Finite Group Databasebeta

← #17037402

Subgroups — #17037402

Order 4352.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762911C211yesmaximal, characteristic
2#14531436C211yesmaximal, characteristic, complemented
3#14531146C211yesmaximal, 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
13C4×#10587487C211yesmaximal, characteristic, complemented
14C4×#10587492C211yesmaximal, characteristic
15C17×#1746C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21#11762911C418yescomplemented
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587492C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31#11762715C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762691—21no—
36#11762691—21no—
37#11762676—21no—
38#11762673—21no—
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43#11762656—21no—
44#11762656—21no—
45#11762656—21no—
46#95×D17—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C2×C17×#97C2211yescharacteristic
54C2×C4×C8×C17C2211yescharacteristic
55C17×#376C2211yescharacteristic
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×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×#63219—22no—
90C2×#63219—22no—
91C2×#63219C2×C414yes—
92C4×D68—22no—
93C4×D68—22no—
94C4×D68—22no—
95C4×D68—22no—
96C4×D68—22no—
97C4×D68C2×C414yes—
98C4×Dic34—22no—
99C4×Dic34C2×C414yescomplemented
100C17×#97C2×C414yescomplemented
101C4×C8×C17—22no—
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104C17×#95D411yescharacteristic
105C17×#95D411yescharacteristic
106#10587502—21no—
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#10587492—22no—
117#10587492—22no—
118#10587492C2×C414yes—
119#10587489—22no—
120#10587489C2×C414yescomplemented
121#10587488—22no—
122#10587487—22no—
123#10587487—22no—
124#10587487C2×C414yescomplemented
125C23×D17—21no—
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68C22×C412yes—
144C2×D68#4112yes—
145C2×D68#4112yes—
146C2×C4×D17—24no—
147C2×C4×D17—24no—
148C2×C4×D17—24no—
149C2×C4×D17—81no—
150C2×C4×D17—81no—
151C2×C4×D17—81no—
152C2×C4×D17—81no—
153C2×C4×D17—81no—
154C2×C4×D17—81no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34C22×C412yes—
161C2×C8×C17—21no—
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×C17C22×C412yes—
172C2×C8×C17C22×C412yes—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17C22×C412yes—
178C42×C17#4112yes—
179#63227—24no—
180#63227—24no—
181#63227—24no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17C22×C412yes—
191C4⋊Dic17C22×C412yes—
192C4⋊Dic17#4112yes—
193C4⋊Dic17#4112yes—
194#63225—24no—
195#63225—81no—
196#63225—81no—
197C4×Dic17—24no—
198C4×Dic17—81no—
199C4×Dic17—81no—
200C4×Dic17—81no—
201C4×Dic17—81no—
202#63219—28no—
203#21125—171nomaximal, Sylow
204C23×C17C22×D411yescharacteristic
205C22×D17—22no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—41no—
211C22×D17—41no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
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×C17C4×D412yes—
229C2×C4×C17C4×D412yes—
230C2×C4×C17C23×C411yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
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—
257D68—22no—
258D68—22no—
259D68—22no—
260D68#11714yes—
261C4×D17—84no—
262C4×D17—84no—
263C4×D17—84no—
264Dic34—22no—
265Dic34—22no—
266Dic34—41no—
267Dic34—41no—
268C8×C17—22no—
269C8×C17—22no—
270C8×C17—41no—
271C8×C17—41no—
272C2×#440—171no—
273#2320—171no—
274#2275—171no—
275#1928—171no—
276#1927—171no—
277#1921—171no—
278#1918—171no—
279#1877—171no—
280#1877—171no—
281#1864—171no—
282#1862—171no—
283#1776—171no—
284C4×#103—171no—
285C4×#102—171no—
286#1746D1711yescharacteristic, complemented
287C22×C17C2×C4×D411yescharacteristic
288C22×C17#52511yescharacteristic
289C22×C17#52511yescharacteristic
290C22×C17#57611yescharacteristic
291C22×C17#57711yescharacteristic
292C22×C17#57711yescharacteristic
293C22×C17#57811yescharacteristic
294C2×D17—24no—
295C2×D17—42no—
296C2×D17—42no—
297C2×D17—42no—
298C2×D17—42no—
299C2×D17—42no—
300C2×D17—42no—
301C4×C17—22no—
302C4×C17—22no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17—41no—
307C4×C17C2×#11711yescharacteristic
308C4×C17C4×#4111yescharacteristic
309C4×C17#52911yescharacteristic
310C4×C17#52911yescharacteristic
311Dic17—42no—
312Dic17—42no—
313Dic17—42no—
314Dic17—42no—
315Dic17—42no—
316Dic17—81no—
317Dic17—81no—
318C22×SD16—171no—
319C2×C4×Q8—171no—
320C2×C4×D4—171no—
321C2×C4×D4—171no—
322C2×C4×D4—171no—
323#440—178no—
324C2×#107—171no—
325C2×#107—171no—
326C2×#106—171no—
327C2×#103—171no—
328C2×#103—171no—
329C2×#103—171no—
330C2×#102—171no—
331C2×#102—171no—
332C2×#102—171no—
333C2×#97C2×D1711yescharacteristic
334C2×C4×C8C2×D1711yescharacteristic
335#391—171no—
336#391—171no—
337#389—171no—
338#389—171no—
339#387—341no—
340#387—341no—
341#383—341no—
342#382—341no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C22⋊C4—341no—
348C4×C22⋊C4—341no—
349C4×C22⋊C4—341no—
350#376C2×D1711yescharacteristic
351C2×#95—341no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#336C2×D1711yescharacteristic
359#336C2×D1711yescharacteristic
360#336C2×D1711yescharacteristic
361#336C2×D1711yescharacteristic
362C2×C17#229011yescharacteristic
363C2×C17#294611yescharacteristic
364C2×C17#294711yescharacteristic
365C2×C17#311811yescharacteristic
366C2×C17#312011yescharacteristic
367C2×C17#312311yescharacteristic
368C2×C17#312411yescharacteristic
369D17—44no—
370C22×Q8—171no—
371C22×D4—171no—
372C23×C4—341no—
373C23×C4—341no—
374C23×C4—341no—
375C2×SD16—174no—
376C2×SD16—342no—
377C2×SD16—342no—
378C22×C8C22×D1711yescharacteristic
379C22×C8C22×D1711yescharacteristic
380C22×C8C22×D1711yescharacteristic
381C22×C8C22×D1711yescharacteristic
382C4×Q8—174no—
383C4×Q8—342no—
384C4×D4—174no—
385C4×D4—342no—
386C4×D4—342no—
387C4×D4—342no—
388C4×D4—342no—
389C4×D4—342no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—341no—
402C2×C42—341no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42—341no—
407C2×C42C22×D1711yescharacteristic
408C2×C42C22×D1711yescharacteristic
409C2×C42C22×D1711yescharacteristic
410#107—174no—
411#107—342no—
412#106—342no—
413#103—174no—
414#103—342no—
415#103—342no—
416#102—174no—
417#102—342no—
418#102—342no—
419#97C4×D1714yescomplemented
420C4×C8—22no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95D6811yescharacteristic
432#95D6811yescharacteristic
433#95D6811yescharacteristic
434#95D6811yescharacteristic
435C17#2112511yescharacteristic, complemented, Sylow
436C24—341no—
437C2×Q8—172no—
438C2×Q8—341no—
439C2×Q8—341no—
440C2×Q8—341no—
441C2×Q8—341no—
442C2×Q8—341no—
443C2×D4—172no—
444C2×D4—172no—
445C2×D4—172no—
446C2×D4—341no—
447C2×D4—341no—
448C2×D4—341no—
449C22×C4—344no—
450C22×C4—344no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—344no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—1361no—
460C22×C4—1361no—
461C22×C4—1361no—
462C22×C4—1361no—
463C22×C4—1361no—
464C22×C4—1361no—
465C22×C4C2×D6811yescharacteristic
466C22×C4C2×D6811yescharacteristic
467C22×C4C2×D6811yescharacteristic
468C22×C4C2×D6811yescharacteristic
469C22×C4C2×D6811yescharacteristic
470C22×C4C2×D6811yescharacteristic
471C22×C4C23×D1711yescharacteristic
472SD16—348no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8C2×C4×D1712yes—
484C2×C8C2×C4×D1712yes—
485C4⋊C4—172no—
486C4⋊C4—172no—
487C4⋊C4—172no—
488C4⋊C4—172no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—344no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—1361no—
499C4⋊C4—1361no—
500C22⋊C4—344no—
501C22⋊C4—344no—
502C22⋊C4—344no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—344no—
508C42—1361no—
509C42—1361no—
510C42—1361no—
511C42—1361no—
512C42C2×C4×D1712yes—
513C42#6325212yes—
514C23—342no—
515C23—342no—
516C23—342no—
517C23—681no—
518C23—681no—
519C23—681no—
520C23—681no—
521C23C22×D6811yescharacteristic
522Q8—342no—
523Q8—342no—
524Q8—681no—
525Q8—681no—
526D4—174no—
527D4—342no—
528D4—342no—
529D4—342no—
530C2×C4—21no—
531C2×C4—21no—
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—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—342no—
547C2×C4—342no—
548C2×C4—342no—
549C2×C4—342no—
550C2×C4—342no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—1364no—
568C2×C4—1364no—
569C2×C4—1364no—
570C2×C4C4×D6812yes—
571C2×C4C4×D6812yes—
572C2×C4C22×C4×D1711yescharacteristic
573C2×C4C2×#6325211yescharacteristic
574C2×C4#41×D1711yescharacteristic
575C2×C4#41×D1711yescharacteristic
576C2×C4#41×D1711yescharacteristic
577C2×C4#41×D1711yescharacteristic
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C22—344no—
583C22—682no—
584C22—682no—
585C22—682no—
586C22—682no—
587C22—682no—
588C22—682no—
589C22C2×C4×D6811yescharacteristic
590C22#1176332111yescharacteristic
591C22#1176332111yescharacteristic
592C22#1176361411yescharacteristic
593C22#1176362111yescharacteristic
594C22#1176362111yescharacteristic
595C22#1176362611yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—682no—
603C4—682no—
604C4—682no—
605C4—682no—
606C4—682no—
607C4—1361no—
608C4—1361no—
609C4#117×D1711yescharacteristic
610C4#1176327711yescharacteristic
611C4#1176333711yescharacteristic
612C4#1176333711yescharacteristic
613C2—684no—
614C2#1453122811yescharacteristic
615C2#1453761811yescharacteristic
616C2#1453762911yescharacteristic
617C2#1453897011yescharacteristic
618C2#1453900011yescharacteristic
619C2#1453903711yescharacteristic
620C2#1453904811yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1746C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1746D17
hypocenterC17#21125
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
2. #14531436C2semidirect
4. #14529426C2semidirect
5. #14529419C2semidirect
6. #14529377C2semidirect
7. #14529356C2semidirect
12. #14528681C2semidirect
13. C4×#10587487C2semidirect
15. C17×#1746C2semidirect
21. #11762911C4semidirect
99. C4×Dic34C2×C4semidirect
100. C17×#97C2×C4semidirect
120. #10587489C2×C4semidirect
124. #10587487C2×C4semidirect
286. #1746D17semidirect
419. #97C4×D17semidirect
435. C17#21125semidirect

Maximal subgroups

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

subgroupindex
C2×#117629112
#145314362
#145311462
#145294262
#145294192
#145293772
#145293562
#145291172
#145290482
#145290342
#145286812
C4×#105874872
C4×#105874922
C17×#17462
#2112517