Finite Group Databasebeta

← #17003992

Subgroups — #17003992

Order 4352.

726 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537822C211yesmaximal, characteristic, complemented
2#14537805C211yesmaximal, characteristic, complemented
3C4×D8×D17C211yesmaximal, characteristic
4#14527650C211yesmaximal, characteristic, complemented
5#14527649C211yesmaximal, characteristic, complemented
6#14527647C211yesmaximal, characteristic, complemented
7#14527616C211yesmaximal, characteristic, complemented
8#14527613C211yesmaximal, characteristic, complemented
9#14527611C211yesmaximal, characteristic, complemented
10#14527610C211yesmaximal, characteristic, complemented
11#14527529C211yesmaximal, characteristic, complemented
12#14527173C211yesmaximal, characteristic, complemented
13#14527173C211yesmaximal, characteristic, complemented
14#14527075C211yesmaximal, characteristic, complemented
15C17×#1586C211yesmaximal, characteristic, complemented
16C2×D8×D17C411yescharacteristic
17C4×D4×D17C2211yescharacteristic
18C4×D4×D17C2211yescharacteristic
19C4×D136C2211yescharacteristic
20D8×Dic17C411yescharacteristic
21#11762891C411yescharacteristic
22C4×#63228C2211yescharacteristic
23C4×#63228C2211yescharacteristic
24#11762889C411yescharacteristic
25#11762889C411yescharacteristic
26#11762888C2211yescharacteristic, complemented
27#11762887C2211yescharacteristic, complemented
28#11762886C2211yescharacteristic, complemented
29#11762885—21no—
30#11762884—21no—
31C8×D68C2211yescharacteristic, complemented
32D4×#3622C2211yescharacteristic, complemented
33#11762874C2211yescharacteristic, complemented
34C8×C22⋊D17—21no—
35#11762872—21no—
36#11762861C2211yescharacteristic
37#11762861C2211yescharacteristic
38#11762860C2211yescharacteristic
39#11762860C2211yescharacteristic
40#107×D17C411yescharacteristic
41#102×D17C411yescharacteristic
42#102×D17C411yescharacteristic
43#11762772—21no—
44#11762771—21no—
45C2×#10587491—21no—
46C22⋊C8×D17—21no—
47C4×C8×D17C2211yescharacteristic
48#11762561C2211yescharacteristic, complemented
49#11762561C2211yescharacteristic, complemented
50#11762560C2211yescharacteristic, complemented
51#11762538C2211yescharacteristic, complemented
52#11762538C2211yescharacteristic, complemented
53D4⋊#3622C2211yescharacteristic, complemented
54D4⋊#3622C2211yescharacteristic, complemented
55#11762536C2211yescharacteristic, complemented
56#11762536C2211yescharacteristic, complemented
57#11762531C2211yescharacteristic, complemented
58#11762531C2211yescharacteristic, complemented
59#11762530C2211yescharacteristic, complemented
60C4×D8×C17C2211yescharacteristic
61C17×#436C2211yescharacteristic, complemented
62C8×D4×C17C2211yescharacteristic, complemented
63C17×#334C2211yescharacteristic, complemented
64C17×D4⋊C8C2211yescharacteristic, complemented
65C17×D4⋊C8C2211yescharacteristic, complemented
66C17×#322C2211yescharacteristic, complemented
67C2×D4×D17C2×C411yescharacteristic
68C2×D4×D17C2×C411yescharacteristic
69C22×C4×D17—21no—
70C22×C4×D17—21no—
71C2×D8×C17C2×C411yescharacteristic
72C2×M16×C17—21no—
73C22×C8×C17—21no—
74C4×D4×C17C2311yescharacteristic
75C4×D4×C17C2311yescharacteristic
76D4×Dic17C2×C411yescharacteristic
77D4×Dic17C2×C411yescharacteristic
78C2×#63228C2×C411yescharacteristic
79C2×#63228C2×C411yescharacteristic
80C4×C22⋊D17—21no—
81C4×C22⋊D17—21no—
82C2×#63223—21no—
83C22×#3622—21no—
84D8×D17—24no—
85C2×D136C2×C411yescharacteristic
86C2×#63218—21no—
87C2×#63218—21no—
88C2×C8×D17—21no—
89C2×C8×D17—21no—
90C2×C8×D17C2×C411yescharacteristic
91C2×C8×D17C2×C411yescharacteristic
92#10587579C2×C411yescharacteristic
93#10587579C2×C411yescharacteristic
94C4⋊C4×D17C2×C411yescharacteristic
95C4⋊C4×D17C2×C411yescharacteristic
96#10587567—21no—
97#10587567—21no—
98C22⋊C4×D17—21no—
99C22⋊C4×D17—21no—
100C4×D68C2311yescharacteristic
101C4×D68C2311yescharacteristic
102C42×D17C2311yescharacteristic
103C17×#107C2×C411yescharacteristic
104C17×C4⋊C8C2311yescharacteristic
105C17×C4⋊C8C2311yescharacteristic
106C17×#102C2×C411yescharacteristic
107C17×#102C2×C411yescharacteristic
108C17×C22⋊C8—21no—
109C17×C22⋊C8—21no—
110C4×C8×C17C2311yescharacteristic
111C4×C8×C17C2311yescharacteristic
112C4×C8×C17C2311yescharacteristic
113#10587503C2×C411yescharacteristic
114#10587503C2×C411yescharacteristic
115C22⋊#3622—21no—
116C22⋊#3622—21no—
117#10587492C2×C411yescharacteristic
118#10587492C2×C411yescharacteristic
119#10587491—21no—
120#10587491—21no—
121#10587491—21no—
122#10587491—21no—
123#10587491—21no—
124#10587491—21no—
125#10587491—21no—
126#10587491D411yescharacteristic
127#10587491D411yescharacteristic
128#10587489C2×C411yescharacteristic
129#10587485—21no—
130#10587485D411yescharacteristic
131#10587485D411yescharacteristic
132C8×Dic17—21no—
133C8×Dic17—21no—
134C8×Dic17C2×C411yescharacteristic
135C8×Dic17C2×C411yescharacteristic
136#10587480C2×C411yescharacteristic
137#10587480C2×C411yescharacteristic
138#10587478C2×C411yescharacteristic
139#10587478C2×C411yescharacteristic
140C4⋊#3622C2311yescharacteristic
141C4⋊#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C23×D17—21no—
146C23×D17—21no—
147C2×D4×C17C22×C411yescharacteristic
148C2×D4×C17C22×C411yescharacteristic
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C2×C22⋊D17—21no—
152C2×C22⋊D17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155D4×D17—22no—
156D4×D17—22no—
157D4×D17—22no—
158D4×D17M1614yescomplemented
159C2×D68C22×C411yescharacteristic
160C2×D68C22×C411yescharacteristic
161C2×C4×D17—21no—
162C2×C4×D17—21no—
163C2×C4×D17—21no—
164C2×C4×D17—21no—
165C2×C4×D17—21no—
166C2×C4×D17—21no—
167C2×C4×D17—81no—
168C2×C4×D17—81no—
169C2×C4×D17—81no—
170C2×C4×D17—81no—
171C2×C4×D17C22×C411yescharacteristic
172C2×C4×D17C2×D411yescharacteristic
173C2×C4×D17#4111yescharacteristic
174D8×C17—22no—
175M16×C17—81no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—81no—
180C2×C8×C17C22×C411yescharacteristic
181C2×C8×C17C22×C411yescharacteristic
182C2×C8×C17C2×D411yescharacteristic
183C2×C8×C17C2×D411yescharacteristic
184C4⋊C4×C17C22×C411yescharacteristic
185C4⋊C4×C17C22×C411yescharacteristic
186C22⋊C4×C17—21no—
187C22⋊C4×C17—21no—
188C42×C17C2411yescharacteristic
189C22⋊Dic17—21no—
190C22⋊Dic17—21no—
191#63228—22no—
192#63228—22no—
193#63227—21no—
194#63227—21no—
195#63227—21no—
196#63227—21no—
197C4⋊Dic17C22×C411yescharacteristic
198C4⋊Dic17C22×C411yescharacteristic
199#63225—21no—
200#63225—21no—
201C4×Dic17C22×C411yescharacteristic
202C4×Dic17C2×D411yescharacteristic
203C4×Dic17#4111yescharacteristic
204#63223—81no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—81no—
209C2×#3622C22×C411yescharacteristic
210C2×#3622C22×C411yescharacteristic
211C2×#3622C2×D411yescharacteristic
212C2×#3622C2×D411yescharacteristic
213D136—22no—
214#63218—81no—
215#63218—81no—
216C8×D17—22no—
217C8×D17—22no—
218C8×D17—81no—
219C8×D17—81no—
220#17877—171nomaximal, Sylow
221C23×C17—21no—
222C23×C17—21no—
223C22×D17—21no—
224C22×D17—24no—
225C22×D17—21no—
226C22×D17—42no—
227C22×D17—81no—
228C22×D17—81no—
229C22×D17—81no—
230C22×D17—81no—
231C22×D17C4×D411yescharacteristic
232D4×C17—21no—
233D4×C17—21no—
234D4×C17—21no—
235D4×C17C2×M1612yes—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17C23×C411yescharacteristic
243C2×C4×C17C22×D411yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C22⋊D17—24no—
246C22⋊D17—42no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—81no—
250C2×Dic17—81no—
251C2×Dic17C4×D411yescharacteristic
252C2×Dic17C4×D411yescharacteristic
253C2×Dic17C4×D411yescharacteristic
254D68—21no—
255D68—21no—
256D68—21no—
257D68C2×M1612yes—
258C4×D17—21no—
259C4×D17—21no—
260C4×D17—41no—
261C4×D17—41no—
262C4×D17—41no—
263C4×D17—41no—
264C4×D17—41no—
265C4×D17—81no—
266C4×D17—81no—
267C4×D17—81no—
268C4×D17—81no—
269C4×D17C2×M1612yes—
270C4×D17#13112yes—
271C8×C17—21no—
272C8×C17—21no—
273C8×C17—21no—
274C8×C17—21no—
275C8×C17—41no—
276C8×C17—41no—
277C8×C17—41no—
278#3622—21no—
279#3622—21no—
280#3622—21no—
281#3622—21no—
282#3622—41no—
283#3622—41no—
284#3622—41no—
285#2975—171no—
286#2972—171no—
287C2×C4×D8—171no—
288#1586—171no—
289#1586—171no—
290#1586—171no—
291#1586D1711yescharacteristic, complemented
292#1581—171no—
293#1581—171no—
294#1581—171no—
295#1581—171no—
296#1564—171no—
297#1478—171no—
298#1478—171no—
299#1452—171no—
300C22×C17—22no—
301C22×C17—41no—
302C22×C17—41no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17C2×C4×D411yescharacteristic
307C2×D17—21no—
308C2×D17—21no—
309C2×D17—22no—
310C2×D17—41no—
311C2×D17—41no—
312C2×D17—41no—
313C2×D17—41no—
314C2×D17—41no—
315C2×D17—84no—
316C2×D17—84no—
317C2×D17#43512yes—
318C4×C17—21no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17C22×M1611yescharacteristic
322C4×C17C2×#13111yescharacteristic
323C4×C17#57611yescharacteristic
324C4×C17#57611yescharacteristic
325Dic17—21no—
326Dic17—21no—
327Dic17—41no—
328Dic17—41no—
329Dic17#43512yes—
330C22×D8—171no—
331C2×C4×D4—171no—
332C2×C4×D4—171no—
333C4×D8—171no—
334C4×D8—171no—
335C4×D8—171no—
336C4×D8—171no—
337C4×D8—171no—
338C4×D8—171no—
339C4×D8—171no—
340C4×D8C2×D1711yescharacteristic
341#436—171no—
342#436—171no—
343#436—171no—
344#436—341no—
345#436—341no—
346#436C2×D1711yescharacteristic, complemented
347C8×D4—171no—
348C8×D4—171no—
349C8×D4—171no—
350C8×D4—341no—
351C8×D4—341no—
352C8×D4C2×D1711yescharacteristic, complemented
353#431—171no—
354#431—171no—
355#431—171no—
356#431—171no—
357C2×#107—171no—
358C2×#102—171no—
359C2×#102—171no—
360#407—341no—
361#407—341no—
362C2×C22⋊C8—341no—
363C2×C22⋊C8—341no—
364C2×C4×C8—171no—
365#335—171no—
366#335—171no—
367#334—171no—
368#334C2×D1711yescharacteristic, complemented
369D4⋊C8—171no—
370D4⋊C8—171no—
371D4⋊C8—171no—
372D4⋊C8—171no—
373D4⋊C8—171no—
374D4⋊C8—171no—
375D4⋊C8C2×D1711yescharacteristic, complemented
376D4⋊C8C2×D1711yescharacteristic, complemented
377#322—171no—
378#322—171no—
379#322—171no—
380#322C2×D1711yescharacteristic, complemented
381C2×C17—42no—
382C2×C17—81no—
383C2×C17C2×#43511yescharacteristic
384C2×C17#294811yescharacteristic
385C2×C17#295711yescharacteristic
386D17—22no—
387D17—42no—
388D17—81no—
389C22×D4—171no—
390C22×D4—171no—
391C23×C4—341no—
392C23×C4—341no—
393C2×D8—171no—
394C2×D8—171no—
395C2×D8—171no—
396C2×D8—344no—
397C2×D8C4×D1711yescharacteristic
398C2×M16—21no—
399C2×M16—341no—
400C2×M16—341no—
401C2×M16—341no—
402C22×C8—21no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—341no—
406C22×C8—341no—
407C22×C8—341no—
408C4×D4—171no—
409C4×D4—171no—
410C4×D4—171no—
411C4×D4—171no—
412C4×D4—171no—
413C4×D4—171no—
414C4×D4—341no—
415C4×D4—341no—
416C4×D4—341no—
417C4×D4—341no—
418C4×D4C22×D1711yescharacteristic
419C4×D4C22×D1711yescharacteristic
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C42—171no—
425#107—171no—
426#107—171no—
427#107—171no—
428#107C4×D1711yescharacteristic
429C4⋊C8—171no—
430C4⋊C8—171no—
431C4⋊C8—171no—
432C4⋊C8—171no—
433C4⋊C8—341no—
434C4⋊C8C22×D1711yescharacteristic
435C4⋊C8C22×D1711yescharacteristic
436#102—171no—
437#102—171no—
438#102—171no—
439#102—171no—
440#102—171no—
441#102—171no—
442#102C4×D1711yescharacteristic
443#102C4×D1711yescharacteristic
444C22⋊C8—21no—
445C22⋊C8—21no—
446C22⋊C8—171no—
447C22⋊C8—171no—
448C22⋊C8—341no—
449C22⋊C8—341no—
450C22⋊C8—341no—
451C22⋊C8—341no—
452C22⋊C8—341no—
453C22⋊C8—341no—
454C22⋊C8—341no—
455C22⋊C8—341no—
456C22⋊C8—341no—
457C4×C8—171no—
458C4×C8—171no—
459C4×C8—171no—
460C4×C8—171no—
461C4×C8—171no—
462C4×C8—341no—
463C4×C8—341no—
464C4×C8C22×D1711yescharacteristic
465C4×C8C22×D1711yescharacteristic
466C4×C8C22×D1711yescharacteristic
467C17#1787711yescharacteristic, complemented, Sylow
468C24—341no—
469C24—341no—
470C2×D4—174no—
471C2×D4—171no—
472C2×D4—171no—
473C2×D4—342no—
474C2×D4—342no—
475C2×D4—342no—
476C2×D4—341no—
477C2×D4—341no—
478C2×D4C2×C4×D1711yescharacteristic
479C2×D4C2×C4×D1711yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—171no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—1361no—
494C22×C4—1361no—
495C22×C4—1361no—
496C22×C4—1361no—
497D8—22no—
498D8—342no—
499D8—342no—
500D8—342no—
501M16—81no—
502M16—1361no—
503M16—1361no—
504M16—1361no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—81no—
509C2×C8—171no—
510C2×C8—171no—
511C2×C8—171no—
512C2×C8—171no—
513C2×C8—342no—
514C2×C8—342no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—1361no—
519C2×C8—1361no—
520C2×C8—1361no—
521C2×C8C2×C4×D1711yescharacteristic
522C2×C8C2×C4×D1711yescharacteristic
523C2×C8D4×D1711yescharacteristic
524C2×C8D4×D1711yescharacteristic
525C4⋊C4—171no—
526C4⋊C4—171no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4C2×C4×D1711yescharacteristic
530C4⋊C4C2×C4×D1711yescharacteristic
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—341no—
534C22⋊C4—341no—
535C22⋊C4—341no—
536C22⋊C4—341no—
537C22⋊C4—341no—
538C22⋊C4—341no—
539C42—171no—
540C42—171no—
541C42—171no—
542C42C23×D1711yescharacteristic
543C23—21no—
544C23—21no—
545C23—171no—
546C23—344no—
547C23—341no—
548C23—341no—
549C23—682no—
550C23—1361no—
551C23—1361no—
552C23—1361no—
553C23—1361no—
554D4—21no—
555D4—21no—
556D4—21no—
557D4—172no—
558D4—344no—
559D4—341no—
560D4—341no—
561D4—341no—
562D4—682no—
563D4M16×D1712yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—171no—
571C2×C4—172no—
572C2×C4—172no—
573C2×C4—171no—
574C2×C4—171no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—1361no—
585C2×C4—1361no—
586C2×C4—1361no—
587C2×C4—1361no—
588C2×C4—1361no—
589C2×C4—1361no—
590C2×C4C22×C4×D1711yescharacteristic
591C2×C4C2×D4×D1711yescharacteristic
592C2×C4#41×D1711yescharacteristic
593C8—21no—
594C8—21no—
595C8—21no—
596C8—21no—
597C8—41no—
598C8—41no—
599C8—41no—
600C8—341no—
601C8—341no—
602C8—341no—
603C8—341no—
604C8—681no—
605C8—681no—
606C8—681no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—172no—
614C22—341no—
615C22—341no—
616C22—342no—
617C22—681no—
618C22—681no—
619C22—681no—
620C22—681no—
621C22—681no—
622C22—1364no—
623C22—1364no—
624C22C4×D4×D1711yescharacteristic
625C4—21no—
626C4—41no—
627C4—41no—
628C4—172no—
629C4—341no—
630C4—341no—
631C4—681no—
632C4—681no—
633C4C2×M16×D1711yescharacteristic
634C4#131×D1711yescharacteristic
635C4#1176361311yescharacteristic
636C4#1176361311yescharacteristic
637C2—42no—
638C2—81no—
639C2—342no—
640C2—682no—
641C2—1361no—
642C2D17×#43511yescharacteristic
643C2#1453763311yescharacteristic
644C2#1453769611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1586C2
socleC22×C17C2×C4×D4
prime radicalC2×D8×D17C4
cosocleC42×C17C24
hypercenter#1586D17
hypocenterC17#17877
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
1. #14537822C2semidirect
2. #14537805C2semidirect
4. #14527650C2semidirect
5. #14527649C2semidirect
6. #14527647C2semidirect
7. #14527616C2semidirect
8. #14527613C2semidirect
9. #14527611C2semidirect
10. #14527610C2semidirect
11. #14527529C2semidirect
12. #14527173C2semidirect
13. #14527173C2semidirect
14. #14527075C2semidirect
15. C17×#1586C2semidirect
26. #11762888C22semidirect
27. #11762887C22semidirect
28. #11762886C22semidirect
31. C8×D68C22semidirect
32. D4×#3622C22semidirect
33. #11762874C22semidirect
48. #11762561C22semidirect
49. #11762561C22semidirect
50. #11762560C22semidirect
51. #11762538C22semidirect
52. #11762538C22semidirect
53. D4⋊#3622C22semidirect
54. D4⋊#3622C22semidirect
55. #11762536C22semidirect
56. #11762536C22semidirect
57. #11762531C22semidirect
58. #11762531C22semidirect
59. #11762530C22semidirect
61. C17×#436C22semidirect
62. C8×D4×C17C22semidirect
63. C17×#334C22semidirect
64. C17×D4⋊C8C22semidirect
65. C17×D4⋊C8C22semidirect
66. C17×#322C22semidirect
158. D4×D17M16semidirect
291. #1586D17semidirect
346. #436C2×D17semidirect
352. C8×D4C2×D17semidirect
368. #334C2×D17semidirect
375. D4⋊C8C2×D17semidirect
376. D4⋊C8C2×D17semidirect
380. #322C2×D17semidirect
467. C17#17877semidirect

Maximal subgroups

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

subgroupindex
#145378222
#145378052
C4×D8×D172
#145276502
#145276492
#145276472
#145276162
#145276132
#145276112
#145276102
#145275292
#145271732
#145270752
C17×#15862
#1787717