Finite Group Databasebeta

← #17042448

Subgroups — #17042448

Order 4352.

886 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×#11763167C211yesmaximal, characteristic
2C4×#10587574C211yesmaximal, characteristic, complemented
3C4×#10587622C211yesmaximal, characteristic
4#14529755C211yesmaximal, characteristic, complemented
5#14529599C211yesmaximal, characteristic
6#14529469C211yesmaximal, characteristic
7#14529464C211yesmaximal, characteristic
8#14529461C211yesmaximal, characteristic, complemented
9#14529458C211yesmaximal, characteristic, complemented
10#14528916C211yesmaximal, characteristic, complemented
11#14528617C211yesmaximal, characteristic, complemented
12#14528615C211yesmaximal, characteristic, complemented
13#14528609C211yesmaximal, characteristic
14#97×Dic17C211yesmaximal, characteristic, complemented
15C17×#1762C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587622C2211yescharacteristic
18C2×C4×Dic34—21no—
19C2×C4×Dic34C2211yescharacteristic
20#11763167C418yescomplemented
21C2×#10587487C2211yescharacteristic
22C2×#10587487C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587474C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762714C2211yescharacteristic
33#11762713—21no—
34#11762712C2211yescharacteristic
35#11762708C2211yescharacteristic
36#11762706—21no—
37#11762705C2211yescharacteristic
38#11762703—21no—
39#11762703—21no—
40#11762678C2211yescharacteristic
41#11762674C2211yescharacteristic
42C4×#63225—21no—
43C4×C4⋊Dic17C2211yescharacteristic
44C4×C22⋊Dic17—21no—
45C4×C22⋊Dic17—21no—
46C42×Dic17C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×#97C2211yescharacteristic
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×Dic17D411yescharacteristic
92C2×C4×Dic17D411yescharacteristic
93C2×C4×Dic17D411yescharacteristic
94C2×C4×Dic17D411yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587574—22no—
100#10587574C2×C414yescomplemented
101C4×Dic34—22no—
102C4×Dic34—22no—
103C4×Dic34—44no—
104C17×#102—22no—
105C17×#102C2×C414yes—
106C17×#97—22no—
107C17×#95—21no—
108#10587503—22no—
109#10587503C2×C414yes—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587487—22no—
118#10587487C2×C414yescomplemented
119#10587486—22no—
120#10587486C2×C414yescomplemented
121C8×Dic17—22no—
122C8×Dic17—22no—
123#10587481—22no—
124#10587481C2×C414yescomplemented
125#10587474—22no—
126C24×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17C22×C412yes—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—42no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2411yescharacteristic
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34—21no—
155C2×Dic34—24no—
156C2×Dic34—42no—
157C2×Dic34—42no—
158C2×Dic34C22×C412yes—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17C22×C412yes—
164C2×C8×C17#4112yes—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17C22×C412yes—
171C22⋊C4×C17—42no—
172C42×C17—42no—
173C42×C17#4112yes—
174C42×C17#4112yes—
175C22⋊Dic17—24no—
176C22⋊Dic17—24no—
177C22⋊Dic17—42no—
178C22⋊Dic17—42no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—42no—
185C4⋊Dic17C22×C412yes—
186#63225—24no—
187#63225—42no—
188#63225—42no—
189#63225—42no—
190#63225—42no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—22no—
194C4×Dic17—22no—
195C4×Dic17—41no—
196C4×Dic17—41no—
197C4×Dic17—42no—
198C4×Dic17—42no—
199C4×Dic17C22×C412yes—
200C4×Dic17#4112yes—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622C22×C412yes—
206C2×#3622#4112yes—
207#21655—171nomaximal, Sylow
208C23×C17—22no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17C22×D411yescharacteristic
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—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×C17—41no—
233C2×C4×C17—84no—
234C2×C4×C17#12412yes—
235C2×C4×C17#12412yes—
236C2×C4×C17C23×C411yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
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—22no—
251C2×Dic17—22no—
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×Dic17C4×D412yes—
263C2×Dic17C4×D412yes—
264C2×Dic17AGL(1,ℤ/8)12yes—
265C2×Dic17#13712yes—
266Dic34—22no—
267Dic34—22no—
268Dic34—48no—
269Dic34—41no—
270Dic34—41no—
271C8×C17—22no—
272C8×C17—22no—
273#3622—22no—
274#3622—22no—
275C2×#489—171no—
276C4×C4⋊Q8—171no—
277C4×#124—171no—
278#1993—171no—
279#1959—171no—
280#1934—171no—
281#1934—171no—
282#1933—171no—
283#1933—171no—
284#1836—171no—
285#1763—171no—
286#1763—171no—
287#1762—171no—
288#1762D1711yescharacteristic
289C4×#97—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×C17C2×#12411yescharacteristic
299C22×C17#53211yescharacteristic
300C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C17C2×#13711yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57511yescharacteristic
304C4×C17—22no—
305C4×C17—22no—
306C4×C17—42no—
307C4×C17—81no—
308C4×C17—81no—
309C4×C17C4×#4111yescharacteristic
310C4×C17C4×#4111yescharacteristic
311C4×C17#52911yescharacteristic
312C4×C17#52911yescharacteristic
313Dic17—22no—
314Dic17—22no—
315Dic17—22no—
316Dic17—22no—
317Dic17—42no—
318Dic17—42no—
319Dic17—81no—
320Dic17—81no—
321C2×C4⋊Q8—171no—
322C2×#124—171no—
323C2×C4×Q8—171no—
324C2×C4×Q8—341no—
325C2×C4×D4C2×D1711yescharacteristic
326#489—178no—
327C2×#103—171no—
328C2×#103—171no—
329C2×#103—171no—
330C2×#103—171no—
331C2×#102—171no—
332C2×#102—171no—
333C2×#102C2×D1711yescharacteristic
334C2×#102C2×D1711yescharacteristic
335C2×#97—171no—
336C2×#97—171no—
337C2×#97—171no—
338C2×#97C2×D1711yescharacteristic
339C2×C4×C8—171no—
340C2×C4×C8—171no—
341#391—171no—
342#391—341no—
343#390—171no—
344#389—171no—
345#389—171no—
346#389—341no—
347#388—341no—
348#388—341no—
349#384—171no—
350#382—171no—
351C4×C4⋊C4—171no—
352C4×C4⋊C4—341no—
353C4×C4⋊C4C2×D1711yescharacteristic
354C4×C22⋊C4—341no—
355C4×C22⋊C4—341no—
356C43—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340C2×D1711yescharacteristic
364#340C2×D1711yescharacteristic
365C2×C17—82no—
366C2×C17C4×#12411yescharacteristic
367C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
368C2×C17C4×#13711yescharacteristic
369C2×C17#314011yescharacteristic
370C2×C17#314111yescharacteristic
371C2×C17#318011yescharacteristic
372C2×C17#318111yescharacteristic
373C22×Q8—171no—
374C22×Q8—341no—
375C22×D4C22×D1711yescharacteristic
376C23×C4—21no—
377C22×C8—171no—
378C22×C8—171no—
379C22×C8C22×D1711yescharacteristic
380C22×C8C22×D1711yescharacteristic
381C4⋊Q8—174no—
382C4⋊Q8—342no—
383#124—174no—
384#124—342no—
385C4×Q8—342no—
386C4×Q8—342no—
387C4×Q8—684no—
388C4×D4—22no—
389C4×D4—22no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4C22×D1711yescharacteristic
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—341no—
404C2×C22⋊C4—341no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C42—21no—
408C2×C42—171no—
409C2×C42—171no—
410C2×C42—171no—
411C2×C42—171no—
412C2×C42—171no—
413C2×C42—171no—
414C2×C42—341no—
415C2×C42—341no—
416C2×C42C22×D1711yescharacteristic
417#103—174no—
418#103—174no—
419#103—342no—
420#103—342no—
421#102—22no—
422#102—174no—
423#102—342no—
424#102C4×D1714yes—
425#97—22no—
426#97—174no—
427#97—342no—
428#97—342no—
429C4×C8—342no—
430C4×C8—342no—
431#95—21no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439C17#2165511yescharacteristic, complemented, Sylow
440C24—21no—
441C2×Q8—172no—
442C2×Q8—344no—
443C2×Q8—341no—
444C2×Q8—341no—
445C2×Q8—341no—
446C2×Q8—341no—
447C2×Q8—341no—
448C2×Q8—682no—
449C2×Q8—682no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×C4×D1712yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—171no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4D4×D1711yescharacteristic
471C22×C4D4×D1711yescharacteristic
472C22×C4C23×D1711yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—172no—
478C2×C8—172no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8C2×C4×D1712yes—
484C2×C8#6325212yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—172no—
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—682no—
500C4⋊C4—682no—
501C4⋊C4—682no—
502C4⋊C4C2×C4×D1712yes—
503C22⋊C4—42no—
504C22⋊C4—344no—
505C22⋊C4—344no—
506C22⋊C4—682no—
507C22⋊C4—682no—
508C42—42no—
509C42—172no—
510C42—172no—
511C42—342no—
512C42—342no—
513C42—341no—
514C42—341no—
515C42—681no—
516C42—681no—
517C42—682no—
518C42—682no—
519C42#6325212yes—
520C42#6325412yes—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×D4×D1711yescharacteristic
529Q8—342no—
530Q8—342no—
531Q8—688no—
532Q8—681no—
533Q8—681no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—172no—
553C2×C4—172no—
554C2×C4—172no—
555C2×C4—172no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—342no—
565C2×C4—342no—
566C2×C4—681no—
567C2×C4—681no—
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#1058757112yes—
577C2×C4#1058757112yes—
578C2×C4C22×C4×D1711yescharacteristic
579C2×C4C2×#6325211yescharacteristic
580C2×C4C2×#6325211yescharacteristic
581C2×C4C2×#6325411yescharacteristic
582C2×C4#41×D1711yescharacteristic
583C2×C4#41×D1711yescharacteristic
584C8—22no—
585C8—22no—
586C8—342no—
587C8—342no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D1711yescharacteristic
596C22C2×#1058757111yescharacteristic
597C22#1176335611yescharacteristic
598C22AGL(1,ℤ/8)×D1711yescharacteristic
599C22#137×D1711yescharacteristic
600C22#1176360511yescharacteristic
601C22#1176360511yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—342no—
608C4—342no—
609C4—342no—
610C4—342no—
611C4—682no—
612C4—682no—
613C4—1361no—
614C4—1361no—
615C4C4×#6325411yescharacteristic
616C4C4×#6325211yescharacteristic
617C4#1176333711yescharacteristic
618C4#1176333911yescharacteristic
619C2—82no—
620C2C4×#1058757111yescharacteristic
621C2#1453759511yescharacteristic
622C2#1453760611yescharacteristic
623C2#1453916811yescharacteristic
624C2#1453917511yescharacteristic
625C2#1453948011yescharacteristic
626C2#1453948911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1762C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1762D17
hypocenterC17#21655
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C4×#10587574C2semidirect
4. #14529755C2semidirect
8. #14529461C2semidirect
9. #14529458C2semidirect
10. #14528916C2semidirect
11. #14528617C2semidirect
12. #14528615C2semidirect
14. #97×Dic17C2semidirect
20. #11763167C4semidirect
100. #10587574C2×C4semidirect
118. #10587487C2×C4semidirect
120. #10587486C2×C4semidirect
124. #10587481C2×C4semidirect
439. C17#21655semidirect

Maximal subgroups

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

subgroupindex
C2×#117631672
C4×#105875742
C4×#105876222
#145297552
#145295992
#145294692
#145294642
#145294612
#145294582
#145289162
#145286172
#145286152
#145286092
#97×Dic172
C17×#17622
#2165517