Finite Group Databasebeta

← #17054199

Subgroups — #17054199

Order 4352.

838 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762911C211yesmaximal, characteristic
2#14531747C211yesmaximal, characteristic, complemented
3#14531731C211yesmaximal, characteristic
4#14529665C211yesmaximal, characteristic
5#14529638C211yesmaximal, characteristic
6#14529426C211yesmaximal, characteristic, complemented
7#14529419C211yesmaximal, characteristic, complemented
8#14529412C211yesmaximal, characteristic
9#14529405C211yesmaximal, characteristic
10#14528952C211yesmaximal, characteristic, complemented
11#14528795C211yesmaximal, characteristic
12#14528790C211yesmaximal, characteristic, complemented
13#14528681C211yesmaximal, characteristic, complemented
14C4×#10587489C211yesmaximal, characteristic, complemented
15C17×#1771C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587578—21no—
20#11762911C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31C2×#10587492C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762715C2211yescharacteristic
34#11762711C2211yescharacteristic
35#11762711C2211yescharacteristic
36#11762701—21no—
37#11762701—21no—
38#11762690—21no—
39#11762690—21no—
40#11762679C2211yescharacteristic
41#11762676—21no—
42#11762675C2211yescharacteristic
43#11762666—21no—
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47C4×C4⋊Dic17C2211yescharacteristic
48#11762654—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×#97C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
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×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×#63219—22no—
92C2×#63219—22no—
93C2×#63219C2×C414yes—
94#10587578—44no—
95C4×D68—22no—
96C4×D68C2×C414yes—
97C4×Dic34—22no—
98C4×Dic34C2×C414yes—
99C17×C4⋊C8—22no—
100C17×#97—22no—
101C17×#97C2×C414yes—
102C4×C8×C17—22no—
103C17×#95D411yescharacteristic
104C17×#95D411yescharacteristic
105#10587502—21no—
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#10587492—41no—
116#10587492—41no—
117#10587492C2×C414yes—
118#10587489—22no—
119#10587489—41no—
120#10587489C2×C414yes—
121#10587488—22no—
122#10587488—41no—
123#10587487—41no—
124#10587487—41no—
125#10587487C2×C414yes—
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×D68—21no—
145C2×D68—21no—
146C2×D68C22×C412yes—
147C2×C4×D17—24no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34—21no—
155C2×Dic34C22×C412yes—
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×C17C22×C412yes—
165C2×C8×C17C22×C412yes—
166C2×C8×C17#4112yes—
167C2×C8×C17#4112yes—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—41no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17C22×C412yes—
175C42×C17#4112yes—
176#63227—24no—
177#63227—42no—
178#63227—42no—
179#63227—81no—
180#63227—81no—
181C4⋊Dic17—21no—
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⋊Dic17—21no—
191C4⋊Dic17—42no—
192C4⋊Dic17—81no—
193C4⋊Dic17C22×C412yes—
194C4⋊Dic17C22×C412yes—
195#63225—24no—
196C4×Dic17—24no—
197C4×Dic17—42no—
198C4×Dic17—42no—
199C4×Dic17—42no—
200C4×Dic17—42no—
201#63219—28no—
202#22820—171nomaximal, Sylow
203C23×C17C22×D411yescharacteristic
204C22×D17—22no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—41no—
211C2×C4×C17—21no—
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×C17C4×D412yes—
227C2×C4×C17C4×D412yes—
228C2×C4×C17AGL(1,ℤ/8)12yes—
229C2×C4×C17C23×C411yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17H(5,𝔽2)11yescharacteristic
234C2×C4×C172−1+411yescharacteristic
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
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—
256D68—22no—
257D68—22no—
258D68—41no—
259D68—41no—
260C4×D17—48no—
261Dic34—22no—
262Dic34—22no—
263Dic34—41no—
264Dic34—41no—
265C8×C17—22no—
266C8×C17—41no—
267C8×C17—41no—
268C8×C17#11714yescomplemented
269C2×#440—171no—
270#2356—171no—
271#2354—171no—
272#1973—171no—
273#1968—171no—
274#1928—171no—
275#1927—171no—
276#1926—171no—
277#1925—171no—
278#1847—171no—
279#1804—171no—
280#1803—171no—
281#1776—171no—
282C4×#107—171no—
283#1771D1711yescharacteristic, complemented
284C22×C17C2×C4×D411yescharacteristic
285C22×C17#54711yescharacteristic
286C22×C17#55011yescharacteristic
287C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C17#57511yescharacteristic
289C22×C17#57711yescharacteristic
290C22×C17#57811yescharacteristic
291C2×D17—42no—
292C2×D17—42no—
293C2×D17—42no—
294C2×D17—81no—
295C2×D17—81no—
296C2×D17—81no—
297C2×D17—81no—
298C4×C17—22no—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17C2×#11711yescharacteristic
305C4×C17#52011yescharacteristic
306C4×C17#55511yescharacteristic
307C4×C17#55611yescharacteristic
308Dic17—42no—
309Dic17—42no—
310Dic17—42no—
311Dic17—81no—
312Dic17—81no—
313Dic17—81no—
314Dic17—81no—
315C22×SD16—171no—
316C2×C4×Q8—171no—
317C2×C4×D4—171no—
318C2×#117—341no—
319#440—178no—
320C2×#107—171no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#106—171no—
324C2×#106—171no—
325C2×C4⋊C8C2×D1711yescharacteristic
326C2×#103—171no—
327C2×#103—171no—
328C2×#103—171no—
329C2×#102—171no—
330C2×#102—171no—
331C2×#102—171no—
332C2×#97C2×D1711yescharacteristic
333C2×#97C2×D1711yescharacteristic
334C2×C4×C8C2×D1711yescharacteristic
335#391—171no—
336#391—171no—
337#390—171no—
338#390—171no—
339#388—341no—
340#388—341no—
341#387—341no—
342#387—341no—
343#384—171no—
344#383—341no—
345#382—171no—
346#380—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351C4×C4⋊C4C2×D1711yescharacteristic
352#376—341no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#336C2×D1711yescharacteristic
358#336C2×D1711yescharacteristic
359C2×C17#235411yescharacteristic
360C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
361C2×C17#294711yescharacteristic
362C2×C17#334311yescharacteristic
363C2×C17#334811yescharacteristic
364C2×C17#339711yescharacteristic
365C2×C17#340211yescharacteristic
366D17—82no—
367C22×Q8—171no—
368C22×D4—171no—
369C23×C4—341no—
370C2×SD16—174no—
371C2×SD16—342no—
372C2×SD16—342no—
373C22×C8C22×D1711yescharacteristic
374C22×C8C22×D1711yescharacteristic
375C22×C8C22×D1711yescharacteristic
376C22×C8C22×D1711yescharacteristic
377C4×Q8—174no—
378C4×Q8—342no—
379C4×D4—174no—
380C4×D4—342no—
381#117—684no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4D6811yescharacteristic
392C2×C4⋊C4D6811yescharacteristic
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42—341no—
401C2×C42—341no—
402C2×C42—341no—
403C2×C42C22×D1711yescharacteristic
404C2×C42C22×D1711yescharacteristic
405C2×C42C22×D1711yescharacteristic
406#107—174no—
407#107—342no—
408#107—681no—
409#106—342no—
410#106—681no—
411C4⋊C8—22no—
412#103—174no—
413#103—681no—
414#103—681no—
415#102—174no—
416#102—681no—
417#102—681no—
418#97—22no—
419#97C4×D1714yes—
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
433C17#2282011yescharacteristic, complemented, Sylow
434C24—341no—
435C2×Q8—172no—
436C2×Q8—341no—
437C2×Q8—341no—
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—344no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—682no—
456C22×C4—682no—
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×C8C2×C4×D1712yes—
474C2×C8C2×C4×D1712yes—
475C2×C8#6325412yes—
476C2×C8#6325612yes—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—172no—
480C4⋊C4—172no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—344no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—682no—
493C4⋊C4—1361no—
494C22⋊C4—344no—
495C22⋊C4—682no—
496C22⋊C4—682no—
497C22⋊C4—1361no—
498C22⋊C4—1361no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—344no—
504C42—682no—
505C42—682no—
506C42—682no—
507C42—682no—
508C42C2×C4×D1712yes—
509C42#6325212yes—
510C23—342no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23—681no—
515C23—681no—
516C23—681no—
517C23C22×D6811yescharacteristic
518Q8—342no—
519Q8—342no—
520Q8—681no—
521Q8—681no—
522D4—342no—
523D4—342no—
524D4—681no—
525D4—681no—
526C2×C4—21no—
527C2×C4—21no—
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—342no—
542C2×C4—342no—
543C2×C4—342no—
544C2×C4—688no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
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×C4C4×D6812yes—
564C2×C4C4×D6812yes—
565C2×C4#1058759412yes—
566C2×C4C22×C4×D1711yescharacteristic
567C2×C4C2×#6325211yescharacteristic
568C2×C4C2×#6325411yescharacteristic
569C2×C4C2×#6325611yescharacteristic
570C2×C4#1058769611yescharacteristic
571C2×C4#1058769711yescharacteristic
572C8—22no—
573C8—41no—
574C8—41no—
575C8#1058757814yescomplemented
576C22—682no—
577C22—682no—
578C22—682no—
579C22—1361no—
580C22—1361no—
581C22—1361no—
582C22—1361no—
583C22C2×C4×D6811yescharacteristic
584C22#1176346311yescharacteristic
585C22#1176348111yescharacteristic
586C22C2×#1058759411yescharacteristic
587C22#1176360911yescharacteristic
588C22#1176362111yescharacteristic
589C22#1176362611yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—682no—
597C4—682no—
598C4—682no—
599C4—1361no—
600C4—1361no—
601C4—1361no—
602C4—1361no—
603C4C2×#1058757811yescharacteristic
604C4#1176328911yescharacteristic
605C4#1176352411yescharacteristic
606C4#1176353211yescharacteristic
607C2—1362no—
608C2#1453173111yescharacteristic
609C2C4×#1058759411yescharacteristic
610C2#1453762911yescharacteristic
611C2#1454122611yescharacteristic
612C2#1454130111yescharacteristic
613C2#1454180611yescharacteristic
614C2#1454186511yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1771C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1771D17
hypocenterC17#22820
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #14531747C2semidirect
6. #14529426C2semidirect
7. #14529419C2semidirect
10. #14528952C2semidirect
12. #14528790C2semidirect
13. #14528681C2semidirect
14. C4×#10587489C2semidirect
15. C17×#1771C2semidirect
20. #11762911C4semidirect
268. C8×C17#117semidirect
283. #1771D17semidirect
433. C17#22820semidirect
575. C8#10587578semidirect

Maximal subgroups

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

subgroupindex
C2×#117629112
#145317472
#145317312
#145296652
#145296382
#145294262
#145294192
#145294122
#145294052
#145289522
#145287952
#145287902
#145286812
C4×#105874892
C17×#17712
#2282017