Finite Group Databasebeta

← #17042435

Subgroups — #17042435

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×#11763167C211yesmaximal, characteristic
2#14531528C211yesmaximal, characteristic, complemented
3#14531496C211yesmaximal, characteristic
4#14529743C211yesmaximal, characteristic, complemented
5#14529594C211yesmaximal, characteristic
6#14529483C211yesmaximal, characteristic
7#14529472C211yesmaximal, characteristic, complemented
8#14529464C211yesmaximal, characteristic
9#14529461C211yesmaximal, characteristic, complemented
10#14528916C211yesmaximal, characteristic, complemented
11#14528617C211yesmaximal, characteristic, complemented
12#14528609C211yesmaximal, characteristic
13C4×#10587487C211yesmaximal, characteristic, complemented
14#14528470C211yesmaximal, characteristic, complemented
15C4×C17×#102C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587622C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19#11763167C418yescomplemented
20C2×#10587487C2211yescharacteristic
21C2×#10587487C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31#11762715—21no—
32#11762714C2211yescharacteristic
33#11762713—21no—
34#11762712C2211yescharacteristic
35#11762708C2211yescharacteristic
36#11762705C2211yescharacteristic
37#11762697—21no—
38#11762697—21no—
39#11762696—21no—
40#11762693—21no—
41#11762682—21no—
42#11762682—21no—
43#11762678C2211yescharacteristic
44#11762674C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46#11762655C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C22×Dic34—21no—
63C22×Dic34C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C42×C17—21no—
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78#10587622—22no—
79#10587622C2×C414yes—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95#10587574—22no—
96#10587574C2×C414yescomplemented
97C4×Dic34—22no—
98C4×Dic34—22no—
99C17×#102—22no—
100C17×#102C2×C414yes—
101C4×C8×C17—22no—
102C17×#95—21no—
103#10587503—22no—
104#10587503C2×C414yes—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502D411yescharacteristic
113#10587502D411yescharacteristic
114#10587502D411yescharacteristic
115#10587502D411yescharacteristic
116#10587487—22no—
117#10587487C2×C414yescomplemented
118#10587486—22no—
119#10587486C2×C414yescomplemented
120C8×Dic17—22no—
121C8×Dic17—22no—
122#10587481—22no—
123#10587481C2×C414yescomplemented
124#10587474—22no—
125C24×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17C22×C412yes—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—42no—
136C22×C4×C17—81no—
137C22×C4×C17—81no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17C2×D411yescharacteristic
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34—24no—
155C2×Dic34—81no—
156C2×Dic34—81no—
157C2×Dic34C22×C412yes—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C2×C8×C17#4112yes—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17C22×C412yes—
170C22⋊C4×C17—42no—
171C42×C17—81no—
172C42×C17#4112yes—
173C42×C17#4112yes—
174C22⋊Dic17—24no—
175C22⋊Dic17—24no—
176C22⋊Dic17—81no—
177C22⋊Dic17—81no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—81no—
184C4⋊Dic17C22×C412yes—
185#63225—24no—
186#63225—42no—
187#63225—81no—
188#63225—81no—
189#63225—81no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—42no—
193C4×Dic17—81no—
194C4×Dic17C22×C412yes—
195C4×Dic17#4112yes—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622C22×C412yes—
201C2×#3622#4112yes—
202#21654—171nomaximal, Sylow
203C23×C17—22no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17C22×D411yescharacteristic
211D4×C17—22no—
212D4×C17—22no—
213D4×C17—41no—
214D4×C17—41no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—22no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—84no—
229C2×C4×C17#12412yes—
230C2×C4×C17#12412yes—
231C2×C4×C17C23×C411yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—22no—
242C2×Dic17—22no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
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×Dic17C4×D412yes—
260C2×Dic17C4×D412yes—
261Dic34—22no—
262Dic34—22no—
263Dic34—41no—
264Dic34—41no—
265Dic34—84no—
266C8×C17—22no—
267C8×C17—22no—
268#3622—22no—
269#3622—22no—
270C2×#489—171no—
271#2331—171no—
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×#102D1711yescharacteristic
284#1726—171no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17—81no—
291C22×C17—81no—
292C22×C17C2×C4×D411yescharacteristic
293C22×C17C2×#12411yescharacteristic
294C22×C17#53211yescharacteristic
295C22×C17#57611yescharacteristic
296C22×C17#57711yescharacteristic
297C22×C17#57711yescharacteristic
298C22×C17#57811yescharacteristic
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—42no—
302C4×C17—81no—
303C4×C17—81no—
304C4×C17C4×#4111yescharacteristic
305C4×C17C4×#4111yescharacteristic
306C4×C17#52911yescharacteristic
307C4×C17#52911yescharacteristic
308Dic17—22no—
309Dic17—22no—
310Dic17—41no—
311Dic17—41no—
312Dic17—42no—
313Dic17—42no—
314Dic17—81no—
315Dic17—81no—
316C2×C4⋊Q8—171no—
317C2×#124—171no—
318C2×C4×Q8—171no—
319C2×C4×D4C2×D1711yescharacteristic
320#489—178no—
321C2×#103—171no—
322C2×#103—171no—
323C2×#103—171no—
324C2×#103—171no—
325C2×#102—171no—
326C2×#102—171no—
327C2×#102C2×D1711yescharacteristic
328C2×#102C2×D1711yescharacteristic
329C2×#97—171no—
330C2×#97—171no—
331C2×#97—171no—
332C2×C4×C8—171no—
333C2×C4×C8—171no—
334C2×C4×C8C2×D1711yescharacteristic
335#391—171no—
336#391—341no—
337#391—341no—
338#390—171no—
339#389—171no—
340#389—171no—
341#388—341no—
342#388—341no—
343#387—341no—
344#387—341no—
345#385—341no—
346#385—341no—
347#384—171no—
348#382—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4C2×D1711yescharacteristic
351#376—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340C2×D1711yescharacteristic
359#340C2×D1711yescharacteristic
360C2×C17—82no—
361C2×C17C4×#12411yescharacteristic
362C2×C17#294611yescharacteristic
363C2×C17#294711yescharacteristic
364C2×C17#313611yescharacteristic
365C2×C17#313711yescharacteristic
366C2×C17#317311yescharacteristic
367C2×C17#317511yescharacteristic
368C22×Q8—171no—
369C22×Q8—341no—
370C22×D4C22×D1711yescharacteristic
371C23×C4—21no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8C22×D1711yescharacteristic
375C22×C8C22×D1711yescharacteristic
376C4⋊Q8—174no—
377C4⋊Q8—342no—
378#124—174no—
379#124—342no—
380C4×Q8—342no—
381C4×Q8—342no—
382C4×D4—22no—
383C4×D4—22no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4C22×D1711yescharacteristic
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C42—21no—
402C2×C42—171no—
403C2×C42—171no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42C22×D1711yescharacteristic
407#103—174no—
408#103—174no—
409#103—342no—
410#103—342no—
411#102—22no—
412#102—174no—
413#102—342no—
414#102C4×D1714yes—
415#97—174no—
416#97—342no—
417#97—342no—
418C4×C8—22no—
419C4×C8—342no—
420C4×C8—342no—
421#95—21no—
422#95—171no—
423#95—171no—
424#95—171no—
425#95—171no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433C17#2165411yescharacteristic, complemented, Sylow
434C24—21no—
435C2×Q8—172no—
436C2×Q8—344no—
437C2×Q8—341no—
438C2×Q8—341no—
439C2×Q8—341no—
440C2×Q8—341no—
441C2×Q8—341no—
442C2×Q8—1361no—
443C2×Q8—1361no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×C4×D1712yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—171no—
457C22×C4—171no—
458C22×C4—171no—
459C22×C4—171no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4D4×D1711yescharacteristic
465C22×C4D4×D1711yescharacteristic
466C22×C4C23×D1711yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—172no—
472C2×C8—172no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8C2×C4×D1712yes—
478C2×C8#6325212yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—172no—
485C4⋊C4—344no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—682no—
492C4⋊C4—1361no—
493C4⋊C4—1361no—
494C4⋊C4—1361no—
495C4⋊C4—1361no—
496C4⋊C4C2×C4×D1712yes—
497C22⋊C4—42no—
498C22⋊C4—344no—
499C22⋊C4—344no—
500C22⋊C4—1361no—
501C22⋊C4—1361no—
502C42—81no—
503C42—172no—
504C42—172no—
505C42—341no—
506C42—341no—
507C42—682no—
508C42—1361no—
509C42#6325212yes—
510C42#6325412yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D1711yescharacteristic
519Q8—342no—
520Q8—342no—
521Q8—681no—
522Q8—681no—
523Q8—1364no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—172no—
543C2×C4—172no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—342no—
549C2×C4—342no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
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#1058757112yes—
567C2×C4#1058757112yes—
568C2×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325211yescharacteristic
570C2×C4C2×#6325211yescharacteristic
571C2×C4C2×#6325411yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C8—22no—
575C8—22no—
576C8—342no—
577C8—342no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C4×D4×D1711yescharacteristic
586C22C2×#1058757111yescharacteristic
587C22#1176335611yescharacteristic
588C22#1176361011yescharacteristic
589C22#1176361511yescharacteristic
590C22#1176361611yescharacteristic
591C22#1176362311yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—342no—
598C4—342no—
599C4—681no—
600C4—682no—
601C4—681no—
602C4—682no—
603C4—1361no—
604C4—1361no—
605C4C4×#6325411yescharacteristic
606C4C4×#6325211yescharacteristic
607C4#1176333711yescharacteristic
608C4#1176333911yescharacteristic
609C2—82no—
610C2C4×#1058757111yescharacteristic
611C2#1453760911yescharacteristic
612C2#1453762011yescharacteristic
613C2#1453913611yescharacteristic
614C2#1453915311yescharacteristic
615C2#1453939711yescharacteristic
616C2#1453941911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#102C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC4×#102D17
hypocenterC17#21654
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #14531528C2semidirect
4. #14529743C2semidirect
7. #14529472C2semidirect
9. #14529461C2semidirect
10. #14528916C2semidirect
11. #14528617C2semidirect
13. C4×#10587487C2semidirect
14. #14528470C2semidirect
19. #11763167C4semidirect
96. #10587574C2×C4semidirect
117. #10587487C2×C4semidirect
119. #10587486C2×C4semidirect
123. #10587481C2×C4semidirect
433. C17#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#117631672
#145315282
#145314962
#145297432
#145295942
#145294832
#145294722
#145294642
#145294612
#145289162
#145286172
#145286092
C4×#105874872
#145284702
C4×C17×#1022
#2165417