Finite Group Databasebeta

← #17068675

Subgroups — #17068675

Order 4352.

1394 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×#11763359C211yesmaximal, characteristic
2C2×#11763094C211yesmaximal, characteristic, complemented
3C2×#11763009C211yesmaximal, characteristic, complemented
4#14529973C212yesmaximal, complemented
5#14529785C212yesmaximal, complemented
6#14529736C211yesmaximal, characteristic, complemented
7#14529711C211yesmaximal, characteristic, complemented
8#14529121C212yesmaximal, complemented
9#14529044C212yesmaximal, complemented
10#14528764C211yesmaximal, characteristic, complemented
11C4×C17×C22⋊Q8C211yesmaximal, characteristic, complemented
12C22×#63252—21no—
13C22×#63230C2212yes—
14#11763359—24no—
15#11763359—24no—
16C2×#10587555C2211yescharacteristic
17C2×#10587560C2211yescharacteristic
18C2×#10587561—21no—
19C2×#10587616—21no—
20C2×#10587616C2212yes—
21C2×C4×D68—21no—
22C2×C4×D68C2212yes—
23#11763094—24no—
24#11763009—24no—
25C2×#10587479C2212yes—
26C2×C4⋊#3622C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587506C2212yes—
29C2×#10587480C2212yes—
30C2×#10587480C2212yes—
31C2×#10587480C2212yes—
32C2×C22⋊#3622C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35#11762564C2212yes—
36C2×C17×C22⋊Q8C2211yescharacteristic
37C2×C4×Q8×C17C2211yescharacteristic
38C22×C42×C17C2211yescharacteristic
39C17×#391C2211yescharacteristic
40C17×#389C2211yescharacteristic
41C17×#387—21no—
42C17×#385—21no—
43C17×#380C2211yescharacteristic
44C4×C4⋊C4×C17—21no—
45C4×C4⋊C4×C17C2211yescharacteristic
46C4×C22⋊C4×C17—21no—
47C22×Q8×C17C2311yescharacteristic
48C23×C4×C17D411yescharacteristic
49C23×C4×C17D411yescharacteristic
50C23×C4×C17C2311yescharacteristic
51C22×C22⋊D17—22no—
52C2×#63252—28no—
53C2×#63252—44no—
54C2×#63252—44no—
55C22×D68—21no—
56C22×D68—21no—
57C22×D68C2312yes—
58C22×C4×D17—22no—
59C22×Dic34—21no—
60C17×C22⋊Q8—22no—
61C17×C22⋊Q8—22no—
62C4×Q8×C17—22no—
63C4×Q8×C17—22no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C42×C17—22no—
73C2×C42×C17—22no—
74C2×C42×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17D411yescharacteristic
77C2×C42×C17D411yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×#63230—24no—
81C2×#63230—24no—
82C2×#63230—42no—
83#10587616—28no—
84#10587616—24no—
85#10587616—24no—
86C2×#63227—22no—
87C2×#63227—22no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2312yes—
90C22×#3622C2312yes—
91C22×#3622C2312yes—
92#10587561—28no—
93#10587560—22no—
94#10587560D414yes—
95C4×D68—28no—
96C4×D68—24no—
97C4×D68—24no—
98#10587555—22no—
99#10587555D414yes—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105#10587506—24no—
106C22⋊#3622—42no—
107#10587480—24no—
108#10587480—24no—
109#10587480—42no—
110#10587479—24no—
111C4⋊#3622—22no—
112C4⋊#3622—22no—
113C24×C17C2×D411yescharacteristic
114C23×D17—22no—
115C2×Q8×C17—21no—
116C2×Q8×C17—21no—
117C2×Q8×C17—21no—
118C2×Q8×C17—21no—
119C2×Q8×C17—21no—
120C2×Q8×C17#4112yescomplemented
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—41no—
135C22×C4×C17—41no—
136C22×C4×C17SD1614yescomplemented
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2411yescharacteristic
143C2×C22⋊D17—28no—
144C2×C22⋊D17—44no—
145C2×C22⋊D17—44no—
146C22×Dic17—22no—
147#63252—432no—
148C2×D68—22no—
149C2×D68—22no—
150C2×D68—22no—
151C2×D68—22no—
152C2×D68—24no—
153C2×D68—24no—
154C2×D68—22no—
155C2×D68—42no—
156C2×D68—42no—
157C2×D68—42no—
158C2×D68—42no—
159C2×D68C2×D414yes—
160C2×C4×D17—28no—
161C2×C4×D17—44no—
162C2×C4×D17—44no—
163C2×Dic34—24no—
164C2×Dic34—42no—
165C2×Dic34—42no—
166C4⋊C4×C17—24no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—42no—
173C4⋊C4×C17—42no—
174C4⋊C4×C17#4112yescomplemented
175C22⋊C4×C17—24no—
176C22⋊C4×C17—42no—
177C42×C17—22no—
178C42×C17—22no—
179C42×C17—21no—
180C42×C17—21no—
181C42×C17—42no—
182C42×C17—42no—
183C42×C17C2×D412yes—
184C42×C17C2×D412yes—
185#63230—48no—
186#63227—28no—
187#63227—28no—
188C4⋊Dic17—24no—
189C4⋊Dic17—22no—
190C4⋊Dic17C2×D414yes—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197#24154—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—21no—
204C23×C17C2×SD1612yes—
205C23×C17C22×D411yescharacteristic
206C22×D17—24no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213Q8×C17—22no—
214Q8×C17—22no—
215Q8×C17—41no—
216Q8×C17—41no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—28no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—44no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17#12012yes—
246C2×C4×C17#12012yes—
247C2×C4×C17C2×SD1612yes—
248C2×C4×C17C2×SD1612yes—
249C2×C4×C17AGL(1,ℤ/8)12yes—
250C2×C4×C17C22×D411yescharacteristic
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17H(5,𝔽2)11yescharacteristic
255C2×C4×C172−1+411yescharacteristic
256C22⋊D17—416no—
257C2×Dic17—24no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260D68—24no—
261D68—24no—
262D68—48no—
263D68—42no—
264D68—42no—
265D68—48no—
266C4×D17—416no—
267Dic34—48no—
268#3622—42no—
269#3622—42no—
270C2×#532—171no—
271C2×#475—171no—
272C2×#460—171no—
273C4×C22⋊Q8D1711yescharacteristic, complemented
274#2034—172no—
275#2000—172no—
276#1988—171no—
277#1982—171no—
278#1877—172no—
279#1864—172no—
280#1797—171no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—22no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17C2×#12011yescharacteristic
289C22×C17#52511yescharacteristic
290C22×C17#53711yescharacteristic
291C22×C17C22×SD1611yescharacteristic
292C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C17#57511yescharacteristic
294C22×C17#57511yescharacteristic
295C2×D17—44no—
296C2×D17—44no—
297C2×D17—44no—
298C2×D17—82no—
299C2×D17—82no—
300C2×D17—82no—
301C2×D17—82no—
302C4×C17—22no—
303C4×C17—22no—
304C4×C17—42no—
305C4×C17—41no—
306C4×C17—42no—
307C4×C17—41no—
308C4×C17—81no—
309C4×C17—81no—
310C4×C17#45014yes—
311C4×C17#54611yescharacteristic
312C4×C17#54711yescharacteristic
313C4×C17#54811yescharacteristic
314C4×C17#55011yescharacteristic
315Dic17—44no—
316C22×#41—341no—
317C22×SD16—172no—
318#532—344no—
319#532—344no—
320C2×C4⋊Q8—171no—
321C2×#127—171no—
322C2×#124—341no—
323C2×C22⋊Q8C2×D1711yescharacteristic
324C2×#121—172no—
325C2×#121—341no—
326C2×C4×Q8C2×D1711yescharacteristic
327C2×C4×D4—172no—
328C2×C4×D4—341no—
329C22×C42C2×D1711yescharacteristic
330#475—344no—
331#460—344no—
332C2×#106—172no—
333C2×C4⋊C8—171no—
334C2×C4⋊C8—171no—
335C2×#103—172no—
336C2×#102—172no—
337C2×#102—172no—
338C2×#102—172no—
339C2×C22⋊C8—172no—
340#391C2×D1711yescharacteristic
341#389C2×D1711yescharacteristic
342#387—21no—
343#385—21no—
344#380C2×D1711yescharacteristic
345C4×C4⋊C4—21no—
346C4×C4⋊C4C2×D1711yescharacteristic
347C4×C22⋊C4—21no—
348#340—172no—
349#340—172no—
350#336—172no—
351C2×C17—42no—
352C2×C17#240711yescharacteristic
353C2×C17C2×#45011yescharacteristic
354C2×C17#300511yescharacteristic
355C2×C17#310311yescharacteristic
356C2×C17#311511yescharacteristic
357C2×C17#324611yescharacteristic
358C2×C17#325111yescharacteristic
359D17—84no—
360C2×#41—348no—
361C2×#41—684no—
362C2×#41—684no—
363C22×Q8—341no—
364C22×Q8C22×D1711yescharacteristic
365C22×D4—172no—
366C22×D4—342no—
367C22×D4—341no—
368C22×D4—341no—
369C23×C4—342no—
370C23×C4C22⋊D1711yescharacteristic
371C23×C4C22⋊D1711yescharacteristic
372C23×C4C22×D1711yescharacteristic
373C2×SD16—344no—
374C2×SD16—344no—
375C2×SD16—682no—
376C22×C8—172no—
377C22×C8—172no—
378C4⋊Q8—174no—
379C4⋊Q8—342no—
380#127—174no—
381#127—342no—
382#124—348no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385#121—344no—
386#121—344no—
387#121—348no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×D4—344no—
391C4×D4—344no—
392C4×D4—348no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—172no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C4⋊C4C22×D1711yescharacteristic
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—342no—
404C2×C22⋊C4—342no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42C22⋊D1711yescharacteristic
410C2×C42C22⋊D1711yescharacteristic
411C2×C42C22×D1711yescharacteristic
412C2×C42C22×D1711yescharacteristic
413#106—344no—
414C4⋊C8—342no—
415C4⋊C8—342no—
416#103—344no—
417#102—344no—
418#102—344no—
419#102—682no—
420C22⋊C8—682no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426C17#2415411yescharacteristic, complemented, Sylow
427C24—342no—
428C24C2×C22⋊D1711yescharacteristic
429#41—6832no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—344no—
436C2×Q8—682no—
437C2×Q8—682no—
438C2×Q8#6325212yescomplemented
439C2×D4—174no—
440C2×D4—348no—
441C2×D4—342no—
442C2×D4—342no—
443C2×D4—342no—
444C2×D4—342no—
445C2×D4—342no—
446C2×D4—344no—
447C2×D4—344no—
448C2×D4—682no—
449C2×D4—682no—
450C2×D4—684no—
451C2×D4—684no—
452C2×D4—682no—
453C2×D4—682no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—348no—
470C22×C4—342no—
471C22×C4—684no—
472C22×C4—684no—
473C22×C4#6323014yescomplemented
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C2×C22⋊D1711yescharacteristic
478C22×C4C2×C22⋊D1711yescharacteristic
479C22×C4C23×D1711yescharacteristic
480SD16—688no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—342no—
484C2×C8—342no—
485C2×C8—342no—
486C2×C8—342no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—174no—
496C4⋊C4—342no—
497C4⋊C4—344no—
498C4⋊C4#6325212yescomplemented
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—348no—
502C22⋊C4—348no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42D4×D1712yes—
510C42D4×D1712yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—344no—
518C23—682no—
519C23—682no—
520C23—682no—
521C23—682no—
522C23—682no—
523C23—682no—
524C23C2×#6323012yes—
525C23C22×C22⋊D1711yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—688no—
531D4—344no—
532D4—344no—
533D4—688no—
534D4—6816no—
535D4—682no—
536D4—688no—
537D4—682no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—344no—
567C2×C4—6816no—
568C2×C4—682no—
569C2×C4—682no—
570C2×C4#1058762312yes—
571C2×C4#1058762312yes—
572C2×C4C2×#6323012yes—
573C2×C4C2×#6323012yes—
574C2×C4#1058763512yes—
575C2×C4C2×#6325211yescharacteristic
576C2×C4C2×#6325211yescharacteristic
577C2×C4C2×D4×D1711yescharacteristic
578C2×C4C2×D4×D1711yescharacteristic
579C2×C4#1058769611yescharacteristic
580C2×C4#1058769711yescharacteristic
581C8—682no—
582C8—682no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—684no—
591C22—684no—
592C22—684no—
593C22—1362no—
594C22—1362no—
595C22—1362no—
596C22—1362no—
597C22C2×#1058762311yescharacteristic
598C22#1176332011yescharacteristic
599C22#1176338711yescharacteristic
600C22C22×#6323011yescharacteristic
601C22C2×#1058763511yescharacteristic
602C22#1176360711yescharacteristic
603C22#1176360711yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4—684no—
613C4#1176295914yes—
614C4#1176345511yescharacteristic
615C4#1176346411yescharacteristic
616C4#1176346811yescharacteristic
617C4#1176348011yescharacteristic
618C2—42no—
619C2—1364no—
620C2#1453212911yescharacteristic
621C2C2×#1176295911yescharacteristic
622C2#1453802411yescharacteristic
623C2#1453881511yescharacteristic
624C2#1453893511yescharacteristic
625C2#1454009611yescharacteristic
626C2#1454014611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×C22⋊Q8C2
socleC23×C17C22×D4
prime radicalC2×#11763359C2
hypercenterC4×C22⋊Q8D17
hypocenterC17#24154
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11763094C2semidirect
3. C2×#11763009C2semidirect
4. #14529973C2semidirect
5. #14529785C2semidirect
6. #14529736C2semidirect
7. #14529711C2semidirect
8. #14529121C2semidirect
9. #14529044C2semidirect
10. #14528764C2semidirect
11. C4×C17×C22⋊Q8C2semidirect
120. C2×Q8×C17#41semidirect
136. C22×C4×C17SD16semidirect
174. C4⋊C4×C17#41semidirect
273. C4×C22⋊Q8D17semidirect
426. C17#24154semidirect
438. C2×Q8#63252semidirect
473. C22×C4#63230semidirect
498. C4⋊C4#63252semidirect

Maximal subgroups

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

subgroupindex
C2×#117633592
C2×#117630942
C2×#117630092
#145299732
#145297852
#145297362
#145297112
#145291212
#145290442
#145287642
C4×C17×C22⋊Q82
#2415417