Finite Group Databasebeta

← #17075685

Subgroups — #17075685

Order 4352.

746 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×#11763105C211yesmaximal, characteristic, complemented
2#14534605C211yesmaximal, characteristic
3#14531466C211yesmaximal, characteristic
4#14530212C211yesmaximal, characteristic
5#14530207C211yesmaximal, characteristic
6#14529988C211yesmaximal, characteristic
7#14529981C211yesmaximal, characteristic
8#14529735C211yesmaximal, characteristic, complemented
9#14529554C211yesmaximal, characteristic, complemented
10#14529527C211yesmaximal, characteristic, complemented
11#14529517C211yesmaximal, characteristic, complemented
12#14529474C211yesmaximal, characteristic, complemented
13#14529460C211yesmaximal, characteristic, complemented
14#14528800C211yesmaximal, characteristic, complemented
15C2×C17×#433C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19#11763105—24no—
20C2×#10587489C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587489C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28#11762748—21no—
29#11762745—21no—
30#11762729—21no—
31#11762721—21no—
32#11762715C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762704—21no—
37#11762703—21no—
38#11762697—21no—
39#11762690—21no—
40#11762675C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762669C2211yescharacteristic
43#11762669C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C2×C17×#117C2211yescharacteristic
54C17×#433—24no—
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C2×C17×C22⋊C8C2211yescharacteristic
59C2×C17×#97C2211yescharacteristic
60C2×C4×C8×C17C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C22×Dic34—21no—
63C22×Dic34C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C17×#117—22no—
69C17×#117—22no—
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17D411yescharacteristic
72C2×C22⋊C4×C17D411yescharacteristic
73C2×C22⋊C4×C17D411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88#10587610—22no—
89#10587610—22no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97#10587555—22no—
98#10587555—22no—
99C4×Dic34—22no—
100C4×Dic34—22no—
101C17×C4⋊C8—22no—
102C17×C4⋊C8D414yes—
103C17×C22⋊C8—22no—
104C17×C22⋊C8—22no—
105C17×#97—22no—
106C4×C8×C17—22no—
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#10587502—21no—
118#10587489—22no—
119#10587489—22no—
120#10587489—22no—
121#10587488—22no—
122#10587487—22no—
123#10587487—22no—
124#10587487—22no—
125#10587487—22no—
126C24×C17C2×D411yescharacteristic
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—42no—
153C2×Dic34—81no—
154C2×Dic34—81no—
155C2×Dic34#4112yes—
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×C17—21no—
166C2×C8×C17C2×D412yes—
167C2×C8×C17C2×D412yes—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C22⋊C4×C17—22no—
171C22⋊C4×C17—22no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17C2×D412yes—
175C42×C17#4112yes—
176C22⋊Dic17—24no—
177C22⋊Dic17—81no—
178C22⋊Dic17—81no—
179C22⋊Dic17—81no—
180C4⋊Dic17—21no—
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—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—42no—
194C4⋊Dic17#4112yes—
195#63225—24no—
196#63225—42no—
197#63225—81no—
198#63225—81no—
199#63225—81no—
200C4×Dic17—42no—
201C4×Dic17—81no—
202#24714—171nomaximal, Sylow
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17C22×D411yescharacteristic
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—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17#12112yes—
229C2×C4×C17#12112yes—
230C2×C4×C17C22×D411yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C172−1+411yescharacteristic
235C2×C4×C172−1+411yescharacteristic
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—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260Dic34—22no—
261Dic34—22no—
262Dic34—41no—
263Dic34—41no—
264Dic34—84no—
265C8×C17—22no—
266C8×C17—22no—
267C8×C17—41no—
268C8×C17—41no—
269C2×#477—171no—
270C2×#433D1711yescharacteristic
271#2628—171no—
272#2323—171no—
273#2076—171no—
274#2075—171no—
275#2036—171no—
276#2035—171no—
277#1988—171no—
278#1949—171no—
279#1944—171no—
280#1942—171no—
281#1935—171no—
282#1933—171no—
283#1805—171no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17C2×#12111yescharacteristic
292C22×C17#52511yescharacteristic
293C22×C17#53611yescharacteristic
294C22×C17#57611yescharacteristic
295C22×C17#57711yescharacteristic
296C22×C17#57811yescharacteristic
297C22×C17#57811yescharacteristic
298C4×C17—22no—
299C4×C17—22no—
300C4×C17—41no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17#54711yescharacteristic
304C4×C17#54811yescharacteristic
305C4×C17#55211yescharacteristic
306C4×C17#55611yescharacteristic
307Dic17—42no—
308Dic17—42no—
309Dic17—81no—
310Dic17—81no—
311Dic17—81no—
312Dic17—81no—
313Dic17—81no—
314Dic17—81no—
315C2×C4⋊Q8—171no—
316C2×C22⋊Q8—171no—
317C2×C4×Q8—171no—
318C2×#117C2×D1711yescharacteristic
319#477—344no—
320#433—24no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#107—171no—
324C2×#106—171no—
325C2×C4⋊C8C2×D1711yescharacteristic
326C2×C4⋊C8C2×D1711yescharacteristic
327C2×#103—171no—
328C2×#103—171no—
329C2×#103—171no—
330C2×#103—171no—
331C2×C22⋊C8C2×D1711yescharacteristic
332C2×C22⋊C8C2×D1711yescharacteristic
333C2×#97C2×D1711yescharacteristic
334C2×C4×C8C2×D1711yescharacteristic
335#400—341no—
336#399—341no—
337#395—341no—
338#393—341no—
339#391—171no—
340#389—171no—
341#389—171no—
342#389—171no—
343#389—341no—
344#388—341no—
345#388—341no—
346#387—341no—
347#382—171no—
348#380—171no—
349#380—171no—
350#380—171no—
351C4×C4⋊C4—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360C2×C17—81no—
361C2×C17#241511yescharacteristic
362C2×C17#305911yescharacteristic
363C2×C17#306211yescharacteristic
364C2×C17#312111yescharacteristic
365C2×C17#312411yescharacteristic
366C2×C17#324211yescharacteristic
367C2×C17#324311yescharacteristic
368C22×Q8—171no—
369C22×Q8—341no—
370C23×C4C22×D1711yescharacteristic
371C22×C8C22×D1711yescharacteristic
372C22×C8C22×D1711yescharacteristic
373C22×C8C22×D1711yescharacteristic
374C22×C8C22×D1711yescharacteristic
375C4⋊Q8—342no—
376C4⋊Q8—342no—
377C22⋊Q8—342no—
378C22⋊Q8—342no—
379C4×Q8—342no—
380C4×Q8—342no—
381#117—22no—
382#117—22no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4D6811yescharacteristic
397C2×C4⋊C4D6811yescharacteristic
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—341no—
402C2×C22⋊C4D6811yescharacteristic
403C2×C22⋊C4D6811yescharacteristic
404C2×C42—341no—
405C2×C42—341no—
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#107—342no—
409#107—342no—
410#107—342no—
411#106—342no—
412C4⋊C8—22no—
413C4⋊C8C22⋊D1714yes—
414#103—342no—
415#103—342no—
416#103—342no—
417#103—342no—
418C22⋊C8—22no—
419C22⋊C8—22no—
420#97—22no—
421C4×C8—22no—
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#95—341no—
432#95—341no—
433C17#2471411yescharacteristic, complemented, Sylow
434C24C2×D6811yescharacteristic
435C2×Q8—172no—
436C2×Q8—341no—
437C2×Q8—341no—
438C2×Q8—341no—
439C2×Q8—341no—
440C2×Q8—341no—
441C2×Q8—682no—
442C2×Q8—1361no—
443C2×Q8—1361no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4C2×D6811yescharacteristic
459C22×C4C2×D6811yescharacteristic
460C22×C4C2×D6811yescharacteristic
461C22×C4C2×D6811yescharacteristic
462C22×C4C2×D6811yescharacteristic
463C22×C4C23×D1711yescharacteristic
464C2×C8—21no—
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×C8C2×C22⋊D1712yes—
475C2×C8C2×C22⋊D1712yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—172no—
479C4⋊C4—344no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
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—341no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C4⋊C4—1361no—
496C4⋊C4—1361no—
497C4⋊C4—1361no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—344no—
501C22⋊C4—1361no—
502C22⋊C4—1361no—
503C22⋊C4—1361no—
504C42—21no—
505C42—21no—
506C42—682no—
507C42—1361no—
508C42#6325212yes—
509C42C2×C22⋊D1712yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×D6811yescharacteristic
518Q8—342no—
519Q8—342no—
520Q8—681no—
521Q8—681no—
522Q8—1364no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
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—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—342no—
541C2×C4—342no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
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×C4—681no—
564C2×C4#1058761612yes—
565C2×C4#1058761612yes—
566C2×C4C2×#6325211yescharacteristic
567C2×C4#41×D1711yescharacteristic
568C2×C4#41×D1711yescharacteristic
569C2×C4#1058769711yescharacteristic
570C2×C4#1058769711yescharacteristic
571C2×C4C22×C22⋊D1711yescharacteristic
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1058761611yescharacteristic
584C22#1176332111yescharacteristic
585C22#1176338511yescharacteristic
586C22#1176361411yescharacteristic
587C22#1176362111yescharacteristic
588C22#1176362611yescharacteristic
589C22#1176362611yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—682no—
596C4—682no—
597C4—1361no—
598C4—1361no—
599C4—1361no—
600C4—1361no—
601C4—1361no—
602C4—1361no—
603C4#1176345811yescharacteristic
604C4#1176346611yescharacteristic
605C4#1176349811yescharacteristic
606C4#1176352711yescharacteristic
607C2—81no—
608C2#1453220711yescharacteristic
609C2#1453843911yescharacteristic
610C2#1453846411yescharacteristic
611C2#1453901511yescharacteristic
612C2#1453904811yescharacteristic
613C2#1454005411yescharacteristic
614C2#1454006911yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#433C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC2×#433D17
hypocenterC17#24714
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#11763105C2semidirect
8. #14529735C2semidirect
9. #14529554C2semidirect
10. #14529527C2semidirect
11. #14529517C2semidirect
12. #14529474C2semidirect
13. #14529460C2semidirect
14. #14528800C2semidirect
433. C17#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#117631052
#145346052
#145314662
#145302122
#145302072
#145299882
#145299812
#145297352
#145295542
#145295272
#145295172
#145294742
#145294602
#145288002
C2×C17×#4332
#2471417