yescharacteristic
548C2×C4#1058757111yescharacteristic
549C2×C4#1058757111yescharacteristic
550C2×C4#1058757111yescharacteristic
551C2×C4#1058758011yescharacteristic
552C2×C4#1058758011yescharacteristic
553C2×C4#1058758011yescharacteristic
554C2×C4#1058758011yescharacteristic
555C2×C4#1058758411yescharacteristic
556C2×C4#1058758411yescharacteristic
557C2×C4C2×#6325211yescharacteristic
558C2×C4C2×#6325211yescharacteristic
559C2×C4C2×#6325211yescharacteristic
560C2×C4C2×D4×D1711yescharacteristic
561C2×C4C2×D4×D1711yescharacteristic
562C2×C4C2×#6325411yescharacteristic
563C8—42no—
564C8—42no—
565C8—42no—
566C8—682no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—682no—
575C22—1362no—
576C22—1362no—
577C22—1362no—
578C22#1176273811yescharacteristic
579C22#1176273811yescharacteristic
580C22#1176273811yescharacteristic
581C22#1176273811yescharacteristic
582C22C2×#1058758011yescharacteristic
583C22C2×#1058758011yescharacteristic
584C22C2×#1058758411yescharacteristic
585C4—684no—
586C4—682no—
587C4—1362no—
588C4#1176273811yescharacteristic
589C4#1176273811yescharacteristic
590C4#1176273811yescharacteristic
591C4#1176273811yescharacteristic
592C4C2×#1058757011yescharacteristic
593C4C2×#1058756911yescharacteristic
594C4C2×#1058757111yescharacteristic
595C4C2×#1058756211yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—1362no—
604C2C2×#1176273811yescharacteristic
subgroupG/H
centerC4C2×#10587571
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1738C2
socleC2×C17C2×#397
prime radicalC22×#63252C22
hypercenter#1738D17
hypocenterC17#23649
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
1. #14537347C2semidirect
5. #14530240C2semidirect
6. C2×#11762580C2semidirect
7. C2×#11762566C2semidirect
8. C17×#1738C2semidirect
14. #11762835C22semidirect
27. #11762580C22semidirect
28. #11762566C22semidirect
32. C17×#341C22semidirect
271. #1738D17semidirect
324. #341C2×D17semidirect
400. C17#23649semidirect

Maximal subgroups

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

subgroupindex
#145373472
C2×#117628252
#145373042
C2×#117627912
#145302402
C2×#117625802
C2×#117625662
C17×#17382
#2364917