Finite Group Databasebeta

← #17037271

Subgroups — #17037271

Order 4352.

934 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic17C211yesmaximal, characteristic
2#14531246C211yesmaximal, characteristic
3C4×Q8×Dic17C211yesmaximal, characteristic, complemented
4#14529414C211yesmaximal, characteristic
5#14529408C211yesmaximal, characteristic
6#14529363C211yesmaximal, characteristic, complemented
7#14529361C211yesmaximal, characteristic, complemented
8#14529141C211yesmaximal, characteristic
9#14529052C211yesmaximal, characteristic, complemented
10#14529050C211yesmaximal, characteristic, complemented
11#14528672C211yesmaximal, characteristic, complemented
12#103×Dic17C211yesmaximal, characteristic, complemented
13#102×Dic17C211yesmaximal, characteristic
14#14528528C211yesmaximal, characteristic, complemented
15C17×#1880C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19C2×#10587563—21no—
20SD16×Dic17C418yescomplemented
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762712C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762705C2211yescharacteristic
33#11762703—21no—
34#11762697—21no—
35C4⋊C4×Dic17—21no—
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×C4⋊Dic17—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42C22⋊C4×Dic17—21no—
43C42×Dic17—21no—
44C42×Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×SD16×C17C2211yescharacteristic
54C2×C4×Q8×C17C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#390C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C2×SD16×C17—22no—
64C2×SD16×C17—22no—
65C2×SD16×C17C2×C414yes—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C4×Q8×C17—22no—
69C4×Q8×C17—22no—
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×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80Q8×Dic17—22no—
81Q8×Dic17—22no—
82Q8×Dic17—22no—
83Q8×Dic17C2×C414yes—
84D4×Dic17—22no—
85D4×Dic17C2×C414yes—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17D411yescharacteristic
101C2×C4×Dic17D411yescharacteristic
102C2×C4×Dic17D411yescharacteristic
103C2×C4×Dic17D411yescharacteristic
104C2×C4×Dic17C2311yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587563—44no—
109C17×#103—22no—
110C17×#102—22no—
111#10587506—22no—
112#10587506C2×C414yes—
113#10587503—22no—
114#10587503C2×C414yes—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587488C2×C414yes—
122C8×Dic17—22no—
123C8×Dic17C2×C414yes—
124#10587479—22no—
125#10587479—22no—
126C24×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17C22×C412yes—
131C2×Q8×C17#4112yes—
132C2×Q8×C17#4112yes—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17C22×C412yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—24no—
149C22×Dic17—21no—
150C22×Dic17—42no—
151C22×Dic17—42no—
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156SD16×C17—28no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C4⋊C4×C17—24no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17#4112yes—
168C22⋊C4×C17—42no—
169C22⋊C4×C17—42no—
170C42×C17—24no—
171C42×C17—21no—
172C42×C17—21no—
173C22⋊Dic17—24no—
174C22⋊Dic17—42no—
175C22⋊Dic17—42no—
176C4⋊Dic17—24no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—24no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17C22×C412yes—
184C4⋊Dic17C22×C412yes—
185C4⋊Dic17#4112yes—
186#63225—42no—
187#63225—42no—
188C4×Dic17—24no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—24no—
192C4×Dic17—22no—
193C4×Dic17—41no—
194C4×Dic17—41no—
195C4×Dic17—42no—
196C4×Dic17—42no—
197C4×Dic17—42no—
198C4×Dic17—42no—
199C4×Dic17SD1614yescomplemented
200C4×Dic17C22×C412yes—
201C4×Dic17#4112yes—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622C22×C412yes—
208#21115—171nomaximal, Sylow
209C23×C17—22no—
210C23×C17—41no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17C22×D411yescharacteristic
217Q8×C17—22no—
218Q8×C17—22no—
219Q8×C17—22no—
220Q8×C17#11714yescomplemented
221D4×C17—22no—
222D4×C17—22no—
223D4×C17—41no—
224D4×C17—41no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17C2×SD1612yes—
240C2×C4×C17C23×C411yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—48no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17C4×D412yes—
267C2×Dic17C4×D412yes—
268C2×Dic17C2×SD1612yes—
269C2×Dic17C2×SD1612yes—
270C2×Dic17#13512yes—
271C8×C17—22no—
272C8×C17—41no—
273#3622—22no—
274#3622—41no—
275C2×C4×SD16—171no—
276#2293—171no—
277C42×Q8—171no—
278#1926—171no—
279#1926—171no—
280#1919—171no—
281#1919—171no—
282#1880—171no—
283#1880D1711yescharacteristic
284#1865—171no—
285#1865—171no—
286C4×#106—171no—
287C4×#103—171no—
288C4×#102—171no—
289#1744—171no—
290C22×C17—42no—
291C22×C17—42no—
292C22×C17—42no—
293C22×C17—81no—
294C22×C17—81no—
295C22×C17—81no—
296C22×C17—81no—
297C22×C17C2×C4×D411yescharacteristic
298C22×C17#52511yescharacteristic
299C22×C17#52511yescharacteristic
300C22×C17C22×SD1611yescharacteristic
301C22×C17C2×#13511yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57511yescharacteristic
304C4×C17—22no—
305C4×C17—24no—
306C4×C17—41no—
307C4×C17—42no—
308C4×C17—81no—
309C4×C17C2×#11711yescharacteristic
310C4×C17C4×#4111yescharacteristic
311C4×C17#52911yescharacteristic
312C4×C17#52911yescharacteristic
313Dic17—22no—
314Dic17—22no—
315Dic17—41no—
316Dic17—42no—
317Dic17—42no—
318Dic17—42no—
319Dic17—81no—
320Dic17C4×SD1614yes—
321C22×SD16C2×D1711yescharacteristic
322C2×C4×Q8—171no—
323C2×C4×Q8—171no—
324C2×C4×Q8C2×D1711yescharacteristic
325C2×C4×D4—171no—
326C2×#117—341no—
327C4×SD16—178no—
328C2×#106—171no—
329C2×#106—171no—
330C2×#106—171no—
331C2×#103—171no—
332C2×#103—171no—
333C2×#103C2×D1711yescharacteristic
334C2×#102—171no—
335C2×#102—171no—
336C2×#102C2×D1711yescharacteristic
337C2×C4×C8—171no—
338C2×C4×C8—171no—
339#390—171no—
340#390C2×D1711yescharacteristic
341#389—171no—
342#389—171no—
343#388—341no—
344#388—341no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—341no—
350C4×C4⋊C4—341no—
351C4×C4⋊C4—341no—
352C4×C22⋊C4—341no—
353C43—171no—
354C43—341no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340C2×D1711yescharacteristic
361#336—171no—
362#336—171no—
363#336—171no—
364#336C2×D1711yescharacteristic
365C2×C17—82no—
366C2×C17#229011yescharacteristic
367C2×C17C2×C4×SD1611yescharacteristic
368C2×C17#294311yescharacteristic
369C2×C17#310211yescharacteristic
370C2×C17#310211yescharacteristic
371C2×C17#310311yescharacteristic
372C2×C17#310311yescharacteristic
373C22×Q8C22×D1711yescharacteristic
374C22×D4C22×D1711yescharacteristic
375C23×C4—341no—
376C2×SD16—22no—
377C2×SD16—22no—
378C2×SD16C4×D1714yes—
379C22×C8—171no—
380C22×C8—171no—
381C22×C8C22×D1711yescharacteristic
382C22×C8C22×D1711yescharacteristic
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—174no—
386C4×Q8—342no—
387C4×Q8—342no—
388C4×Q8—342no—
389C4×D4—174no—
390C4×D4—342no—
391#117—684no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C22⋊C4—341no—
408C2×C42—21no—
409C2×C42—171no—
410C2×C42—171no—
411C2×C42—171no—
412C2×C42—171no—
413C2×C42—171no—
414C2×C42—171no—
415C2×C42—341no—
416C2×C42—341no—
417C2×C42—341no—
418C2×C42—341no—
419C2×C42—341no—
420C2×C42—341no—
421C2×C42C22×D1711yescharacteristic
422#106—174no—
423#106—342no—
424#106—342no—
425#103—22no—
426#103—174no—
427#103—342no—
428#102—22no—
429#102—174no—
430#102—342no—
431C4×C8—174no—
432C4×C8—342no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439C17#2111511yescharacteristic, complemented, Sylow
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D1712yes—
445C2×Q8#6325412yes—
446C2×Q8#6325412yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D1712yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—171no—
457C22×C4—171no—
458C22×C4—171no—
459C22×C4—171no—
460C22×C4—344no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—682no—
466C22×C4—682no—
467C22×C4D4×D1711yescharacteristic
468C22×C4D4×D1711yescharacteristic
469C22×C4C23×D1711yescharacteristic
470SD16—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—172no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8C2×C4×D1712yes—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—172no—
488C4⋊C4—172no—
489C4⋊C4—172no—
490C4⋊C4—344no—
491C4⋊C4—344no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—682no—
498C4⋊C4—682no—
499C4⋊C4#6325412yes—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—344no—
503C22⋊C4—682no—
504C22⋊C4—682no—
505C42—24no—
506C42—21no—
507C42—21no—
508C42—172no—
509C42—172no—
510C42—174no—
511C42—342no—
512C42—344no—
513C42—344no—
514C42—341no—
515C42—341no—
516C42—681no—
517C42—681no—
518C42—682no—
519C42—682no—
520C42—682no—
521C42—682no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D1711yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1058756314yescomplemented
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—172no—
553C2×C4—172no—
554C2×C4—172no—
555C2×C4—172no—
556C2×C4—172no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—342no—
562C2×C4—342no—
563C2×C4—342no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—688no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4SD16×D1712yes—
578C2×C4C22×C4×D1711yescharacteristic
579C2×C4C2×#6325411yescharacteristic
580C2×C4C2×#6325411yescharacteristic
581C2×C4C2×#6325411yescharacteristic
582C2×C4#41×D1711yescharacteristic
583C2×C4#41×D1711yescharacteristic
584C8—22no—
585C8—41no—
586C8—342no—
587C8—681no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D1711yescharacteristic
596C22#1176331511yescharacteristic
597C22#1176331911yescharacteristic
598C22C2×SD16×D1711yescharacteristic
599C22#135×D1711yescharacteristic
600C22#1176360511yescharacteristic
601C22#1176360811yescharacteristic
602C4—22no—
603C4—24no—
604C4—41no—
605C4—42no—
606C4—81no—
607C4—174no—
608C4—342no—
609C4—342no—
610C4—681no—
611C4—682no—
612C4—682no—
613C4—682no—
614C4—1361no—
615C4C2×#1058756311yescharacteristic
616C4C4×#6325411yescharacteristic
617C4#1176333611yescharacteristic
618C4#1176334011yescharacteristic
619C2—82no—
620C2#1453122411yescharacteristic
621C2C4×SD16×D1711yescharacteristic
622C2#1453757911yescharacteristic
623C2#1453879511yescharacteristic
624C2#1453880111yescharacteristic
625C2#1453881011yescharacteristic
626C2#1453881411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1880C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1880D17
hypocenterC17#21115
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
3. C4×Q8×Dic17C2semidirect
6. #14529363C2semidirect
7. #14529361C2semidirect
9. #14529052C2semidirect
10. #14529050C2semidirect
11. #14528672C2semidirect
12. #103×Dic17C2semidirect
14. #14528528C2semidirect
20. SD16×Dic17C4semidirect
199. C4×Dic17SD16semidirect
220. Q8×C17#117semidirect
439. C17#21115semidirect
533. Q8#10587563semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic172
#145312462
C4×Q8×Dic172
#145294142
#145294082
#145293632
#145293612
#145291412
#145290522
#145290502
#145286722
#103×Dic172
#102×Dic172
#145285282
C17×#18802
#2111517