Finite Group Databasebeta

← #17005141

Subgroups — #17005141

Order 4352.

606 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537933C211yesmaximal, characteristic, complemented
2#14537860C211yesmaximal, characteristic, complemented
3#14527663C211yesmaximal, characteristic, complemented
4#14527649C211yesmaximal, characteristic, complemented
5#14527649C211yesmaximal, characteristic, complemented
6#14527642C211yesmaximal, characteristic, complemented
7#14527628C211yesmaximal, characteristic, complemented
8#14527621C211yesmaximal, characteristic, complemented
9#14527614C211yesmaximal, characteristic, complemented
10#14527614C211yesmaximal, characteristic, complemented
11#14527551C211yesmaximal, characteristic, complemented
12#14527244C211yesmaximal, characteristic, complemented
13#14527235C211yesmaximal, characteristic, complemented
14#14527071C211yesmaximal, characteristic, complemented
15C17×#3171C211yesmaximal, characteristic
16#11762940C2211yescharacteristic
17Q8×#3622C2211yescharacteristic
18#11762887—21no—
19#11762887C2211yescharacteristic
20#11762887C2211yescharacteristic
21#11762887C2211yescharacteristic
22#11762881—21no—
23#11762881—21no—
24#11762881—21no—
25D4×#3622C2211yescharacteristic
26D4×#3622C2211yescharacteristic
27D4×#3622C2211yescharacteristic
28#11762870—21no—
29#11762866C2211yescharacteristic
30#11762862C2211yescharacteristic
31#11762836—21no—
32#11762833C2211yescharacteristic
33#11762776—21no—
34#11762776—21no—
35C2×C4×#3622C2211yescharacteristic
36#11762562C2211yescharacteristic
37#11762562C2211yescharacteristic
38#11762560C2211yescharacteristic
39#11762560C2211yescharacteristic
40Q8⋊#3622C2211yescharacteristic
41Q8⋊#3622C2211yescharacteristic
42D4⋊#3622C2211yescharacteristic
43D4⋊#3622C2211yescharacteristic
44D4⋊#3622C2211yescharacteristic
45D4⋊#3622C2211yescharacteristic
46D4⋊#3622C2211yescharacteristic
47D4⋊#3622C2211yescharacteristic
48#11762531C2211yescharacteristic
49#11762531C2211yescharacteristic
50#11762531C2211yescharacteristic
51#11762531C2211yescharacteristic
52C17×#532C2211yescharacteristic
53C17×#529C2211yescharacteristic
54C17×#501C411yescharacteristic
55C17×#496C411yescharacteristic
56C17×#490C411yescharacteristic
57C17×#489C411yescharacteristic
58C17×#469C411yescharacteristic
59C17×#469C411yescharacteristic
60C17×#468C411yescharacteristic
61C17×#468C411yescharacteristic
62C4×SD16×C17C2211yescharacteristic
63C4×SD16×C17C2211yescharacteristic
64C4×D8×C17C2211yescharacteristic
65C4×D8×C17C2211yescharacteristic
66C4×M16×C17C2211yescharacteristic
67C2×#41×C17—21no—
68C2×SD16×C17C2×C411yescharacteristic
69C2×SD16×C17C2×C411yescharacteristic
70C2×D8×C17C2×C411yescharacteristic
71C2×D8×C17C2×C411yescharacteristic
72C2×M16×C17C2×C411yescharacteristic
73C2×M16×C17C2×C411yescharacteristic
74C17×C4⋊Q8C2×C411yescharacteristic
75C17×#127C2×C411yescharacteristic
76C17×#126—21no—
77C17×#125C2×C411yescharacteristic
78C17×#124—21no—
79C17×#124C2×C411yescharacteristic
80C17×#123—21no—
81C17×C22⋊Q8C2×C411yescharacteristic
82C17×#121—21no—
83C17×#121C2×C411yescharacteristic
84C17×#121C2×C411yescharacteristic
85C17×#121C2×C411yescharacteristic
86C4×Q8×C17C2311yescharacteristic
87C4×D4×C17—21no—
88C4×D4×C17—21no—
89C4×D4×C17C2311yescharacteristic
90C4×D4×C17C2311yescharacteristic
91C4×D4×C17C2311yescharacteristic
92C17×#117—21no—
93C2×C42×C17C2311yescharacteristic
94C2×#63223—21no—
95C2×#63223—21no—
96C2×#63223—21no—
97C2×#63223—21no—
98C22×#3622—21no—
99C22×#3622—21no—
100C22×#3622—21no—
101C22×#3622—21no—
102C17×#107C2×C411yescharacteristic
103C17×#107C2×C411yescharacteristic
104C17×#106C2×C411yescharacteristic
105C17×#106C2×C411yescharacteristic
106C17×#103C2×C411yescharacteristic
107C17×#103C2×C411yescharacteristic
108C17×#102C2×C411yescharacteristic
109C17×#102C2×C411yescharacteristic
110C17×#102C2×C411yescharacteristic
111C17×#102C2×C411yescharacteristic
112C17×#102C2×C411yescharacteristic
113C17×#102C2×C411yescharacteristic
114C17×#97C2×C411yescharacteristic
115C17×#97C2×C411yescharacteristic
116C4×C8×C17C2311yescharacteristic
117C4×C8×C17C2311yescharacteristic
118C22⋊#3622—21no—
119C22⋊#3622—21no—
120C22⋊#3622—21no—
121C22⋊#3622—21no—
122C22⋊#3622—21no—
123C22⋊#3622—21no—
124C22⋊#3622—21no—
125C22⋊#3622D411yescharacteristic
126C22⋊#3622D411yescharacteristic
127C4⋊#3622—21no—
128C4⋊#3622—21no—
129C4⋊#3622—21no—
130C4⋊#3622D411yescharacteristic
131C4⋊#3622D411yescharacteristic
132C4⋊#3622C2311yescharacteristic
133C4⋊#3622C2311yescharacteristic
134C4⋊#3622C2311yescharacteristic
135C4⋊#3622C2311yescharacteristic
136#10587474—21no—
137#10587474—21no—
138C4×#3622—21no—
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144#41×C17—81no—
145#41×C17—81no—
146C2×Q8×C17—21no—
147C2×Q8×C17C22×C411yescharacteristic
148C2×D4×C17—21no—
149C2×D4×C17—21no—
150C2×D4×C17—21no—
151C2×D4×C17—21no—
152C2×D4×C17—21no—
153C2×D4×C17C22×C411yescharacteristic
154C2×D4×C17C22×C411yescharacteristic
155C2×D4×C17C22×C411yescharacteristic
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17C22×C411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17#4111yescharacteristic
162SD16×C17—41no—
163SD16×C17—41no—
164D8×C17—41no—
165D8×C17—41no—
166M16×C17—41no—
167M16×C17—41no—
168C2×C8×C17C22×C411yescharacteristic
169C2×C8×C17C22×C411yescharacteristic
170C2×C8×C17C22×C411yescharacteristic
171C2×C8×C17C22×C411yescharacteristic
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17C22×C411yescharacteristic
178C4⋊C4×C17C22×C411yescharacteristic
179C4⋊C4×C17C22×C411yescharacteristic
180C4⋊C4×C17C22×C411yescharacteristic
181C22⋊C4×C17—21no—
182C22⋊C4×C17—21no—
183C22⋊C4×C17—21no—
184C22⋊C4×C17—21no—
185C22⋊C4×C17—21no—
186C22⋊C4×C17—21no—
187C22⋊C4×C17—21no—
188C22⋊C4×C17—21no—
189C22⋊C4×C17—21no—
190C42×C17—21no—
191C42×C17C22×C411yescharacteristic
192C42×C17C2×D411yescharacteristic
193C42×C17#4111yescharacteristic
194C42×C17C2411yescharacteristic
195#63223—81no—
196#63223—81no—
197#63223—81no—
198#63223—81no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—42no—
206C2×#3622—81no—
207C2×#3622—81no—
208C2×#3622—81no—
209C2×#3622C2×D411yescharacteristic
210C2×#3622C2×D411yescharacteristic
211C2×#3622C2×D411yescharacteristic
212C2×#3622C2×D411yescharacteristic
213#17987—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17C4×D411yescharacteristic
218Q8×C17—21no—
219Q8×C17—21no—
220Q8×C17—81no—
221D4×C17—21no—
222D4×C17—21no—
223D4×C17—21no—
224D4×C17—21no—
225D4×C17—21no—
226D4×C17—21no—
227D4×C17—81no—
228D4×C17—81no—
229D4×C17—81no—
230D4×C17—81no—
231D4×C17—81no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17C4×D411yescharacteristic
249C2×C4×C17C4×D411yescharacteristic
250C2×C4×C17C4×D411yescharacteristic
251C2×C4×C17C23×C411yescharacteristic
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C8×C17—21no—
255C8×C17—21no—
256C8×C17—21no—
257C8×C17—21no—
258#3622—22no—
259#3622—21no—
260#3622—21no—
261#3622—21no—
262#3622—21no—
263#3622—41no—
264#3622—41no—
265#3622—41no—
266#3622—41no—
267#3622—41no—
268#3171D1711yescharacteristic
269#2993—171no—
270#2981—171no—
271#1588—171no—
272#1586—171no—
273#1586—171no—
274#1585—171no—
275#1583—171no—
276#1582—171no—
277#1581—171no—
278#1581—171no—
279#1569—171no—
280#1494—171no—
281#1492—171no—
282#1451—171no—
283C22×C17—21no—
284C22×C17—21no—
285C22×C17—21no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17C2×C4×D411yescharacteristic
296C4×C17—21no—
297C4×C17—21no—
298C4×C17—21no—
299C4×C17—21no—
300C4×C17—41no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17#56811yescharacteristic
306C4×C17#56811yescharacteristic
307C4×C17#57611yescharacteristic
308C4×C17#57711yescharacteristic
309#532C2×D1711yescharacteristic
310#529C2×D1711yescharacteristic
311#501Dic1711yescharacteristic
312#496Dic1711yescharacteristic
313#490Dic1711yescharacteristic
314#489Dic1711yescharacteristic
315#469Dic1711yescharacteristic
316#469Dic1711yescharacteristic
317#468Dic1711yescharacteristic
318#468Dic1711yescharacteristic
319#446—171no—
320C8×Q8—171no—
321C4×SD16C2×D1711yescharacteristic
322C4×SD16C2×D1711yescharacteristic
323C4×D8C2×D1711yescharacteristic
324C4×D8C2×D1711yescharacteristic
325#436—171no—
326#436—171no—
327#436—171no—
328#436—341no—
329#435—341no—
330#435—341no—
331#435—341no—
332C8×D4—171no—
333C8×D4—171no—
334C8×D4—171no—
335#433—341no—
336#432—171no—
337#431—171no—
338#424—341no—
339#423—171no—
340#408—341no—
341#408—341no—
342C4×M16C2×D1711yescharacteristic
343C2×C4×C8—171no—
344#335—171no—
345#335—171no—
346#334—171no—
347#334—171no—
348Q8⋊C8—171no—
349Q8⋊C8—171no—
350D4⋊C8—171no—
351D4⋊C8—171no—
352D4⋊C8—171no—
353D4⋊C8—171no—
354D4⋊C8—171no—
355D4⋊C8—171no—
356#322—171no—
357#322—171no—
358#322—171no—
359#322—171no—
360C2×C17—41no—
361C2×C17—81no—
362C2×C17—81no—
363C2×C17—81no—
364C2×C17#293211yescharacteristic
365C2×C17#294611yescharacteristic
366C2×C17#295611yescharacteristic
367C2×#41—21no—
368C2×SD16C2×Dic1711yescharacteristic
369C2×SD16C2×Dic1711yescharacteristic
370C2×D8C2×Dic1711yescharacteristic
371C2×D8C2×Dic1711yescharacteristic
372C2×M16—341no—
373C2×M16—341no—
374C2×M16—341no—
375C2×M16—341no—
376C2×M16C2×Dic1711yescharacteristic
377C2×M16C2×Dic1711yescharacteristic
378C22×C8—341no—
379C22×C8—341no—
380C22×C8—341no—
381C22×C8—341no—
382C4⋊Q8C2×Dic1711yescharacteristic
383#127C2×Dic1711yescharacteristic
384#126—21no—
385#125C2×Dic1711yescharacteristic
386#124—21no—
387#124C2×Dic1711yescharacteristic
388#123—21no—
389C22⋊Q8C2×Dic1711yescharacteristic
390#121—21no—
391#121C2×Dic1711yescharacteristic
392#121C2×Dic1711yescharacteristic
393#121C2×Dic1711yescharacteristic
394C4×Q8C22×D1711yescharacteristic
395C4×D4—21no—
396C4×D4—21no—
397C4×D4C22×D1711yescharacteristic
398C4×D4C22×D1711yescharacteristic
399C4×D4C22×D1711yescharacteristic
400#117—21no—
401C2×C42C22×D1711yescharacteristic
402#107C2×Dic1711yescharacteristic
403#107C2×Dic1711yescharacteristic
404#106C2×Dic1711yescharacteristic
405#106C2×Dic1711yescharacteristic
406C4⋊C8—171no—
407C4⋊C8—171no—
408C4⋊C8—171no—
409C4⋊C8—171no—
410C4⋊C8—171no—
411C4⋊C8—171no—
412C4⋊C8—341no—
413C4⋊C8—341no—
414C4⋊C8—341no—
415#103C2×Dic1711yescharacteristic
416#103C2×Dic1711yescharacteristic
417#102C2×Dic1711yescharacteristic
418#102C2×Dic1711yescharacteristic
419#102C2×Dic1711yescharacteristic
420#102C2×Dic1711yescharacteristic
421#102C2×Dic1711yescharacteristic
422#102C2×Dic1711yescharacteristic
423C22⋊C8—171no—
424C22⋊C8—171no—
425C22⋊C8—341no—
426C22⋊C8—341no—
427C22⋊C8—341no—
428C22⋊C8—341no—
429C22⋊C8—341no—
430C22⋊C8—341no—
431C22⋊C8—341no—
432#97—341no—
433#97—341no—
434#97C2×Dic1711yescharacteristic
435#97C2×Dic1711yescharacteristic
436C4×C8—171no—
437C4×C8—171no—
438C4×C8—171no—
439C4×C8—171no—
440C4×C8—341no—
441C4×C8—341no—
442C4×C8C22×D1711yescharacteristic
443C4×C8C22×D1711yescharacteristic
444C17#1798711yescharacteristic, complemented, Sylow
445#41—81no—
446#41—81no—
447C2×Q8—21no—
448C2×Q8C22×Dic1711yescharacteristic
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4C22×Dic1711yescharacteristic
455C2×D4C22×Dic1711yescharacteristic
456C2×D4C22×Dic1711yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4D4×D1711yescharacteristic
461C22×C4#6325411yescharacteristic
462C22×C4C22×Dic1711yescharacteristic
463SD16—41no—
464SD16—41no—
465D8—41no—
466D8—41no—
467M16—41no—
468M16—41no—
469M16—1361no—
470M16—1361no—
471M16—1361no—
472M16—1361no—
473C2×C8—171no—
474C2×C8—171no—
475C2×C8—171no—
476C2×C8—171no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—682no—
484C2×C8—1361no—
485C2×C8—1361no—
486C2×C8—1361no—
487C2×C8C22×Dic1711yescharacteristic
488C2×C8C22×Dic1711yescharacteristic
489C2×C8C22×Dic1711yescharacteristic
490C2×C8C22×Dic1711yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C22×Dic1711yescharacteristic
497C4⋊C4C22×Dic1711yescharacteristic
498C4⋊C4C22×Dic1711yescharacteristic
499C4⋊C4C22×Dic1711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C42—21no—
510C42D4×D1711yescharacteristic
511C42#6325411yescharacteristic
512C42C22×Dic1711yescharacteristic
513C42C23×D1711yescharacteristic
514C23—21no—
515C23—21no—
516C23—21no—
517C23D4×Dic1711yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—81no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531D4—81no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4D4×Dic1711yescharacteristic
549C2×C4D4×Dic1711yescharacteristic
550C2×C4D4×Dic1711yescharacteristic
551C2×C4C2×D4×D1711yescharacteristic
552C2×C4C2×#6325411yescharacteristic
553C2×C4C23×Dic1711yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—342no—
559C8—341no—
560C8—341no—
561C8—341no—
562C8—341no—
563C8—681no—
564C8—681no—
565C8—681no—
566C8—681no—
567C8—681no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22C2×D4×Dic1711yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4#1176357611yescharacteristic
591C4#1176357611yescharacteristic
592C4#1176361311yescharacteristic
593C4#1176361911yescharacteristic
594C2—41no—
595C2—81no—
596C2—81no—
597C2—81no—
598C2#1453750211yescharacteristic
599C2#1453761711yescharacteristic
600C2#1453769411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
Fitting, prime radicalC17×#3171C2
socleC22×C17C2×C4×D4
cosocleC42×C17C24
hypercenter#3171D17
hypocenterC17#17987
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3171

Complements

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

NH ≅ G/Ntype
1. #14537933C2semidirect
2. #14537860C2semidirect
3. #14527663C2semidirect
4. #14527649C2semidirect
5. #14527649C2semidirect
6. #14527642C2semidirect
7. #14527628C2semidirect
8. #14527621C2semidirect
9. #14527614C2semidirect
10. #14527614C2semidirect
11. #14527551C2semidirect
12. #14527244C2semidirect
13. #14527235C2semidirect
14. #14527071C2semidirect
444. C17#17987semidirect

Maximal subgroups

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

subgroupindex
#145379332
#145378602
#145276632
#145276492
#145276422
#145276282
#145276212
#145276142
#145275512
#145272442
#145272352
#145270712
C17×#31712
#1798717