Finite Group Databasebeta

← #17040552

Subgroups — #17040552

Order 4352.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763031C211yesmaximal, characteristic
2C2×#11762901C211yesmaximal, characteristic, complemented
3C2×#11762894C211yesmaximal, characteristic
4#14531323C211yesmaximal, characteristic
5C4×Q8×Dic17C211yesmaximal, characteristic, complemented
6#14529714C211yesmaximal, characteristic
7#14529550C211yesmaximal, characteristic, complemented
8#14529416C211yesmaximal, characteristic, complemented
9#14529402C211yesmaximal, characteristic
10#14529137C211yesmaximal, characteristic
11#14529123C211yesmaximal, characteristic, complemented
12C4⋊C8×Dic17C211yesmaximal, characteristic, complemented
13#14528603C211yesmaximal, characteristic, complemented
14#14528598C211yesmaximal, characteristic
15C17×#1805C211yesmaximal, characteristic, complemented
16C22×#63231C2211yescharacteristic
17C22×#63230C2211yescharacteristic
18C2×#10587561C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20C2×Q8×Dic17C2211yescharacteristic
21C2×#10587573C2211yescharacteristic
22C2×#10587579C2211yescharacteristic
23#11763031C418yescomplemented
24#11762901—24no—
25#11762894—24no—
26C2×#10587478C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33C2×#10587492C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37#11762714C2211yescharacteristic
38#11762710C2211yescharacteristic
39#11762698—21no—
40#11762698—21no—
41C4⋊C4×Dic17—21no—
42C4⋊C4×Dic17—21no—
43C4⋊C4×Dic17C2211yescharacteristic
44C4⋊C4×Dic17C2211yescharacteristic
45C4×C4⋊Dic17—21no—
46C4×C4⋊Dic17C2211yescharacteristic
47#11762656—21no—
48#11762656—21no—
49C42×Dic17—21no—
50C42×Dic17C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762563C2211yescharacteristic
54#11762563C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C17×#382C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×Dic34C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C4×Q8×C17—22no—
69C4×Q8×C17C2×C414yescomplemented
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C42×C17—21no—
75C2×C42×C17C2311yescharacteristic
76Q8×Dic17—22no—
77Q8×Dic17—22no—
78Q8×Dic17—22no—
79Q8×Dic17D414yes—
80C2×#63231—22no—
81C2×#63231—22no—
82C2×#63231C2×C414yescomplemented
83C2×#63230—22no—
84C2×#63230—22no—
85C2×#63230C2×C414yes—
86C2×#63227—21no—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17D411yescharacteristic
103C2×C4×Dic17D411yescharacteristic
104C2×C4×Dic17D411yescharacteristic
105C2×C4×Dic17D411yescharacteristic
106C2×C4×Dic17C2311yescharacteristic
107C2×C4×Dic17C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110#10587579—22no—
111#10587579—22no—
112#10587573—22no—
113#10587573—22no—
114#10587561—22no—
115#10587561C2×C414yes—
116C17×C4⋊C8—22no—
117C17×#103—22no—
118C17×#103—22no—
119C17×#95—21no—
120C17×#95—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587492—22no—
127#10587492—22no—
128#10587487—22no—
129#10587487—22no—
130C8×Dic17—22no—
131C8×Dic17—22no—
132#10587481C2×C414yescomplemented
133#10587480C2×C414yes—
134#10587479—22no—
135#10587478—22no—
136C4⋊#3622C2×C414yescomplemented
137C23×D17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17C22×C412yes—
142C2×Q8×C17C2×D412yes—
143C2×Q8×C17#4112yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156C22×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68C22×C412yes—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C2×C4×D17—42no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34C22×C412yes—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C4⋊C4×C17—24no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17C22×C412yes—
186C42×C17—24no—
187C42×C17C22×C412yes—
188C42×C17#4112yes—
189#63231—28no—
190#63230—28no—
191#63227—24no—
192#63227—24no—
193#63227—42no—
194#63227—42no—
195#63227—42no—
196C4⋊Dic17—24no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—24no—
200C4⋊Dic17C2×D412yes—
201C4⋊Dic17#4112yes—
202#63225—42no—
203C4×Dic17—24no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—24no—
207C4×Dic17—22no—
208C4×Dic17—22no—
209C4×Dic17—22no—
210C4×Dic17—41no—
211C4×Dic17—42no—
212C4×Dic17—42no—
213C4×Dic17—42no—
214C4×Dic17—42no—
215C4×Dic17C2×D412yes—
216C4×Dic17#4112yes—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622C22×C412yes—
222C2×#3622C22×C412yes—
223C4×#464—171nomaximal, Sylow
224C23×C17C22×D411yescharacteristic
225C22×D17—22no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231C22×D17—41no—
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17—22no—
235Q8×C17C4×D414yes—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17C4×D412yes—
253C2×C4×C17C4×D412yes—
254C2×C4×C17C23×C411yescharacteristic
255C2×C4×C17C22×D411yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17#12112yes—
277C2×Dic17#12112yes—
278C2×Dic17#13512yes—
279C2×Dic17#13512yes—
280C2×Dic17#13712yes—
281D68—22no—
282D68—22no—
283D68—41no—
284D68—41no—
285C4×D17—84no—
286Dic34—22no—
287Dic34—22no—
288Dic34—41no—
289Dic34—41no—
290C8×C17—41no—
291C8×C17—41no—
292#3622—22no—
293#3622—22no—
294C2×#464—171no—
295C2×C4×Q16—171no—
296C2×C4×SD16—171no—
297C4×#124—171no—
298C42×Q8—171no—
299#1983—171no—
300#1949—171no—
301#1927—171no—
302#1925—171no—
303#1880—171no—
304#1878—171no—
305#1805D1711yescharacteristic, complemented
306C4×C4⋊C8—171no—
307C4×#103—171no—
308C4×#102—171no—
309C22×C17C2×C4×D411yescharacteristic
310C22×C17C2×#12111yescharacteristic
311C22×C17#52511yescharacteristic
312C22×C17C2×#13511yescharacteristic
313C22×C17C2×#13511yescharacteristic
314C22×C17C2×#13711yescharacteristic
315C22×C17#57511yescharacteristic
316C2×D17—42no—
317C2×D17—42no—
318C2×D17—42no—
319C2×D17—81no—
320C2×D17—81no—
321C2×D17—81no—
322C2×D17—81no—
323C4×C17—22no—
324C4×C17—22no—
325C4×C17—24no—
326C4×C17—42no—
327C4×C17—81no—
328C4×C17—81no—
329C4×C17C2×C4×D411yescharacteristic
330C4×C17C4×#4111yescharacteristic
331C4×C17#52911yescharacteristic
332C4×C17#53211yescharacteristic
333Dic17—22no—
334Dic17—22no—
335Dic17—41no—
336Dic17—42no—
337Dic17—42no—
338Dic17—42no—
339Dic17#46414yescomplemented
340C22×Q16—171no—
341C22×SD16—171no—
342C2×#124—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8—171no—
345C2×C4×Q8—171no—
346C2×C4×Q8C2×D1711yescharacteristic
347C2×C4×D4—171no—
348#464—178no—
349C4×Q16—344no—
350C4×SD16—344no—
351C2×#107—171no—
352C2×#106—171no—
353C2×C4⋊C8—171no—
354C2×C4⋊C8C2×D1711yescharacteristic
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103C2×D1711yescharacteristic
359C2×#103C2×D1711yescharacteristic
360C2×#102—171no—
361C2×#102—171no—
362C2×#102—171no—
363C2×C4×C8—171no—
364C2×C4×C8—171no—
365#391—171no—
366#390—171no—
367#388—341no—
368#388—341no—
369#382C2×D1711yescharacteristic
370C4×C4⋊C4—171no—
371C4×C4⋊C4—171no—
372C4×C4⋊C4—171no—
373C4×C4⋊C4—341no—
374C4×C4⋊C4—341no—
375C4×C4⋊C4—341no—
376C4×C22⋊C4—341no—
377C4×C22⋊C4—341no—
378C43—171no—
379C43—341no—
380#340—171no—
381#340—171no—
382#340C2×D1711yescharacteristic
383#340C2×D1711yescharacteristic
384#336—171no—
385#336—171no—
386C2×C17C4×#12111yescharacteristic
387C2×C17C4×#13511yescharacteristic
388C2×C17C4×#13711yescharacteristic
389C2×C17C2×#46411yescharacteristic
390C2×C17#303511yescharacteristic
391C2×C17#310211yescharacteristic
392C2×C17#310811yescharacteristic
393D17—82no—
394C22×Q8—171no—
395C22×Q8C22×D1711yescharacteristic
396C22×D4—171no—
397C23×C4—341no—
398C2×Q16—174no—
399C2×Q16—342no—
400C2×Q16—342no—
401C2×SD16—174no—
402C2×SD16—342no—
403C2×SD16—342no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408#124—174no—
409#124—342no—
410C4×Q8—22no—
411C4×Q8—174no—
412C4×Q8—342no—
413C4×Q8—342no—
414C4×Q8—342no—
415C4×Q8—342no—
416C4×Q8—342no—
417C4×Q8C4×D1714yescomplemented
418C4×D4—342no—
419C4×D4—342no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—21no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—171no—
438C2×C42—171no—
439C2×C42—171no—
440C2×C42—171no—
441C2×C42—341no—
442C2×C42—341no—
443C2×C42—341no—
444C2×C42—341no—
445C2×C42—341no—
446C2×C42—341no—
447C2×C42C22×D1711yescharacteristic
448#107—342no—
449#106—342no—
450C4⋊C8—22no—
451C4⋊C8—174no—
452#103—22no—
453#103—22no—
454#103—174no—
455#103—342no—
456#103—342no—
457#102—174no—
458#102—342no—
459#102—342no—
460C4×C8—342no—
461C4×C8—342no—
462#95—21no—
463#95—21no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469C17C4×#46411yescharacteristic, complemented, Sylow
470C24—341no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—172no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8C2×C4×D1712yes—
481C2×Q8D4×D1712yes—
482C2×Q8#6325612yes—
483C2×D4—172no—
484C2×D4—341no—
485C2×D4—341no—
486C2×D4—341no—
487C2×D4—341no—
488C2×D4—341no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—171no—
494C22×C4—171no—
495C22×C4—171no—
496C22×C4—171no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—682no—
501C22×C4—682no—
502C22×C4—682no—
503C22×C4D4×D1711yescharacteristic
504C22×C4D4×D1711yescharacteristic
505C22×C4C23×D1711yescharacteristic
506Q16—348no—
507SD16—348no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—172no—
515C2×C8—172no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C4⋊C4—21no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—172no—
527C4⋊C4—172no—
528C4⋊C4—344no—
529C4⋊C4—344no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—682no—
533C4⋊C4C2×C4×D1712yes—
534C22⋊C4—344no—
535C22⋊C4—344no—
536C22⋊C4—682no—
537C22⋊C4—682no—
538C22⋊C4—682no—
539C42—24no—
540C42—172no—
541C42—172no—
542C42—342no—
543C42—342no—
544C42—342no—
545C42—344no—
546C42—344no—
547C42—341no—
548C42—341no—
549C42—681no—
550C42—682no—
551C42—682no—
552C42—682no—
553C42—682no—
554C42C2×C4×D1712yes—
555C42#6325412yes—
556C23—342no—
557C23—681no—
558C23—681no—
559C23—681no—
560C23—681no—
561C23—681no—
562C23—681no—
563C23C2×D4×D1711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—342no—
568Q8—342no—
569Q8—681no—
570Q8—681no—
571Q8#1058757914yes—
572D4—342no—
573D4—342no—
574D4—681no—
575D4—681no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—172no—
593C2×C4—172no—
594C2×C4—172no—
595C2×C4—172no—
596C2×C4—172no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—342no—
602C2×C4—342no—
603C2×C4—342no—
604C2×C4—341no—
605C2×C4—341no—
606C2×C4—341no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—1364no—
614C2×C4#1058756712yes—
615C2×C4#1058756712yes—
616C2×C4C22×C4×D1711yescharacteristic
617C2×C4C2×D4×D1711yescharacteristic
618C2×C4C2×#6325411yescharacteristic
619C2×C4C2×#6325611yescharacteristic
620C2×C4#41×D1711yescharacteristic
621C2×C4#41×D1711yescharacteristic
622C8—41no—
623C8—41no—
624C8—342no—
625C8—342no—
626C22—682no—
627C22—682no—
628C22—682no—
629C22—1361no—
630C22—1361no—
631C22—1361no—
632C22—1361no—
633C22C2×#1058756711yescharacteristic
634C22#121×D1711yescharacteristic
635C22#1176332211yescharacteristic
636C22#135×D1711yescharacteristic
637C22#135×D1711yescharacteristic
638C22#137×D1711yescharacteristic
639C22#1176360811yescharacteristic
640C4—22no—
641C4—22no—
642C4—24no—
643C4—42no—
644C4—81no—
645C4—81no—
646C4—174no—
647C4—342no—
648C4—342no—
649C4—681no—
650C4—682no—
651C4—682no—
652C4—682no—
653C4C2×#1058757911yescharacteristic
654C4#1176327711yescharacteristic
655C4#1176334511yescharacteristic
656C4#1176335411yescharacteristic
657C2—1362no—
658C2#1453129011yescharacteristic
659C2#1453755211yescharacteristic
660C2#1453759911yescharacteristic
661C2D17×#46411yescharacteristic
662C2#1453823811yescharacteristic
663C2#1453880911yescharacteristic
664C2#1453886911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1805C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1805D17
hypocenterC17C4×#464
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#11762901C2semidirect
5. C4×Q8×Dic17C2semidirect
7. #14529550C2semidirect
8. #14529416C2semidirect
11. #14529123C2semidirect
12. C4⋊C8×Dic17C2semidirect
13. #14528603C2semidirect
15. C17×#1805C2semidirect
23. #11763031C4semidirect
69. C4×Q8×C17C2×C4semidirect
82. C2×#63231C2×C4semidirect
132. #10587481C2×C4semidirect
136. C4⋊#3622C2×C4semidirect
305. #1805D17semidirect
339. Dic17#464semidirect
417. C4×Q8C4×D17semidirect
469. C17C4×#464semidirect

Maximal subgroups

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

subgroupindex
C2×#117630312
C2×#117629012
C2×#117628942
#145313232
C4×Q8×Dic172
#145297142
#145295502
#145294162
#145294022
#145291372
#145291232
C4⋊C8×Dic172
#145286032
#145285982
C17×#18052
C4×#46417