Finite Group Databasebeta

← #17036727

Subgroups — #17036727

Order 4352.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762852C211yesmaximal, characteristic, complemented
2C2×#11762811C211yesmaximal, characteristic
3C2×#11762808C211yesmaximal, characteristic
4#14531272C211yesmaximal, characteristic
5C2×C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
6#14529423C212yesmaximal, complemented
7#14529416C212yesmaximal, complemented
8#14529409C212yesmaximal, complemented
9#14529402C212yesmaximal, complemented
10#14528477C211yesmaximal, characteristic, complemented
11C17×#1736C211yesmaximal, characteristic, complemented
12C22×#63252C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4×Dic17—21no—
16C22×C4×Dic17C2211yescharacteristic
17#11762852—24no—
18C2×#10587478C2212yes—
19C2×#10587479C2212yes—
20#11762811—24no—
21#11762808C418yescomplemented
22C2×#10587487C2212yes—
23C2×#10587481C2212yes—
24C2×#10587492C2212yes—
25C2×#10587480C2212yes—
26#11762714C2211yescharacteristic
27#11762714C2211yescharacteristic
28#11762710C2211yescharacteristic
29#11762710C2211yescharacteristic
30#11762686—21no—
31#11762686—21no—
32C4⋊C4×Dic17—21no—
33C4⋊C4×Dic17—21no—
34C4⋊C4×Dic17C2212yes—
35C4⋊C4×Dic17C2212yes—
36C4×#63227—21no—
37C4×#63227—21no—
38C2×#10587502—21no—
39#11762563C2212yes—
40#11762563C2212yes—
41C22×M16×C17C2211yescharacteristic
42C2×C17×#117C2211yescharacteristic
43C22×C4⋊C4×C17C2211yescharacteristic
44C17×#340C2212yes—
45C17×#340C2212yes—
46C23×C4×C17—21no—
47C23×C4×C17C2311yescharacteristic
48C22×C22⋊D17—21no—
49C23×Dic17—21no—
50C23×Dic17D411yescharacteristic
51C23×Dic17D411yescharacteristic
52C2×#63252—22no—
53C2×#63252—22no—
54C2×#63252—22no—
55C2×#63252—22no—
56C2×#63252—22no—
57C2×#63252C2×C414yes—
58C22×D68C2311yescharacteristic
59C22×C4×D17—21no—
60C22×Dic34C2311yescharacteristic
61C2×M16×C17—22no—
62C2×M16×C17—22no—
63C2×M16×C17—22no—
64C22×C8×C17C2312yes—
65C17×#117—22no—
66C17×#117—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17C2×C414yes—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2312yes—
73C2×C22⋊C4×C17—21no—
74C2×C42×C17—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17D414yes—
82C2×C4⋊Dic17C2312yes—
83C2×C4×Dic17—22no—
84C2×C4×Dic17—22no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—22no—
87C2×C4×Dic17—44no—
88C2×C4×Dic17D411yescharacteristic
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223C2×C414yes—
95C22×#3622C2312yes—
96#10587502—21no—
97#10587502—21no—
98#10587502—21no—
99#10587502—21no—
100#10587502—22no—
101#10587502—22no—
102#10587492—24no—
103#10587487—24no—
104#10587481C2×C418yescomplemented
105#10587480C2×C418yescomplemented
106#10587479—24no—
107#10587478—24no—
108C24×C17C2×D411yescharacteristic
109C23×D17—21no—
110C22×C4×C17—24no—
111C22×C4×C17—22no—
112C22×C4×C17—21no—
113C22×C4×C17—21no—
114C22×C4×C17—42no—
115C22×C4×C17—42no—
116C22×C4×C17C22×C412yes—
117C22×C4×C17C2×D411yescharacteristic
118C22×C4×C17C2×D412yes—
119C22×C4×C17#4112yes—
120C22×C4×C17#4112yes—
121C22×C4×C17#4112yes—
122C22×C4×C17#4112yes—
123C22×C4×C17C2411yescharacteristic
124C2×C22⋊D17—24no—
125C2×C22⋊D17—42no—
126C2×C22⋊D17—42no—
127C22×Dic17—22no—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—24no—
131C22×Dic17—22no—
132C22×Dic17—21no—
133C22×Dic17—22no—
134C22×Dic17—41no—
135C22×Dic17—41no—
136C22×Dic17—42no—
137C22×Dic17—42no—
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142#63252—28no—
143#63252—28no—
144#63252—44no—
145#63252—44no—
146C2×D68—21no—
147C2×D68—21no—
148C2×D68—21no—
149C2×D68—21no—
150C2×D68—21no—
151C2×D68C22×C412yes—
152C2×C4×D17—24no—
153C2×C4×D17—42no—
154C2×C4×D17—42no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34C22×C412yes—
161M16×C17—44no—
162C2×C8×C17—22no—
163C2×C8×C17—22no—
164C2×C8×C17—22no—
165C4⋊C4×C17—24no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17C22×C414yes—
172C22⋊C4×C17—42no—
173C42×C17—42no—
174#63227—42no—
175#63227—42no—
176#63227—42no—
177#63227—42no—
178C4⋊Dic17—24no—
179C4⋊Dic17—22no—
180C4⋊Dic17C2×D414yes—
181C4×Dic17—22no—
182C4×Dic17—22no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—44no—
188C4×Dic17—44no—
189#63223—28no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622C22×C414yes—
193#21059—171nomaximal, Sylow
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17C4×D412yes—
199C23×C17#12112yes—
200C23×C17#12312yes—
201C23×C17C22×D411yescharacteristic
202C22×D17—22no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209C2×C4×C17—24no—
210C2×C4×C17—48no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17#11714yes—
220C2×C4×C17C4×D412yes—
221C2×C4×C17C4×D414yes—
222C2×C4×C17#12112yes—
223C2×C4×C17#12312yes—
224C2×C4×C17#12414yes—
225C2×C4×C17#12614yes—
226C2×C4×C17C23×C411yescharacteristic
227C2×C4×C17C22×D411yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C22⋊D17—48no—
233C2×Dic17—28no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—22no—
239C2×Dic17—24no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—44no—
247C2×Dic17—48no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17AGL(1,ℤ/8)12yes—
253C2×Dic17#13712yes—
254D68—22no—
255D68—22no—
256D68—41no—
257D68—41no—
258C4×D17—48no—
259Dic34—22no—
260Dic34—22no—
261Dic34—41no—
262Dic34—41no—
263C8×C17—42no—
264#3622—24no—
265C2×#428—171no—
266C2×#417—171no—
267C2×#417—171no—
268#2297—171no—
269C2×C4×C4⋊C4—171no—
270#1928—172no—
271#1927—172no—
272#1926—172no—
273#1925—172no—
274#1736D1711yescharacteristic, complemented
275#1728—171no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17#38814yes—
283C22×C17C2×C4×D411yescharacteristic
284C22×C17C2×#12111yescharacteristic
285C22×C17C2×#12311yescharacteristic
286C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C17C2×#13711yescharacteristic
288C22×C17#57511yescharacteristic
289C22×C17#57511yescharacteristic
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—42no—
293C2×D17—81no—
294C2×D17—81no—
295C2×D17—81no—
296C2×D17—81no—
297C4×C17—44no—
298C4×C17—81no—
299C4×C17—81no—
300C4×C17#38814yes—
301C4×C17C2×#11711yescharacteristic
302C4×C17C2×C4×D411yescharacteristic
303C4×C17C2×#12411yescharacteristic
304C4×C17C2×#12611yescharacteristic
305Dic17—22no—
306Dic17—22no—
307Dic17—41no—
308Dic17—41no—
309Dic17—44no—
310Dic17—42no—
311C22×#41—171no—
312C22×M16—171no—
313C22×M16C2×D1711yescharacteristic
314C2×#117C2×D1711yescharacteristic
315C22×C4⋊C4—171no—
316C22×C4⋊C4C2×D1711yescharacteristic
317C22×C42—171no—
318C22×C42—341no—
319#428—344no—
320C2×#107—172no—
321C2×#106—172no—
322#417—178no—
323#417—344no—
324C2×#103—172no—
325C2×#103—172no—
326C2×#102—172no—
327C2×#102—172no—
328#391—171no—
329#391—171no—
330#390—171no—
331#390—171no—
332#386—341no—
333#386—341no—
334C4×C4⋊C4—172no—
335C4×C4⋊C4—172no—
336C4×C4⋊C4—341no—
337C4×C4⋊C4—341no—
338C4×C22⋊C4—341no—
339C4×C22⋊C4—341no—
340C2×#95—341no—
341#340C2×D1712yes—
342#340C2×D1712yes—
343#336—172no—
344#336—172no—
345C2×C17—24no—
346C2×C17C2×#38811yescharacteristic
347C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
348C2×C17C4×#13711yescharacteristic
349C2×C17#303611yescharacteristic
350C2×C17#303711yescharacteristic
351C2×C17#309211yescharacteristic
352C2×C17#309311yescharacteristic
353D17—82no—
354C2×#41—174no—
355C2×#41—342no—
356C2×#41—342no—
357C2×#41—342no—
358C2×#41—342no—
359C2×#41—342no—
360C22×Q8—171no—
361C22×D4—171no—
362C22×D4—341no—
363C23×C4—21no—
364C23×C4—171no—
365C23×C4—171no—
366C23×C4—341no—
367C23×C4—341no—
368C23×C4C22×D1711yescharacteristic
369C2×M16—22no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—174no—
373C2×M16—342no—
374C2×M16—342no—
375C22×C8—172no—
376C22×C8C22×D1712yes—
377#117—22no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—174no—
382C2×C4⋊C4—172no—
383C2×C4⋊C4—342no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4C4×D1714yes—
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C4⋊C4C22×D1712yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C42—21no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—342no—
400C2×C42—342no—
401C2×C42—342no—
402C2×C42—342no—
403C2×C42—684no—
404#107—344no—
405#106—344no—
406#103—178no—
407#103—344no—
408#102—178no—
409#102—344no—
410#95—342no—
411#95—342no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416C17#2105911yescharacteristic, complemented, Sylow
417C24—341no—
418C24D4×D1711yescharacteristic
419#41—348no—
420#41—348no—
421#41—684no—
422#41—684no—
423C2×Q8—172no—
424C2×Q8—341no—
425C2×Q8—341no—
426C2×Q8—341no—
427C2×Q8—341no—
428C2×Q8—341no—
429C2×D4—172no—
430C2×D4—344no—
431C2×D4—341no—
432C2×D4—341no—
433C2×D4—341no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—682no—
437C2×D4—682no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—341no—
452C22×C4—342no—
453C22×C4—344no—
454C22×C4—342no—
455C22×C4—344no—
456C22×C4—681no—
457C22×C4—681no—
458C22×C4—682no—
459C22×C4—682no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4C2×C4×D1712yes—
463C22×C4#6325212yes—
464C22×C4#6325212yes—
465C22×C4D4×D1712yes—
466C22×C4D4×D1711yescharacteristic
467C22×C4#6325412yes—
468C22×C4#6325612yes—
469C22×C4C23×D1711yescharacteristic
470M16—44no—
471M16—348no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—174no—
476C2×C8—342no—
477C2×C8—342no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—174no—
485C4⋊C4—342no—
486C4⋊C4—344no—
487C4⋊C4C2×C4×D1714yes—
488C22⋊C4—42no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C22⋊C4—682no—
493C42—42no—
494C42—341no—
495C42—341no—
496C42—341no—
497C42—341no—
498C42—342no—
499C42—342no—
500C42—684no—
501C42—684no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—342no—
507C23—681no—
508C23—681no—
509C23—681no—
510C23—681no—
511C23—681no—
512C23—681no—
513C23#1058756712yes—
514C23#1058757012yes—
515C23#1058758012yes—
516C23C2×D4×D1711yescharacteristic
517Q8—342no—
518Q8—342no—
519Q8—681no—
520Q8—681no—
521D4—342no—
522D4—342no—
523D4—688no—
524D4—681no—
525D4—681no—
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—172no—
537C2×C4—172no—
538C2×C4—348no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—342no—
544C2×C4—344no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—688no—
552C2×C4—684no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—682no—
556C2×C4—682no—
557C2×C4—688no—
558C2×C4#1058755814yes—
559C2×C4#1058756712yes—
560C2×C4#1058757012yes—
561C2×C4#1058757114yes—
562C2×C4#1058757914yes—
563C2×C4#1058758012yes—
564C2×C4#1058758414yes—
565C2×C4C22×C4×D1711yescharacteristic
566C2×C4C2×#6325211yescharacteristic
567C2×C4C2×#6325211yescharacteristic
568C2×C4C2×D4×D1711yescharacteristic
569C2×C4C2×#6325411yescharacteristic
570C2×C4C2×#6325611yescharacteristic
571C8—42no—
572C8—344no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—682no—
580C22—682no—
581C22—682no—
582C22—1361no—
583C22—1361no—
584C22—1361no—
585C22—1361no—
586C22#1176269814yes—
587C22C2×#1058756711yescharacteristic
588C22C2×#1058757011yescharacteristic
589C22C2×#1058758011yescharacteristic
590C22AGL(1,ℤ/8)×D1711yescharacteristic
591C22#137×D1711yescharacteristic
592C22#1176360511yescharacteristic
593C22#1176360811yescharacteristic
594C4—44no—
595C4—81no—
596C4—81no—
597C4—342no—
598C4—342no—
599C4—684no—
600C4—682no—
601C4—681no—
602C4—681no—
603C4#1176269814yes—
604C4C2×#1058755811yescharacteristic
605C4C2×#1058757911yescharacteristic
606C4C2×#1058757111yescharacteristic
607C4C2×#1058758411yescharacteristic
608C2—24no—
609C2—1362no—
610C2C2×#1176269811yescharacteristic
611C2#1453758811yescharacteristic
612C2#1453759911yescharacteristic
613C2#1453824711yescharacteristic
614C2#1453825511yescharacteristic
615C2#1453871011yescharacteristic
616C2#1453871811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1736C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#1736D17
hypocenterC17#21059
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
1. C2×#11762852C2semidirect
5. C2×C4⋊C4×Dic17C2semidirect
6. #14529423C2semidirect
7. #14529416C2semidirect
8. #14529409C2semidirect
9. #14529402C2semidirect
10. #14528477C2semidirect
11. C17×#1736C2semidirect
21. #11762808C4semidirect
104. #10587481C2×C4semidirect
105. #10587480C2×C4semidirect
274. #1736D17semidirect
416. C17#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#117628522
C2×#117628112
C2×#117628082
#145312722
C2×C4⋊C4×Dic172
#145294232
#145294162
#145294092
#145294022
#145284772
C17×#17362
#2105917