Finite Group Databasebeta

← #17037342

Subgroups — #17037342

Order 4352.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762911C211yesmaximal, characteristic
2#14531261C211yesmaximal, characteristic, complemented
3C42×D68C211yesmaximal, characteristic
4#14529426C211yesmaximal, characteristic, complemented
5#14529419C211yesmaximal, characteristic, complemented
6#14529377C211yesmaximal, characteristic, complemented
7#14529356C211yesmaximal, characteristic, complemented
8#14529117C211yesmaximal, characteristic
9#14529117C211yesmaximal, characteristic
10#14529048C211yesmaximal, characteristic
11#14529034C211yesmaximal, characteristic
12#14528681C211yesmaximal, characteristic, complemented
13#14528615C211yesmaximal, characteristic, complemented
14#14528610C211yesmaximal, characteristic
15C17×#1745C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21C2×C42×D17—21no—
22#11762911C418yescomplemented
23C2×#10587489C2211yescharacteristic
24C2×#10587489C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31C2×#10587492C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762715C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762693—21no—
37#11762693—21no—
38C4×#63225—21no—
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43C4×#63227—21no—
44C4×#63227—21no—
45C4×#63227—21no—
46#11762655—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C2×C17×#97C2211yescharacteristic
54C2×C17×#97C2211yescharacteristic
55C43×C17C2211yescharacteristic
56C17×#336C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×Dic34C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C2×C42×C17D411yescharacteristic
70C2×C42×C17D411yescharacteristic
71C2×C42×C17D411yescharacteristic
72C2×C42×C17D411yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×#63219—22no—
94C2×#63219—22no—
95C2×#63219C2×C414yes—
96C4×D68—22no—
97C4×D68—22no—
98C4×D68—22no—
99C4×D68—22no—
100C4×D68—22no—
101C4×D68C2×C414yes—
102C42×D17—44no—
103C4×Dic34—22no—
104C4×Dic34C2×C414yescomplemented
105C17×#97—22no—
106C17×#97C2×C414yescomplemented
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587492—22no—
118#10587492—22no—
119#10587492C2×C414yes—
120#10587489—22no—
121#10587489C2×C414yescomplemented
122#10587488—22no—
123#10587487—22no—
124#10587487—22no—
125#10587487C2×C414yescomplemented
126C23×D17—21no—
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2411yescharacteristic
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68C22×C412yes—
145C2×D68#4112yes—
146C2×D68#4112yes—
147C2×C4×D17—24no—
148C2×C4×D17—24no—
149C2×C4×D17—24no—
150C2×C4×D17—42no—
151C2×C4×D17—42no—
152C2×C4×D17—42no—
153C2×C4×D17—42no—
154C2×C4×D17—42no—
155C2×C4×D17—42no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34C22×C412yes—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17C22×C412yes—
173C2×C8×C17C22×C412yes—
174C42×C17—22no—
175C42×C17—22no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—41no—
181C42×C17—41no—
182C42×C17C22×C412yes—
183C42×C17#4112yes—
184#63227—24no—
185#63227—24no—
186#63227—24no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17C22×C412yes—
196C4⋊Dic17C22×C412yes—
197C4⋊Dic17#4112yes—
198C4⋊Dic17#4112yes—
199#63225—24no—
200#63225—42no—
201#63225—42no—
202C4×Dic17—24no—
203C4×Dic17—42no—
204C4×Dic17—42no—
205C4×Dic17—42no—
206C4×Dic17—42no—
207#63219—28no—
208#21121—171nomaximal, Sylow
209C23×C17C22×D411yescharacteristic
210C22×D17—22no—
211C22×D17—22no—
212C22×D17—22no—
213C22×D17—41no—
214C22×D17—41no—
215C22×D17—41no—
216C22×D17—41no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17C4×D412yes—
232C2×C4×C17C4×D412yes—
233C2×C4×C17AGL(1,ℤ/8)12yes—
234C2×C4×C17#13712yes—
235C2×C4×C17C23×C411yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×Dic17—22no—
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262D68—22no—
263D68—22no—
264D68—22no—
265D68#11714yes—
266C4×D17—48no—
267C4×D17—48no—
268C4×D17—48no—
269Dic34—22no—
270Dic34—22no—
271Dic34—41no—
272Dic34—41no—
273C8×C17—22no—
274C8×C17—22no—
275C8×C17—41no—
276C8×C17—41no—
277C2×#440—171no—
278#2295—171no—
279C42×D4—171no—
280#1928—171no—
281#1927—171no—
282#1921—171no—
283#1918—171no—
284#1877—171no—
285#1877—171no—
286#1864—171no—
287#1862—171no—
288#1776—171no—
289#1763—171no—
290#1762—171no—
291#1745D1711yescharacteristic, complemented
292C22×C17C2×C4×D411yescharacteristic
293C22×C17#52511yescharacteristic
294C22×C17#52511yescharacteristic
295C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C17C2×#13711yescharacteristic
297C22×C17#57511yescharacteristic
298C22×C17#57511yescharacteristic
299C2×D17—24no—
300C2×D17—42no—
301C2×D17—42no—
302C2×D17—42no—
303C2×D17—42no—
304C2×D17—42no—
305C2×D17—42no—
306C4×C17—22no—
307C4×C17—22no—
308C4×C17—22no—
309C4×C17—22no—
310C4×C17—41no—
311C4×C17—41no—
312C4×C17C2×#11711yescharacteristic
313C4×C17C4×#4111yescharacteristic
314C4×C17#52911yescharacteristic
315C4×C17#52911yescharacteristic
316Dic17—42no—
317Dic17—42no—
318Dic17—42no—
319Dic17—42no—
320Dic17—42no—
321Dic17—81no—
322Dic17—81no—
323C22×SD16—171no—
324C2×C4×Q8—171no—
325C2×C4×D4—171no—
326C2×C4×D4—171no—
327C2×C4×D4—171no—
328C22×C42—341no—
329#440—178no—
330C2×#107—171no—
331C2×#107—171no—
332C2×#106—171no—
333C2×#103—171no—
334C2×#103—171no—
335C2×#103—171no—
336C2×#102—171no—
337C2×#102—171no—
338C2×#102—171no—
339C2×#97C2×D1711yescharacteristic
340C2×#97C2×D1711yescharacteristic
341#391—171no—
342#391—171no—
343#389—171no—
344#389—171no—
345#387—341no—
346#387—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351C4×C4⋊C4—341no—
352C4×C22⋊C4—341no—
353C4×C22⋊C4—341no—
354C4×C22⋊C4—341no—
355#376—341no—
356C43C2×D1711yescharacteristic
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#336C2×D1711yescharacteristic
364#336C2×D1711yescharacteristic
365#336C2×D1711yescharacteristic
366#336C2×D1711yescharacteristic
367C2×C17#229011yescharacteristic
368C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
369C2×C17C4×#13711yescharacteristic
370C2×C17#311011yescharacteristic
371C2×C17#311111yescharacteristic
372C2×C17#311411yescharacteristic
373C2×C17#311611yescharacteristic
374D17—44no—
375C22×Q8—171no—
376C22×D4—171no—
377C23×C4—341no—
378C23×C4—341no—
379C23×C4—341no—
380C2×SD16—174no—
381C2×SD16—342no—
382C2×SD16—342no—
383C22×C8C22×D1711yescharacteristic
384C22×C8C22×D1711yescharacteristic
385C22×C8C22×D1711yescharacteristic
386C22×C8C22×D1711yescharacteristic
387C4×Q8—174no—
388C4×Q8—342no—
389C4×D4—174no—
390C4×D4—342no—
391C4×D4—342no—
392C4×D4—342no—
393C4×D4—342no—
394C4×D4—342no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C22⋊C4—341no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C42—341no—
408C2×C42—341no—
409C2×C42—341no—
410C2×C42—341no—
411C2×C42—341no—
412C2×C42—684no—
413C2×C42D6811yescharacteristic
414C2×C42D6811yescharacteristic
415C2×C42D6811yescharacteristic
416C2×C42D6811yescharacteristic
417C2×C42C22×D1711yescharacteristic
418C2×C42C22×D1711yescharacteristic
419C2×C42C22×D1711yescharacteristic
420#107—174no—
421#107—342no—
422#106—342no—
423#103—174no—
424#103—342no—
425#103—342no—
426#102—174no—
427#102—342no—
428#102—342no—
429#97—22no—
430#97C4×D1714yescomplemented
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441C17#2112111yescharacteristic, complemented, Sylow
442C24—341no—
443C2×Q8—172no—
444C2×Q8—341no—
445C2×Q8—341no—
446C2×Q8—341no—
447C2×Q8—341no—
448C2×Q8—341no—
449C2×D4—172no—
450C2×D4—172no—
451C2×D4—172no—
452C2×D4—341no—
453C2×D4—341no—
454C2×D4—341no—
455C22×C4—344no—
456C22×C4—344no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—344no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—682no—
466C22×C4—682no—
467C22×C4—682no—
468C22×C4—682no—
469C22×C4—682no—
470C22×C4—682no—
471C22×C4C2×D6811yescharacteristic
472C22×C4C2×D6811yescharacteristic
473C22×C4C2×D6811yescharacteristic
474C22×C4C2×D6811yescharacteristic
475C22×C4C2×D6811yescharacteristic
476C22×C4C2×D6811yescharacteristic
477C22×C4C23×D1711yescharacteristic
478SD16—348no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8C2×C4×D1712yes—
490C2×C8C2×C4×D1712yes—
491C4⋊C4—172no—
492C4⋊C4—172no—
493C4⋊C4—172no—
494C4⋊C4—172no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—344no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—682no—
505C4⋊C4—682no—
506C22⋊C4—344no—
507C22⋊C4—344no—
508C22⋊C4—344no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—344no—
518C42—682no—
519C42—682no—
520C42—682no—
521C42—682no—
522C42C2×C4×D1712yes—
523C42#6325212yes—
524C23—342no—
525C23—342no—
526C23—342no—
527C23—681no—
528C23—681no—
529C23—681no—
530C23—681no—
531C23C22×D6811yescharacteristic
532Q8—342no—
533Q8—342no—
534Q8—681no—
535Q8—681no—
536D4—174no—
537D4—342no—
538D4—342no—
539D4—342no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—342no—
557C2×C4—342no—
558C2×C4—342no—
559C2×C4—688no—
560C2×C4—688no—
561C2×C4—688no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4C4×D6812yes—
579C2×C4C4×D6812yes—
580C2×C4#1058759412yes—
581C2×C4#1058759512yes—
582C2×C4C22×C4×D1711yescharacteristic
583C2×C4C2×#6325211yescharacteristic
584C2×C4#41×D1711yescharacteristic
585C2×C4#41×D1711yescharacteristic
586C2×C4#41×D1711yescharacteristic
587C2×C4#41×D1711yescharacteristic
588C8—22no—
589C8—22no—
590C8—41no—
591C8—41no—
592C22—344no—
593C22—682no—
594C22—682no—
595C22—682no—
596C22—682no—
597C22—682no—
598C22—682no—
599C22C2×C4×D6811yescharacteristic
600C22#1176332111yescharacteristic
601C22#1176332111yescharacteristic
602C22C2×#1058759411yescharacteristic
603C22C2×#1058759511yescharacteristic
604C22#1176360911yescharacteristic
605C22#1176360911yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—682no—
613C4—682no—
614C4—682no—
615C4—682no—
616C4—682no—
617C4—1361no—
618C4—1361no—
619C4#117×D1711yescharacteristic
620C4#1176327711yescharacteristic
621C4#1176333711yescharacteristic
622C4#1176333711yescharacteristic
623C2—684no—
624C2#1453122811yescharacteristic
625C2C4×#1058759411yescharacteristic
626C2C4×#1058759511yescharacteristic
627C2#1453888411yescharacteristic
628C2#1453889511yescharacteristic
629C2#1453892411yescharacteristic
630C2#1453894711yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1745C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1745D17
hypocenterC17#21121
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. #14531261C2semidirect
4. #14529426C2semidirect
5. #14529419C2semidirect
6. #14529377C2semidirect
7. #14529356C2semidirect
12. #14528681C2semidirect
13. #14528615C2semidirect
15. C17×#1745C2semidirect
22. #11762911C4semidirect
104. C4×Dic34C2×C4semidirect
106. C17×#97C2×C4semidirect
121. #10587489C2×C4semidirect
125. #10587487C2×C4semidirect
291. #1745D17semidirect
430. #97C4×D17semidirect
441. C17#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#117629112
#145312612
C42×D682
#145294262
#145294192
#145293772
#145293562
#145291172
#145290482
#145290342
#145286812
#145286152
#145286102
C17×#17452
#2112117