Finite Group Databasebeta

← #17042445

Subgroups — #17042445

Order 4352.

826 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×#11763171C211yesmaximal, characteristic
2#14531495C211yesmaximal, characteristic
3#14529746C211yesmaximal, characteristic, complemented
4#14529593C211yesmaximal, characteristic
5#14529480C211yesmaximal, characteristic
6#14529473C211yesmaximal, characteristic, complemented
7#14529466C211yesmaximal, characteristic
8#14529459C211yesmaximal, characteristic, complemented
9#14528917C211yesmaximal, characteristic, complemented
10#14528617C211yesmaximal, characteristic, complemented
11#14528612C211yesmaximal, characteristic
12C4×#10587481C211yesmaximal, characteristic, complemented
13C4×#10587480C211yesmaximal, characteristic
14#14528471C211yesmaximal, characteristic, complemented
15C17×#2331C211yesmaximal, characteristic, complemented
16C2×#10587561C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19#11763171C418yescomplemented
20C2×#10587481C2211yescharacteristic
21C2×#10587481C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587480C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587474C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31C2×C4×#3622C2211yescharacteristic
32C2×C4×#3622C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762711C2211yescharacteristic
36#11762701—21no—
37#11762701—21no—
38#11762683—21no—
39#11762683—21no—
40#11762679C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49C2×C17×C4⋊Q8C2211yescharacteristic
50C17×#391—21no—
51C17×#391—21no—
52C17×#389C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#387—21no—
55C17×#387—21no—
56C17×#382C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C17×#376C2211yescharacteristic
60C22×Q8×C17—21no—
61C22×D68C2311yescharacteristic
62C22×C4×D17—21no—
63C22×Dic34C2311yescharacteristic
64C17×C4⋊Q8—22no—
65C17×C4⋊Q8C2×C414yescomplemented
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C42×C17—21no—
74C2×C42×C17—21no—
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93#10587561—22no—
94#10587561C2×C414yes—
95C4×D68—22no—
96C4×D68—22no—
97C4×Dic34—22no—
98C4×Dic34—22no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95D411yescharacteristic
106C17×#95D411yescharacteristic
107C17×#95D411yescharacteristic
108C17×#95D411yescharacteristic
109#10587502—21no—
110#10587502—21no—
111#10587481—22no—
112#10587481—22no—
113#10587481C2×C414yescomplemented
114#10587481C2×C414yescomplemented
115#10587480—22no—
116#10587480—22no—
117#10587480C2×C414yes—
118#10587480C2×C414yes—
119#10587474—22no—
120#10587474—22no—
121#10587474C2×C414yescomplemented
122C4×#3622—22no—
123C4×#3622—22no—
124C4×#3622—22no—
125C23×D17—21no—
126C2×Q8×C17—24no—
127C2×Q8×C17—81no—
128C2×Q8×C17—81no—
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×C17C2×D411yescharacteristic
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68—21no—
148C2×D68C22×C412yes—
149C2×C4×D17—42no—
150C2×C4×D17—81no—
151C2×C4×D17—81no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34C22×C412yes—
158C4⋊C4×C17—24no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—81no—
166C4⋊C4×C17—81no—
167C4⋊C4×C17C22×C412yes—
168C4⋊C4×C17C22×C412yes—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—81no—
172C42×C17—81no—
173C42×C17C22×C412yes—
174C42×C17#4112yes—
175C42×C17#4112yes—
176C42×C17#4112yes—
177#63227—24no—
178#63227—24no—
179#63227—42no—
180#63227—81no—
181#63227—81no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—81no—
187C4⋊Dic17—81no—
188#63225—42no—
189C4×Dic17—42no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C22×C412yes—
199C2×#3622C22×C412yes—
200C2×#3622#4112yes—
201C2×#3622#4112yes—
202#21654—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—
211Q8×C17—84no—
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—22no—
221C2×C4×C17—22no—
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—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17C4×D412yes—
239C2×C4×C17C4×D412yes—
240C2×C4×C17#12412yes—
241C2×C4×C17#12412yes—
242C2×C4×C17C23×C411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×Dic17—22no—
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—41no—
265Dic34—41no—
266#3622—22no—
267#3622—22no—
268#3622—22no—
269#3622—22no—
270C2×#489—171no—
271#2331D1711yescharacteristic, complemented
272#2326—171no—
273#1990—171no—
274#1958—171no—
275#1936—171no—
276#1935—171no—
277#1934—171no—
278#1933—171no—
279#1836—171no—
280#1763—171no—
281#1762—171no—
282C4×#103—171no—
283C4×#102—171no—
284#1726—171no—
285C22×C17C2×C4×D411yescharacteristic
286C22×C17C2×#12411yescharacteristic
287C22×C17#53211yescharacteristic
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—22no—
302C4×C17—22no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—42no—
306C4×C17—42no—
307C4×C17—81no—
308C4×C17—81no—
309C4×C17C4×#4111yescharacteristic
310C4×C17C4×#4111yescharacteristic
311C4×C17#52911yescharacteristic
312C4×C17#52911yescharacteristic
313Dic17—42no—
314Dic17—81no—
315Dic17—81no—
316C2×C4⋊Q8C2×D1711yescharacteristic
317C2×#124—171no—
318C2×C4×Q8—171no—
319C2×C4×D4—171no—
320#489—178no—
321C2×#103—171no—
322C2×#103—171no—
323C2×#103—171no—
324C2×#103—171no—
325C2×#102—171no—
326C2×#102—171no—
327C2×#102—171no—
328C2×#102—171no—
329C2×#97—171no—
330C2×#97—171no—
331C2×#97—171no—
332C2×C4×C8—171no—
333C2×C4×C8—171no—
334C2×C4×C8—171no—
335#391—21no—
336#391—21no—
337#391—171no—
338#390—171no—
339#389C2×D1711yescharacteristic
340#389C2×D1711yescharacteristic
341#388—341no—
342#388—341no—
343#387—21no—
344#387—21no—
345#385—341no—
346#385—341no—
347#384—171no—
348#382C2×D1711yescharacteristic
349C4×C4⋊C4C2×D1711yescharacteristic
350C4×C4⋊C4C2×D1711yescharacteristic
351#376C2×D1711yescharacteristic
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×C17C4×#12411yescharacteristic
361C2×C17#294611yescharacteristic
362C2×C17#294711yescharacteristic
363C2×C17#313611yescharacteristic
364C2×C17#313711yescharacteristic
365C2×C17#317311yescharacteristic
366C2×C17#317511yescharacteristic
367D17—82no—
368C22×Q8—21no—
369C22×Q8—171no—
370C22×D4—171no—
371C23×C4—341no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8—171no—
375C22×C8—171no—
376C4⋊Q8—22no—
377C4⋊Q8C4×D1714yescomplemented
378#124—174no—
379#124—342no—
380C4×Q8—342no—
381C4×Q8—342no—
382C4×D4—342no—
383C4×D4—342no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C4⋊C4C22×D1711yescharacteristic
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—341no—
404C2×C42C22×D1711yescharacteristic
405C2×C42C22×D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407#103—174no—
408#103—174no—
409#103—342no—
410#103—342no—
411#102—174no—
412#102—174no—
413#102—342no—
414#102—342no—
415#97—174no—
416#97—342no—
417#97—342no—
418C4×C8—342no—
419C4×C8—342no—
420C4×C8—342no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—341no—
428#95—341no—
429#95C22⋊D1711yescharacteristic
430#95C22⋊D1711yescharacteristic
431#95C22⋊D1711yescharacteristic
432#95C22⋊D1711yescharacteristic
433C17#2165411yescharacteristic, complemented, Sylow
434C24—341no—
435C2×Q8—24no—
436C2×Q8—81no—
437C2×Q8—81no—
438C2×Q8—172no—
439C2×Q8—341no—
440C2×Q8—341no—
441C2×Q8—341no—
442C2×Q8—341no—
443C2×Q8—341no—
444C2×D4—172no—
445C2×D4—341no—
446C2×D4—341no—
447C2×D4—341no—
448C2×D4—341no—
449C2×D4—341no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—682no—
458C22×C4—1361no—
459C22×C4—1361no—
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C23×D1711yescharacteristic
467C2×C8—172no—
468C2×C8—172no—
469C2×C8—172no—
470C2×C8—172no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—682no—
493C4⋊C4—1361no—
494C4⋊C4—1361no—
495C4⋊C4C2×C4×D1712yes—
496C4⋊C4C2×C4×D1712yes—
497C22⋊C4—344no—
498C22⋊C4—344no—
499C22⋊C4—682no—
500C22⋊C4—1361no—
501C22⋊C4—1361no—
502C42—21no—
503C42—21no—
504C42—81no—
505C42—81no—
506C42—682no—
507C42C2×C4×D1712yes—
508C42#6325212yes—
509C42#6325612yes—
510C42#6325612yes—
511C23—342no—
512C23—681no—
513C23—681no—
514C23—681no—
515C23—681no—
516C23—681no—
517C23—681no—
518C23C22×C22⋊D1711yescharacteristic
519Q8—84no—
520Q8—342no—
521Q8—342no—
522Q8—681no—
523Q8—681no—
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—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—342no—
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—1364no—
564C2×C4C4×C22⋊D1712yes—
565C2×C4C4×C22⋊D1712yes—
566C2×C4#1058763312yes—
567C2×C4#1058763312yes—
568C2×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325211yescharacteristic
570C2×C4C2×#6325611yescharacteristic
571C2×C4C2×#6325611yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C8—342no—
575C8—342no—
576C8—342no—
577C8—342no—
578C22—682no—
579C22—682no—
580C22—682no—
581C22—1361no—
582C22—1361no—
583C22—1361no—
584C22—1361no—
585C22C2×C4×C22⋊D1711yescharacteristic
586C22C2×#1058763311yescharacteristic
587C22#1176335511yescharacteristic
588C22#1176361211yescharacteristic
589C22#1176361711yescharacteristic
590C22#1176362011yescharacteristic
591C22#1176362411yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—682no—
603C4—1361no—
604C4—1361no—
605C4C4×#6325611yescharacteristic
606C4C4×#6325611yescharacteristic
607C4#1176334611yescharacteristic
608C4#1176334611yescharacteristic
609C2—1362no—
610C2C4×#1058763311yescharacteristic
611C2#1453761611yescharacteristic
612C2#1453762711yescharacteristic
613C2#1453914311yescharacteristic
614C2#1453915411yescharacteristic
615C2#1453939511yescharacteristic
616C2#1453941711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2331C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#2331D17
hypocenterC17#21654
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #14529746C2semidirect
6. #14529473C2semidirect
8. #14529459C2semidirect
9. #14528917C2semidirect
10. #14528617C2semidirect
12. C4×#10587481C2semidirect
14. #14528471C2semidirect
15. C17×#2331C2semidirect
19. #11763171C4semidirect
65. C17×C4⋊Q8C2×C4semidirect
113. #10587481C2×C4semidirect
114. #10587481C2×C4semidirect
121. #10587474C2×C4semidirect
271. #2331D17semidirect
377. C4⋊Q8C4×D17semidirect
433. C17#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#117631712
#145314952
#145297462
#145295932
#145294802
#145294732
#145294662
#145294592
#145289172
#145286172
#145286122
C4×#105874812
C4×#105874802
#145284712
C17×#23312
#2165417