Finite Group Databasebeta

← #17005358

Subgroups — #17005358

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538833C211yesmaximal, characteristic
2#14537782C211yesmaximal, characteristic, complemented
3#14537519C211yesmaximal, characteristic, complemented
4#14527697C211yesmaximal, characteristic, complemented
5#14527673C211yesmaximal, characteristic, complemented
6#14527661C211yesmaximal, characteristic, complemented
7#14527647C211yesmaximal, characteristic, complemented
8#14527619C211yesmaximal, characteristic, complemented
9#14527612C211yesmaximal, characteristic, complemented
10#14527595C211yesmaximal, characteristic, complemented
11#14527578C211yesmaximal, characteristic, complemented
12#14527247C211yesmaximal, characteristic, complemented
13#14527174C211yesmaximal, characteristic, complemented
14#14527079C211yesmaximal, characteristic, complemented
15C17×#1574C211yesmaximal, characteristic, complemented
16C2×M16×D17—21no—
17#11763343C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19#11763145C411yescharacteristic
20#11763128C411yescharacteristic
21#11763110C411yescharacteristic
22#11763110C411yescharacteristic
23#11763025C411yescharacteristic
24#11763025C411yescharacteristic
25#11762962C411yescharacteristic
26#11762945C411yescharacteristic
27#11762936C2211yescharacteristic, complemented
28#11762933C2211yescharacteristic, complemented
29C4×#63219C2211yescharacteristic
30C4×#63219C2211yescharacteristic
31C4×D136C2211yescharacteristic
32C4×D136C2211yescharacteristic
33#11762886C2211yescharacteristic
34#11762886C2211yescharacteristic
35#11762886C2211yescharacteristic, complemented
36#11762884—21no—
37#11762883—21no—
38#11762879—21no—
39#11762879—21no—
40#11762874C2211yescharacteristic
41#11762874C2211yescharacteristic
42#11762874C2211yescharacteristic, complemented
43#11762868—21no—
44#11762862C2211yescharacteristic
45#11762772—21no—
46#11762763—21no—
47C4×#63223C2211yescharacteristic
48#11762539C2211yescharacteristic, complemented
49#11762539C2211yescharacteristic, complemented
50#11762536C2211yescharacteristic
51#11762536C2211yescharacteristic
52#11762536C2211yescharacteristic
53#11762536C2211yescharacteristic
54#11762536C2211yescharacteristic, complemented
55#11762536C2211yescharacteristic, complemented
56#11762531C2211yescharacteristic, complemented
57#11762531C2211yescharacteristic, complemented
58#11762530C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C17×#431C2211yescharacteristic
61C17×#431C2211yescharacteristic
62C17×#423C2211yescharacteristic
63C17×#335C2211yescharacteristic, complemented
64C17×#335C2211yescharacteristic, complemented
65C17×#334C2211yescharacteristic, complemented
66C17×#334C2211yescharacteristic, complemented
67C22×D68C2×C411yescharacteristic
68C22×C4×D17—21no—
69C2×M16×C17—21no—
70C2×C42×C17C2311yescharacteristic
71#10587616C2×C411yescharacteristic
72#10587615—21no—
73C4×C22⋊D17—21no—
74C2×#63227—21no—
75C2×C4⋊Dic17C2×C411yescharacteristic
76#10587610C2×C411yescharacteristic
77C2×#63223D411yescharacteristic
78C2×#63223D411yescharacteristic
79M16×D17—44no—
80C2×D136C2×C411yescharacteristic
81C2×D136C2×C411yescharacteristic
82C2×#63219C2×C411yescharacteristic
83C2×#63219C2×C411yescharacteristic
84C2×#63218—21no—
85C2×#63218—21no—
86C2×#63218—21no—
87C2×C8×D17—21no—
88C2×C8×D17—21no—
89C2×C8×D17—21no—
90#10587562—21no—
91#10587561C2×C411yescharacteristic
92C4×D68—21no—
93C4×D68—21no—
94C4×D68C2×C411yescharacteristic
95C4×D68C2×C411yescharacteristic
96C4×D68C2311yescharacteristic
97C4×D68C2311yescharacteristic
98C4×D68C2311yescharacteristic
99#10587558—21no—
100#10587556C2×C411yescharacteristic
101C4×Dic34C2311yescharacteristic
102C17×C4⋊C8—21no—
103C17×C4⋊C8C2×C411yescharacteristic
104C17×C4⋊C8C2×C411yescharacteristic
105C17×C4⋊C8C2311yescharacteristic
106C17×C4⋊C8C2311yescharacteristic
107C17×C4⋊C8C2311yescharacteristic
108C17×C4⋊C8C2311yescharacteristic
109C17×C22⋊C8—21no—
110C17×C22⋊C8C2×C411yescharacteristic
111C17×C22⋊C8C2×C411yescharacteristic
112C4×C8×C17—21no—
113C4×C8×C17C2311yescharacteristic
114C4×C8×C17C2311yescharacteristic
115C22⋊#3622—21no—
116#10587492C2×C411yescharacteristic
117#10587492C2×C411yescharacteristic
118#10587492C2×C411yescharacteristic
119#10587492C2×C411yescharacteristic
120#10587492C2×C411yescharacteristic
121#10587492C2×C411yescharacteristic
122#10587491—21no—
123#10587491—21no—
124#10587491—21no—
125#10587491—21no—
126#10587491—21no—
127#10587491—21no—
128#10587489C2×C411yescharacteristic
129#10587489C2×C411yescharacteristic
130#10587488C2×C411yescharacteristic
131#10587488C2×C411yescharacteristic
132#10587487C2×C411yescharacteristic
133#10587487C2×C411yescharacteristic
134#10587486—21no—
135#10587485—21no—
136#10587485—21no—
137C8×Dic17—21no—
138C4⋊#3622—21no—
139#10587474D411yescharacteristic
140#10587474D411yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C23×D17—21no—
146C22×C4×C17C22×C411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17#4111yescharacteristic
149C2×C22⋊D17—21no—
150C22×Dic17—21no—
151C2×D68—22no—
152C2×D68—21no—
153C2×D68—41no—
154C2×D68C22×C411yescharacteristic
155C2×D68C22×C411yescharacteristic
156C2×D68C22×C411yescharacteristic
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—42no—
163C2×C4×D17—42no—
164C2×Dic34C22×C411yescharacteristic
165M16×C17—42no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17C22×C411yescharacteristic
171C2×C8×C17C22×C411yescharacteristic
172C2×C8×C17C22×C411yescharacteristic
173C2×C8×C17C22×C411yescharacteristic
174C42×C17C22×C411yescharacteristic
175C42×C17C2×D411yescharacteristic
176C42×C17#4111yescharacteristic
177C42×C17C2411yescharacteristic
178C22⋊Dic17—21no—
179#63227—21no—
180#63227—21no—
181#63227—21no—
182#63227—21no—
183#63227—21no—
184#63227—21no—
185#63227—21no—
186#63227—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17C22×C411yescharacteristic
189C4⋊Dic17C22×C411yescharacteristic
190C4⋊Dic17C22×C411yescharacteristic
191C4⋊Dic17C22×C411yescharacteristic
192#63225—21no—
193#63225—21no—
194#63225—21no—
195#63225—21no—
196C4×Dic17—21no—
197#63223—22no—
198#63223—41no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622C2×D411yescharacteristic
202C2×#3622C2×D411yescharacteristic
203C2×#3622C2×D411yescharacteristic
204C2×#3622C2×D411yescharacteristic
205D136—22no—
206D136—41no—
207#63219—22no—
208#63219—41no—
209#63218—42no—
210#63218—42no—
211#63218—42no—
212C8×D17—42no—
213C8×D17—42no—
214C8×D17—42no—
215#18006—171nomaximal, Sylow
216C23×C17C4×D411yescharacteristic
217C22×D17—21no—
218C22×D17—21no—
219C22×D17—21no—
220C22×D17—42no—
221C22×D17—42no—
222C22×D17—81no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—41no—
228C2×C4×C17C4×D411yescharacteristic
229C2×C4×C17C4×D411yescharacteristic
230C2×C4×C17C4×D411yescharacteristic
231C2×C4×C17C23×C411yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C22⋊D17—81no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—42no—
241D68—21no—
242D68—21no—
243D68—21no—
244D68—21no—
245D68—21no—
246D68—42no—
247D68#13112yes—
248C4×D17—42no—
249C4×D17—42no—
250C4×D17—41no—
251C4×D17—41no—
252C4×D17—41no—
253C4×D17—41no—
254C4×D17—41no—
255C4×D17—41no—
256Dic34—21no—
257Dic34—21no—
258C8×C17—21no—
259C8×C17—21no—
260C8×C17—21no—
261C8×C17—21no—
262C8×C17—41no—
263C8×C17—41no—
264C8×C17—41no—
265C8×C17—41no—
266#3622—21no—
267#3622—21no—
268#3622—21no—
269#3622—41no—
270#3622—41no—
271#3622AGL(1,ℤ/8)12yes—
272#3105—171no—
273#2969—171no—
274D4×M16—171no—
275#1594—171no—
276#1590—171no—
277#1588—171no—
278#1586—171no—
279#1582—171no—
280#1581—171no—
281SD16⋊C8—171no—
282C8×D8—171no—
283#1574D1711yescharacteristic, complemented
284#1495—171no—
285#1478—171no—
286#1453—171no—
287C22×C17—21no—
288C22×C17—21no—
289C22×C17—21no—
290C22×C17C2×C4×D411yescharacteristic
291C2×D17—22no—
292C2×D17—41no—
293C2×D17—41no—
294C2×D17—41no—
295C2×D17—41no—
296C2×D17—41no—
297C2×D17—41no—
298C2×D17—41no—
299C2×D17—41no—
300C2×D17—84no—
301C4×C17—21no—
302C4×C17—21no—
303C4×C17—21no—
304C4×C17—21no—
305C4×C17C2×#13111yescharacteristic
306C4×C17#56811yescharacteristic
307C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C17#57511yescharacteristic
309Dic17—41no—
310Dic17—41no—
311Dic17—41no—
312Dic17—41no—
313Dic17—41no—
314C22×M16—341no—
315#529—171no—
316C2×C4×D4—171no—
317#484—171no—
318#481—171no—
319#478—171no—
320#478—171no—
321#463—171no—
322#463—171no—
323#450—171no—
324#447—171no—
325#446—171no—
326C8×Q8—171no—
327C4×SD16—171no—
328C4×SD16—171no—
329C4×D8—171no—
330C4×D8—171no—
331#436—171no—
332#436—171no—
333#436—171no—
334#436—341no—
335#435—341no—
336#435—341no—
337#435—341no—
338C8×D4—171no—
339C8×D4—171no—
340C8×D4—171no—
341#433—341no—
342#431—171no—
343#431C2×D1711yescharacteristic
344#431C2×D1711yescharacteristic
345#423C2×D1711yescharacteristic
346#407—341no—
347#405—341no—
348C4×M16—171no—
349#335C2×D1711yescharacteristic, complemented
350#335C2×D1711yescharacteristic, complemented
351#334C2×D1711yescharacteristic, complemented
352#334C2×D1711yescharacteristic, complemented
353Q8⋊C8—171no—
354Q8⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8—171no—
358D4⋊C8—171no—
359D4⋊C8—171no—
360D4⋊C8—171no—
361#322—171no—
362#322—171no—
363#322—171no—
364C82—171no—
365C2×C17—41no—
366C2×C17#293311yescharacteristic
367C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
368C2×C17#295711yescharacteristic
369D17—42no—
370D17—81no—
371D17—81no—
372C22×D4—171no—
373C23×C4—341no—
374C2×SD16—171no—
375C2×SD16—171no—
376C2×D8—171no—
377C2×D8—171no—
378C2×M16—21no—
379C2×M16—171no—
380C2×M16—171no—
381C2×M16—341no—
382C2×M16—341no—
383C2×M16—341no—
384C2×M16—684no—
385C22×C8—341no—
386C22×C8—341no—
387C22×C8—341no—
388#126—341no—
389#125—171no—
390#124—171no—
391#123—341no—
392C22⋊Q8—171no—
393#121—171no—
394C4×Q8—171no—
395C4×D4—171no—
396C4×D4—171no—
397C4×D4—171no—
398C4×D4—171no—
399C4×D4—171no—
400C4×D4—341no—
401C4×D4—341no—
402C4×D4—341no—
403#117—341no—
404C2×C4⋊C4—171no—
405C2×C22⋊C4—341no—
406C2×C42C22×D1711yescharacteristic
407#107—171no—
408#107—171no—
409#106—171no—
410#106—171no—
411C4⋊C8—21no—
412C4⋊C8—341no—
413C4⋊C8—341no—
414C4⋊C8—341no—
415C4⋊C8C4×D1711yescharacteristic
416C4⋊C8C4×D1711yescharacteristic
417C4⋊C8C22×D1711yescharacteristic
418C4⋊C8C22×D1711yescharacteristic
419C4⋊C8C22×D1711yescharacteristic
420C4⋊C8C22×D1711yescharacteristic
421#103—171no—
422#103—171no—
423#102—171no—
424#102—171no—
425#102—171no—
426#102—171no—
427#102—171no—
428#102—171no—
429C22⋊C8—21no—
430C22⋊C8—341no—
431C22⋊C8—341no—
432C22⋊C8—341no—
433C22⋊C8—341no—
434C22⋊C8—341no—
435C22⋊C8—341no—
436C22⋊C8—341no—
437C22⋊C8C4×D1711yescharacteristic
438C22⋊C8C4×D1711yescharacteristic
439#97—171no—
440#97—171no—
441#97—341no—
442C4×C8—21no—
443C4×C8—171no—
444C4×C8—171no—
445C4×C8—171no—
446C4×C8—171no—
447C4×C8—341no—
448C4×C8C22×D1711yescharacteristic
449C4×C8C22×D1711yescharacteristic
450C17#1800611yescharacteristic, complemented, Sylow
451C24—341no—
452C2×Q8—171no—
453C2×D4—171no—
454C2×D4—171no—
455C2×D4—171no—
456C2×D4—342no—
457C2×D4—341no—
458C2×D4—341no—
459C2×D4—681no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—682no—
467C22×C4—682no—
468C22×C4C2×C4×D1711yescharacteristic
469C22×C4D4×D1711yescharacteristic
470C22×C4#6325611yescharacteristic
471SD16—342no—
472SD16—681no—
473D8—342no—
474D8—681no—
475M16—42no—
476M16—342no—
477M16—682no—
478M16—682no—
479M16—682no—
480M16—681no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—171no—
486C2×C8—171no—
487C2×C8—171no—
488C2×C8—171no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—682no—
492C2×C8—682no—
493C2×C8—682no—
494C2×C8C2×C4×D1711yescharacteristic
495C2×C8C2×C4×D1711yescharacteristic
496C2×C8C2×C4×D1711yescharacteristic
497C2×C8C2×C4×D1711yescharacteristic
498C4⋊C4—171no—
499C4⋊C4—171no—
500C4⋊C4—171no—
501C4⋊C4—171no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C22⋊C4—341no—
508C22⋊C4—341no—
509C22⋊C4—341no—
510C22⋊C4—341no—
511C22⋊C4—341no—
512C22⋊C4—341no—
513C22⋊C4—341no—
514C22⋊C4—341no—
515C22⋊C4—341no—
516C42—341no—
517C42C2×C4×D1711yescharacteristic
518C42D4×D1711yescharacteristic
519C42#6325611yescharacteristic
520C42C23×D1711yescharacteristic
521C23—341no—
522C23—341no—
523C23—341no—
524C23—682no—
525C23—682no—
526C23—1361no—
527C23#1058757911yescharacteristic
528Q8—341no—
529Q8—341no—
530D4—172no—
531D4—341no—
532D4—341no—
533D4—341no—
534D4—341no—
535D4—341no—
536D4—682no—
537D4—1361no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—682no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4#1058757911yescharacteristic
558C2×C4#1058757911yescharacteristic
559C2×C4#1058757911yescharacteristic
560C2×C4C22×C4×D1711yescharacteristic
561C2×C4C2×D4×D1711yescharacteristic
562C2×C4C2×#6325611yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—172no—
572C8—341no—
573C8—341no—
574C8—341no—
575C8—681no—
576C8—681no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—342no—
581C22—681no—
582C22—681no—
583C22—681no—
584C22—681no—
585C22—681no—
586C22—681no—
587C22—681no—
588C22—681no—
589C22—1364no—
590C22C2×#1058757911yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—681no—
596C4—681no—
597C4—681no—
598C4—681no—
599C4—681no—
600C4#131×D1711yescharacteristic
601C4#1176357411yescharacteristic
602C4AGL(1,ℤ/8)×D1711yescharacteristic
603C4#1176360811yescharacteristic
604C2—41no—
605C2—682no—
606C2—1361no—
607C2—1361no—
608C2#1453750811yescharacteristic
609C2#1453759111yescharacteristic
610C2#1453770011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1574C2
socleC22×C17C2×C4×D4
prime radical#11762945C4
cosocleC42×C17C24
hypercenter#1574D17
hypocenterC17#18006
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
2. #14537782C2semidirect
3. #14537519C2semidirect
4. #14527697C2semidirect
5. #14527673C2semidirect
6. #14527661C2semidirect
7. #14527647C2semidirect
8. #14527619C2semidirect
9. #14527612C2semidirect
10. #14527595C2semidirect
11. #14527578C2semidirect
12. #14527247C2semidirect
13. #14527174C2semidirect
14. #14527079C2semidirect
15. C17×#1574C2semidirect
27. #11762936C22semidirect
28. #11762933C22semidirect
35. #11762886C22semidirect
42. #11762874C22semidirect
48. #11762539C22semidirect
49. #11762539C22semidirect
54. #11762536C22semidirect
55. #11762536C22semidirect
56. #11762531C22semidirect
57. #11762531C22semidirect
58. #11762530C22semidirect
59. C8×#3622C22semidirect
63. C17×#335C22semidirect
64. C17×#335C22semidirect
65. C17×#334C22semidirect
66. C17×#334C22semidirect
283. #1574D17semidirect
349. #335C2×D17semidirect
350. #335C2×D17semidirect
351. #334C2×D17semidirect
352. #334C2×D17semidirect
450. C17#18006semidirect

Maximal subgroups

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

subgroupindex
#145388332
#145377822
#145375192
#145276972
#145276732
#145276612
#145276472
#145276192
#145276122
#145275952
#145275782
#145272472
#145271742
#145270792
C17×#15742
#1800617