Finite Group Databasebeta

← #17042446

Subgroups — #17042446

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×#11763168C211yesmaximal, characteristic
2#14531530C211yesmaximal, characteristic, complemented
3#14531497C211yesmaximal, characteristic
4#14529745C211yesmaximal, characteristic, complemented
5#14529478C211yesmaximal, characteristic
6#14529471C211yesmaximal, characteristic, complemented
7#14529464C211yesmaximal, characteristic
8#14529457C211yesmaximal, characteristic, complemented
9#14528917C211yesmaximal, characteristic, complemented
10#14528616C211yesmaximal, characteristic, complemented
11#14528611C211yesmaximal, characteristic
12#103×Dic17C211yesmaximal, characteristic, complemented
13#102×Dic17C211yesmaximal, characteristic
14#14528470C211yesmaximal, characteristic, complemented
15C17×#1958C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19#11763168C418yescomplemented
20C2×#10587506C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587474C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26C2×#10587486C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762716C2211yescharacteristic
30#11762713—21no—
31#11762713—21no—
32#11762712C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762702—21no—
36#11762702—21no—
37#11762694—21no—
38#11762694—21no—
39#11762684—21no—
40#11762684—21no—
41#11762679C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44#11762655C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#124C2211yescharacteristic
54C2×C17×#103C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C2×C4×C8×C17C2211yescharacteristic
59C17×#382C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C17×#124—22no—
67C17×#124C2×C414yes—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C42×C17C2311yescharacteristic
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76Q8×Dic17—22no—
77Q8×Dic17—22no—
78D4×Dic17—22no—
79D4×Dic17—22no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×#63225—21no—
89C2×#63225—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587555—22no—
98#10587555C2×C414yescomplemented
99C17×#103—22no—
100C17×#103—22no—
101C17×#102—22no—
102C17×#102—22no—
103C4×C8×C17—22no—
104C17×#95—21no—
105C17×#95—21no—
106#10587506C2×C414yescomplemented
107#10587506C2×C414yescomplemented
108#10587503C2×C414yes—
109#10587503C2×C414yes—
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—
124#10587474C2×C414yescomplemented
125C24×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17C22×C412yes—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17C22×C412yes—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—42no—
149C22×Dic17—81no—
150C22×Dic17—81no—
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C2×Dic34—24no—
156C2×Dic34—81no—
157C2×Dic34—81no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17#4112yes—
163C2×C8×C17#4112yes—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C22⋊C4×C17—24no—
169C22⋊C4×C17—24no—
170C42×C17C22×C412yes—
171C42×C17#4112yes—
172C22⋊Dic17—42no—
173C22⋊Dic17—81no—
174C22⋊Dic17—81no—
175C4⋊Dic17—24no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—42no—
183C4⋊Dic17—81no—
184C4⋊Dic17—81no—
185C4⋊Dic17C22×C412yes—
186C4⋊Dic17C22×C412yes—
187#63225—81no—
188#63225—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×#3622C22×C412yes—
201C2×#3622C22×C412yes—
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—
215D4×C17—22no—
216D4×C17—22no—
217D4×C17—41no—
218D4×C17—41no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—22no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17C4×D412yes—
233C2×C4×C17C4×D412yes—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
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—84no—
263C2×Dic17#12412yes—
264C2×Dic17#12412yes—
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#1958D1711yescharacteristic
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—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×#124C2×D1711yescharacteristic
318C2×C4×Q8—171no—
319C2×C4×D4—171no—
320#489—178no—
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×#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#382C2×D1711yescharacteristic
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—341no—
369C22×Q8C22×D1711yescharacteristic
370C22×D4C22×D1711yescharacteristic
371C23×C4—341no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8C22×D1711yescharacteristic
375C22×C8C22×D1711yescharacteristic
376C4⋊Q8—174no—
377C4⋊Q8—342no—
378#124—22no—
379#124C4×D1714yes—
380C4×Q8—342no—
381C4×Q8—342no—
382C4×D4—342no—
383C4×D4—342no—
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⋊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—22no—
408#103—22no—
409#103—174no—
410#103—174no—
411#102—22no—
412#102—22no—
413#102—174no—
414#102—174no—
415#97—174no—
416#97—342no—
417#97—342no—
418C4×C8—22no—
419C4×C8—342no—
420C4×C8—342no—
421#95—21no—
422#95—21no—
423#95—171no—
424#95—171no—
425#95—171no—
426#95—171no—
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—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—344no—
441C2×Q8—1361no—
442C2×Q8—1361no—
443C2×Q8C2×C4×D1712yes—
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—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×C8#6325412yes—
478C2×C8#6325412yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—172no—
484C4⋊C4—172no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—344no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—682no—
493C4⋊C4—1361no—
494C4⋊C4—1361no—
495C4⋊C4—1361no—
496C4⋊C4—1361no—
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—
509C42C2×C4×D1712yes—
510C42#6325412yes—
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—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—172no—
542C2×C4—172no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—342no—
548C2×C4—342no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
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×C4#1058756712yes—
567C2×C4#1058756712yes—
568C2×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325411yescharacteristic
570C2×C4C2×#6325411yescharacteristic
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—
585C22C2×#1058756711yescharacteristic
586C22#124×D1711yescharacteristic
587C22#1176335711yescharacteristic
588C22#1176361111yescharacteristic
589C22#1176361611yescharacteristic
590C22#1176361611yescharacteristic
591C22#1176362311yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—342no—
598C4—342no—
599C4—681no—
600C4—682no—
601C4—682no—
602C4—681no—
603C4—1361no—
604C4—1361no—
605C4C4×#6325411yescharacteristic
606C4C4×#6325411yescharacteristic
607C4#1176334011yescharacteristic
608C4#1176334011yescharacteristic
609C2—82no—
610C2#1453133111yescharacteristic
611C2#1453761911yescharacteristic
612C2#1453763011yescharacteristic
613C2#1453913711yescharacteristic
614C2#1453914811yescharacteristic
615C2#1453939811yescharacteristic
616C2#1453942011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1958C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1958D17
hypocenterC17#21654
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. #14531530C2semidirect
4. #14529745C2semidirect
6. #14529471C2semidirect
8. #14529457C2semidirect
9. #14528917C2semidirect
10. #14528616C2semidirect
12. #103×Dic17C2semidirect
14. #14528470C2semidirect
19. #11763168C4semidirect
98. #10587555C2×C4semidirect
106. #10587506C2×C4semidirect
107. #10587506C2×C4semidirect
124. #10587474C2×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×#117631682
#145315302
#145314972
#145297452
#145294782
#145294712
#145294642
#145294572
#145289172
#145286162
#145286112
#103×Dic172
#102×Dic172
#145284702
C17×#19582
#2165417