Finite Group Databasebeta

← #17042440

Subgroups — #17042440

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
1#14531533C211yesmaximal, characteristic, complemented
2#14531497C211yesmaximal, characteristic
3#14529746C211yesmaximal, characteristic, complemented
4#14529596C211yesmaximal, characteristic
5#14529483C211yesmaximal, characteristic
6#14529476C211yesmaximal, characteristic, complemented
7#14529469C211yesmaximal, characteristic
8#14529462C211yesmaximal, characteristic, complemented
9#14528918C211yesmaximal, characteristic, complemented
10#14528616C211yesmaximal, characteristic, complemented
11#14528611C211yesmaximal, characteristic
12#103×Dic17C211yesmaximal, characteristic, complemented
13#102×Dic17C211yesmaximal, characteristic
14#14528472C211yesmaximal, characteristic, complemented
15C2×C17×#489C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C2×#10587506C2211yescharacteristic
19C2×#10587506C2211yescharacteristic
20C2×#10587503C2211yescharacteristic
21C2×#10587503C2211yescharacteristic
22C2×#10587486C2211yescharacteristic
23C2×#10587486C2211yescharacteristic
24C2×C4×#3622C2211yescharacteristic
25C2×C8×Dic17C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762716—21no—
28#11762716—21no—
29#11762716C2211yescharacteristic
30#11762712C2211yescharacteristic
31#11762708C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762702—21no—
34#11762702—21no—
35#11762695—21no—
36#11762695—21no—
37#11762684—21no—
38#11762684—21no—
39#11762679C2211yescharacteristic
40#11762675C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43#11762655C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52C2×C17×C4⋊Q8C2211yescharacteristic
53C2×C17×#124C2211yescharacteristic
54C17×#489C418yescomplemented
55C2×C17×#103C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C2×C17×#97C2211yescharacteristic
60C22×Q8×C17—21no—
61C22×Q8×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×Dic17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C17×C4⋊Q8—22no—
67C17×C4⋊Q8C2×C414yescomplemented
68C17×#124—22no—
69C17×#124C2×C414yes—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79Q8×Dic17—22no—
80Q8×Dic17—22no—
81D4×Dic17—22no—
82D4×Dic17—22no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C17×#103C2×C414yescomplemented
100C17×#103C2×C414yescomplemented
101C17×#102C2×C414yes—
102C17×#102C2×C414yes—
103C17×#97C2×C414yescomplemented
104#10587506—22no—
105#10587506—22no—
106#10587503—22no—
107#10587503—22no—
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#10587502D411yescharacteristic
117#10587502D411yescharacteristic
118#10587502D411yescharacteristic
119#10587502D411yescharacteristic
120#10587486—22no—
121#10587486—22no—
122C8×Dic17—22no—
123C8×Dic17—22no—
124C4×#3622—22no—
125C24×C17—21no—
126C2×Q8×C17—24no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—81no—
133C2×Q8×C17—81no—
134C2×Q8×C17C22×C412yes—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17C22×C412yes—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—42no—
152C22×Dic17—81no—
153C22×Dic17—81no—
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156C22×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C2×C8×C17C22×C412yes—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17C22×C412yes—
168C4⋊C4×C17C22×C412yes—
169C22⋊C4×C17—24no—
170C22⋊C4×C17—24no—
171C42×C17C22×C412yes—
172C42×C17#4112yes—
173C22⋊Dic17—42no—
174C22⋊Dic17—81no—
175C22⋊Dic17—81no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—42no—
185C4⋊Dic17—81no—
186C4⋊Dic17—81no—
187C4⋊Dic17—81no—
188C4⋊Dic17—81no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—42no—
192C4×Dic17—81no—
193C4×Dic17—81no—
194C4×Dic17#4112yes—
195C4×Dic17#4112yes—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622#4112yes—
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
211Q8×C17—22no—
212Q8×C17—22no—
213Q8×C17—41no—
214Q8×C17—41no—
215Q8×C17—84no—
216D4×C17—22no—
217D4×C17—22no—
218D4×C17—41no—
219D4×C17—41no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17C4×D412yes—
234C2×C4×C17C4×D412yes—
235C2×C4×C17C23×C411yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
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—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—84no—
264C2×Dic17#12412yes—
265C2×Dic17#12412yes—
266C8×C17—22no—
267C8×C17—22no—
268#3622—22no—
269#3622—22no—
270C2×#489D1711yescharacteristic
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×#102—171no—
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—42no—
303C4×C17—42no—
304C4×C17C4×#4111yescharacteristic
305C4×C17C4×#4111yescharacteristic
306C4×C17#52911yescharacteristic
307C4×C17#52911yescharacteristic
308Dic17—22no—
309Dic17—22no—
310Dic17—41no—
311Dic17—41no—
312Dic17—81no—
313Dic17—81no—
314Dic17—81no—
315Dic17—81no—
316C2×C4⋊Q8C2×D1711yescharacteristic
317C2×#124C2×D1711yescharacteristic
318C2×C4×Q8—171no—
319C2×C4×D4—171no—
320#489Dic1718yescomplemented
321C2×#103—171no—
322C2×#103—171no—
323C2×#103C2×D1711yescharacteristic
324C2×#103C2×D1711yescharacteristic
325C2×#102—171no—
326C2×#102—171no—
327C2×#102C2×D1711yescharacteristic
328C2×#102C2×D1711yescharacteristic
329C2×#97—171no—
330C2×#97—171no—
331C2×#97C2×D1711yescharacteristic
332C2×C4×C8—171no—
333C2×C4×C8—171no—
334C2×C4×C8—171no—
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⋊C4—171no—
351#376—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—82no—
361C2×C17C4×#12411yescharacteristic
362C2×C17#294611yescharacteristic
363C2×C17#294711yescharacteristic
364C2×C17#313611yescharacteristic
365C2×C17#313711yescharacteristic
366C2×C17#317311yescharacteristic
367C2×C17#317511yescharacteristic
368C22×Q8—21no—
369C22×Q8C22×D1711yescharacteristic
370C22×D4C22×D1711yescharacteristic
371C23×C4—341no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8C22×D1711yescharacteristic
375C22×C8C22×D1711yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8C2×Dic1714yescomplemented
378#124—22no—
379#124C2×Dic1714yes—
380C4×Q8—342no—
381C4×Q8—342no—
382C4×D4—342no—
383C4×D4—342no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C42—171no—
402C2×C42—171no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42C22×D1711yescharacteristic
407#103—342no—
408#103—342no—
409#103C2×Dic1714yescomplemented
410#103C2×Dic1714yescomplemented
411#102—342no—
412#102—342no—
413#102C2×Dic1714yes—
414#102C2×Dic1714yes—
415#97—342no—
416#97—342no—
417#97C2×Dic1714yescomplemented
418C4×C8—342no—
419C4×C8—342no—
420C4×C8—342no—
421#95—171no—
422#95—171no—
423#95—171no—
424#95—171no—
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#2165411yescharacteristic, complemented, Sylow
434C24—21no—
435C2×Q8—21no—
436C2×Q8—24no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—81no—
442C2×Q8—81no—
443C2×Q8C22×Dic1712yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C22×Dic1712yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—171no—
456C22×C4—171no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—682no—
462C22×C4—1361no—
463C22×C4—1361no—
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×C8C22×Dic1712yes—
478C2×C8C22×Dic1712yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
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—682no—
491C4⋊C4—1361no—
492C4⋊C4—1361no—
493C4⋊C4—1361no—
494C4⋊C4—1361no—
495C4⋊C4C22×Dic1712yes—
496C4⋊C4C22×Dic1712yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—682no—
500C22⋊C4—1361no—
501C22⋊C4—1361no—
502C42—172no—
503C42—172no—
504C42—341no—
505C42—341no—
506C42—682no—
507C42—1361no—
508C42—1361no—
509C42#6325412yes—
510C42C22×Dic1712yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D1711yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—84no—
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—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—172no—
542C2×C4—172no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
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×C4D4×Dic1712yes—
567C2×C4D4×Dic1712yes—
568C2×C4C2×#6325411yescharacteristic
569C2×C4#41×D1711yescharacteristic
570C2×C4#41×D1711yescharacteristic
571C2×C4#41×D1711yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4C23×Dic1711yescharacteristic
574C8—22no—
575C8—22no—
576C8—342no—
577C8—342no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×D4×Dic1711yescharacteristic
586C22#124×D1711yescharacteristic
587C22#1176335811yescharacteristic
588C22#1176361311yescharacteristic
589C22#1176361811yescharacteristic
590C22#1176361911yescharacteristic
591C22#1176362511yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—342no—
598C4—342no—
599C4—681no—
600C4—681no—
601C4—1361no—
602C4—1361no—
603C4—1361no—
604C4—1361no—
605C4#41×Dic1711yescharacteristic
606C4#41×Dic1711yescharacteristic
607C4#1176333811yescharacteristic
608C4#1176333811yescharacteristic
609C2—82no—
610C2#124×Dic1711yescharacteristic
611C2#1453761711yescharacteristic
612C2#1453762811yescharacteristic
613C2#1453913511yescharacteristic
614C2#1453914611yescharacteristic
615C2#1453940011yescharacteristic
616C2#1453942211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#489C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#489D17
hypocenterC17#21654
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. #14531533C2semidirect
3. #14529746C2semidirect
6. #14529476C2semidirect
8. #14529462C2semidirect
9. #14528918C2semidirect
10. #14528616C2semidirect
12. #103×Dic17C2semidirect
14. #14528472C2semidirect
54. C17×#489C4semidirect
67. C17×C4⋊Q8C2×C4semidirect
99. C17×#103C2×C4semidirect
100. C17×#103C2×C4semidirect
103. C17×#97C2×C4semidirect
320. #489Dic17semidirect
377. C4⋊Q8C2×Dic17semidirect
409. #103C2×Dic17semidirect
410. #103C2×Dic17semidirect
417. #97C2×Dic17semidirect
433. C17#21654semidirect

Maximal subgroups

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

subgroupindex
#145315332
#145314972
#145297462
#145295962
#145294832
#145294762
#145294692
#145294622
#145289182
#145286162
#145286112
#103×Dic172
#102×Dic172
#145284722
C2×C17×#4892
#2165417