Finite Group Databasebeta

← #17037447

Subgroups — #17037447

Order 4352.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762911C211yesmaximal, characteristic
2#14531428C211yesmaximal, characteristic
3#14531147C211yesmaximal, characteristic, complemented
4#14529412C211yesmaximal, characteristic
5#14529405C211yesmaximal, characteristic
6#14529377C211yesmaximal, characteristic, complemented
7#14529356C211yesmaximal, characteristic, complemented
8#14529138C211yesmaximal, characteristic
9#14529138C211yesmaximal, characteristic
10#14529055C211yesmaximal, characteristic, complemented
11#14529041C211yesmaximal, characteristic, complemented
12#14528681C211yesmaximal, characteristic, complemented
13C4×#10587487C211yesmaximal, characteristic, complemented
14C4×#10587492C211yesmaximal, characteristic
15C17×#1746C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21#11762911C418yescomplemented
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587492C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31#11762711C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762698—21no—
36#11762698—21no—
37#11762671—21no—
38#11762668—21no—
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43#11762660—21no—
44#11762660—21no—
45#11762660—21no—
46#11762654—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×Dic34C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C42×C17C2311yescharacteristic
68C2×C42×C17C2311yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×#63227—21no—
71C2×#63227—21no—
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×#63225—21no—
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—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×D68C2×C414yes—
98C4×Dic34—22no—
99C4×Dic34—22no—
100C4×Dic34—22no—
101C4×Dic34—22no—
102C4×Dic34—22no—
103C4×Dic34C2×C414yes—
104C17×#97C2×C414yes—
105C4×C8×C17—22no—
106C17×#95D411yescharacteristic
107C17×#95D411yescharacteristic
108C17×#95D411yescharacteristic
109C17×#95D411yescharacteristic
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×C414yes—
121#10587488—22no—
122#10587487—22no—
123#10587487—22no—
124#10587487C2×C414yes—
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×D68—21no—
144C2×D68—21no—
145C2×D68C22×C412yes—
146C2×C4×D17—24no—
147C2×C4×D17—81no—
148C2×C4×D17—81no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34C22×C412yes—
153C2×Dic34#4112yes—
154C2×Dic34#4112yes—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17C22×C412yes—
166C2×C8×C17C22×C412yes—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17C22×C412yes—
172C42×C17#4112yes—
173#63227—24no—
174#63227—81no—
175#63227—81no—
176#63227—81no—
177#63227—81no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17C22×C412yes—
187C4⋊Dic17C22×C412yes—
188C4⋊Dic17#4112yes—
189C4⋊Dic17#4112yes—
190#63225—24no—
191#63225—24no—
192#63225—24no—
193#63225—81no—
194#63225—81no—
195C4×Dic17—24no—
196C4×Dic17—24no—
197C4×Dic17—24no—
198C4×Dic17—81no—
199C4×Dic17—81no—
200C4×Dic17—81no—
201C4×Dic17—81no—
202#63219—28no—
203#21128—171nomaximal, Sylow
204C23×C17C22×D411yescharacteristic
205C22×D17—22no—
206C22×D17—41no—
207C22×D17—41no—
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—22no—
242C2×Dic17—22no—
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—41no—
260D68—41no—
261C4×D17—84no—
262Dic34—22no—
263Dic34—22no—
264Dic34—22no—
265Dic34#11714yes—
266C8×C17—22no—
267C8×C17—22no—
268C8×C17—41no—
269C8×C17—41no—
270C2×#440—171no—
271#2319—171no—
272#2276—171no—
273#1926—171no—
274#1925—171no—
275#1921—171no—
276#1918—171no—
277#1880—171no—
278#1880—171no—
279#1865—171no—
280#1863—171no—
281#1776—171no—
282C4×#103—171no—
283C4×#102—171no—
284#1746D1711yescharacteristic, complemented
285C22×C17C2×C4×D411yescharacteristic
286C22×C17#52511yescharacteristic
287C22×C17#52511yescharacteristic
288C22×C17#57611yescharacteristic
289C22×C17#57711yescharacteristic
290C22×C17#57711yescharacteristic
291C22×C17#57811yescharacteristic
292C2×D17—42no—
293C2×D17—42no—
294C2×D17—42no—
295C2×D17—81no—
296C2×D17—81no—
297C2×D17—81no—
298C2×D17—81no—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17C2×#11711yescharacteristic
306C4×C17C4×#4111yescharacteristic
307C4×C17#52911yescharacteristic
308C4×C17#52911yescharacteristic
309Dic17—24no—
310Dic17—42no—
311Dic17—42no—
312Dic17—42no—
313Dic17—42no—
314Dic17—81no—
315Dic17—81no—
316C22×SD16—171no—
317C2×C4×Q8—171no—
318C2×C4×Q8—171no—
319C2×C4×Q8—171no—
320C2×C4×D4—171no—
321#440—178no—
322C2×#107—171no—
323C2×#107—171no—
324C2×#106—171no—
325C2×#103—171no—
326C2×#103—171no—
327C2×#103—171no—
328C2×#102—171no—
329C2×#102—171no—
330C2×#102—171no—
331C2×#97C2×D1711yescharacteristic
332C2×C4×C8C2×D1711yescharacteristic
333#390—171no—
334#390—171no—
335#389—171no—
336#389—171no—
337#388—341no—
338#388—341no—
339#381—341no—
340#380—341no—
341C4×C4⋊C4—171no—
342C4×C4⋊C4—171no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—341no—
346C4×C4⋊C4—341no—
347C4×C4⋊C4—341no—
348#376—341no—
349#376C2×D1711yescharacteristic
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#336C2×D1711yescharacteristic
357#336C2×D1711yescharacteristic
358#336C2×D1711yescharacteristic
359#336C2×D1711yescharacteristic
360C2×C17#229011yescharacteristic
361C2×C17#294611yescharacteristic
362C2×C17#294711yescharacteristic
363C2×C17#311911yescharacteristic
364C2×C17#312111yescharacteristic
365C2×C17#312411yescharacteristic
366C2×C17#312611yescharacteristic
367D17—82no—
368C22×Q8—171no—
369C22×D4—171no—
370C23×C4—341no—
371C2×SD16—174no—
372C2×SD16—342no—
373C2×SD16—342no—
374C22×C8C22×D1711yescharacteristic
375C22×C8C22×D1711yescharacteristic
376C22×C8C22×D1711yescharacteristic
377C22×C8C22×D1711yescharacteristic
378C4×Q8—174no—
379C4×Q8—342no—
380C4×Q8—342no—
381C4×Q8—342no—
382C4×Q8—342no—
383C4×Q8—342no—
384C4×D4—174no—
385C4×D4—342no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C22⋊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×C42—341no—
408C2×C42—341no—
409C2×C42C22×D1711yescharacteristic
410C2×C42C22×D1711yescharacteristic
411C2×C42C22×D1711yescharacteristic
412#107—174no—
413#107—342no—
414#106—342no—
415#103—174no—
416#103—342no—
417#103—342no—
418#102—174no—
419#102—342no—
420#102—342no—
421#97C4×D1714yes—
422C4×C8—22no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95D6811yescharacteristic
430#95D6811yescharacteristic
431#95D6811yescharacteristic
432#95D6811yescharacteristic
433C17#2112811yescharacteristic, complemented, Sylow
434C24—341no—
435C2×Q8—172no—
436C2×Q8—172no—
437C2×Q8—172no—
438C2×Q8—341no—
439C2×Q8—341no—
440C2×Q8—341no—
441C2×D4—172no—
442C2×D4—341no—
443C2×D4—341no—
444C2×D4—341no—
445C2×D4—341no—
446C2×D4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—344no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—1361no—
456C22×C4—1361no—
457C22×C4C2×D6811yescharacteristic
458C22×C4C2×D6811yescharacteristic
459C22×C4C2×D6811yescharacteristic
460C22×C4C2×D6811yescharacteristic
461C22×C4C2×D6811yescharacteristic
462C22×C4C2×D6811yescharacteristic
463C22×C4C23×D1711yescharacteristic
464SD16—348no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8C2×C4×D1712yes—
476C2×C8C2×C4×D1712yes—
477C4⋊C4—172no—
478C4⋊C4—172no—
479C4⋊C4—172no—
480C4⋊C4—172no—
481C4⋊C4—344no—
482C4⋊C4—344no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—344no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—1361no—
493C4⋊C4—1361no—
494C22⋊C4—344no—
495C22⋊C4—1361no—
496C22⋊C4—1361no—
497C22⋊C4—1361no—
498C22⋊C4—1361no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—344no—
504C42—344no—
505C42—344no—
506C42—1361no—
507C42—1361no—
508C42—1361no—
509C42—1361no—
510C42C2×C4×D1712yes—
511C42#6325212yes—
512C23—342no—
513C23—681no—
514C23—681no—
515C23—681no—
516C23—681no—
517C23—681no—
518C23—681no—
519C23C22×D6811yescharacteristic
520Q8—174no—
521Q8—342no—
522Q8—342no—
523Q8—342no—
524D4—342no—
525D4—342no—
526D4—681no—
527D4—681no—
528C2×C4—21no—
529C2×C4—21no—
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—342no—
545C2×C4—342no—
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—1364no—
566C2×C4C4×D6812yes—
567C2×C4C4×D6812yes—
568C2×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325211yescharacteristic
570C2×C4#41×D1711yescharacteristic
571C2×C4#41×D1711yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C8—22no—
575C8—22no—
576C8—41no—
577C8—41no—
578C22—682no—
579C22—682no—
580C22—682no—
581C22—1361no—
582C22—1361no—
583C22—1361no—
584C22—1361no—
585C22C2×C4×D6811yescharacteristic
586C22#1176332111yescharacteristic
587C22#1176332111yescharacteristic
588C22#1176361411yescharacteristic
589C22#1176362111yescharacteristic
590C22#1176362111yescharacteristic
591C22#1176362611yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—344no—
599C4—682no—
600C4—682no—
601C4—682no—
602C4—682no—
603C4—1361no—
604C4—1361no—
605C4#117×D1711yescharacteristic
606C4#1176327711yescharacteristic
607C4#1176333711yescharacteristic
608C4#1176333711yescharacteristic
609C2—1362no—
610C2#1453122811yescharacteristic
611C2#1453761811yescharacteristic
612C2#1453762911yescharacteristic
613C2#1453898511yescharacteristic
614C2#1453901511yescharacteristic
615C2#1453904811yescharacteristic
616C2#1453907611yescharacteristic
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#21128
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
3. #14531147C2semidirect
6. #14529377C2semidirect
7. #14529356C2semidirect
10. #14529055C2semidirect
11. #14529041C2semidirect
12. #14528681C2semidirect
13. C4×#10587487C2semidirect
15. C17×#1746C2semidirect
21. #11762911C4semidirect
284. #1746D17semidirect
433. C17#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#117629112
#145314282
#145311472
#145294122
#145294052
#145293772
#145293562
#145291382
#145290552
#145290412
#145286812
C4×#105874872
C4×#105874922
C17×#17462
#2112817