Finite Group Databasebeta

← #17040637

Subgroups — #17040637

Order 4352.

1004 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×#11763033C211yesmaximal, characteristic
2C2×#11762907C211yesmaximal, characteristic
3C4×#10587633C211yesmaximal, characteristic
4C4×Q8×Dic17C211yesmaximal, characteristic, complemented
5#14529713C211yesmaximal, characteristic
6#14529553C211yesmaximal, characteristic, complemented
7#14529428C211yesmaximal, characteristic, complemented
8#14529410C211yesmaximal, characteristic
9#14529137C211yesmaximal, characteristic
10#14529127C211yesmaximal, characteristic, complemented
11#14528797C211yesmaximal, characteristic, complemented
12#14528638C211yesmaximal, characteristic, complemented
13#14528614C211yesmaximal, characteristic, complemented
14#14528610C211yesmaximal, characteristic
15C2×C17×#441C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×#10587633C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21#11763033C418yescomplemented
22#11762907—24no—
23C2×#10587478C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31C2×#10587480C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33#11762716C2211yescharacteristic
34#11762710C2211yescharacteristic
35#11762699—21no—
36#11762699—21no—
37#11762674C2211yescharacteristic
38C4⋊C4×Dic17—21no—
39C4⋊C4×Dic17—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4×C4⋊Dic17—21no—
43C4×C4⋊Dic17C2211yescharacteristic
44C4×#63227—21no—
45C4×#63227—21no—
46C42×Dic17—21no—
47C42×Dic17C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×Q16×C17C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C4×Q8×C17C2211yescharacteristic
57C17×#441—24no—
58C2×C17×#106C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C2×C17×#103C2211yescharacteristic
61C2×C17×#97C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×Q8×C17C2311yescharacteristic
64C22×D68C2311yescharacteristic
65C22×C4×D17—21no—
66C2×Q16×C17—22no—
67C2×Q16×C17—22no—
68C2×Q16×C17C2×C414yescomplemented
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C4×Q8×C17—22no—
72C4×Q8×C17—22no—
73C4×Q8×C17—22no—
74C4×Q8×C17D414yes—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82#10587633—22no—
83#10587633C2×C414yes—
84Q8×Dic17—22no—
85Q8×Dic17—22no—
86Q8×Dic17—22no—
87Q8×Dic17C2×C414yescomplemented
88C2×#63230—22no—
89C2×#63230—22no—
90C2×#63230C2×C414yes—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×#63227—21no—
95C2×#63227—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—21no—
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17D411yescharacteristic
107C2×C4×Dic17D411yescharacteristic
108C2×C4×Dic17D411yescharacteristic
109C2×C4×Dic17D411yescharacteristic
110C2×C4×Dic17C2311yescharacteristic
111C2×C4×Dic17C2311yescharacteristic
112C22×#3622C2311yescharacteristic
113C22×#3622C2311yescharacteristic
114C4×D68—22no—
115C4×D68—22no—
116C17×#106—22no—
117C17×#103—22no—
118C17×#103—22no—
119C17×#97—22no—
120#10587506—22no—
121#10587506—22no—
122#10587506C2×C414yescomplemented
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587492—22no—
131#10587492C2×C414yes—
132#10587485—22no—
133#10587485C2×C414yescomplemented
134#10587480—22no—
135#10587478—22no—
136#10587474—22no—
137C23×D17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17C22×C412yes—
147C2×Q8×C17C22×C412yes—
148C2×Q8×C17C2×D412yes—
149C2×Q8×C17#4112yes—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2411yescharacteristic
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164C2×D68—21no—
165C2×D68—21no—
166C2×D68—21no—
167C2×D68—21no—
168C2×D68—21no—
169C2×D68C22×C412yes—
170C2×C4×D17—42no—
171C2×C4×D17—42no—
172C2×C4×D17—42no—
173Q16×C17—28no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C22×C412yes—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—42no—
185C4⋊C4×C17C2×D412yes—
186C42×C17—24no—
187C42×C17—42no—
188C42×C17C2×D412yes—
189C42×C17#4112yes—
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⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17C22×C412yes—
203C4⋊Dic17#4112yes—
204C4×Dic17—24no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—24no—
208C4×Dic17—22no—
209C4×Dic17—41no—
210C4×Dic17—41no—
211C4×Dic17—41no—
212C4×Dic17—42no—
213C4×Dic17—42no—
214C4×Dic17—42no—
215C4×Dic17C22×C412yes—
216C4×Dic17#4112yes—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622C22×C412yes—
223#21508—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×C17—22no—
236Q8×C17—22no—
237Q8×C17—41no—
238Q8×C17—41no—
239Q8×C17C4×D414yes—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17#12112yes—
257C2×C4×C17#12112yes—
258C2×C4×C17C23×C411yescharacteristic
259C2×C4×C17C22×D411yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17C4×D412yes—
283C2×Dic17C4×D412yes—
284C2×Dic17#13712yes—
285D68—22no—
286D68—22no—
287D68—41no—
288D68—41no—
289C4×D17—84no—
290C8×C17—22no—
291C8×C17—41no—
292#3622—22no—
293#3622—41no—
294C2×#464—171no—
295C2×#441D1711yescharacteristic, complemented
296C2×#440—171no—
297C4×#124—171no—
298C42×Q8—171no—
299#1983—171no—
300#1949—171no—
301#1928—171no—
302#1926—171no—
303#1880—171no—
304#1878—171no—
305#1805—171no—
306#1767—171no—
307#1763—171no—
308#1762—171no—
309C22×C17C2×C4×D411yescharacteristic
310C22×C17C2×#12111yescharacteristic
311C22×C17#52511yescharacteristic
312C22×C17C2×#13711yescharacteristic
313C22×C17#57511yescharacteristic
314C22×C17#57511yescharacteristic
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—42no—
328C4×C17—81no—
329C4×C17C2×C4×D411yescharacteristic
330C4×C17C4×#4111yescharacteristic
331C4×C17#52911yescharacteristic
332C4×C17#53211yescharacteristic
333Dic17—22no—
334Dic17—22no—
335Dic17—22no—
336Dic17—41no—
337Dic17—42no—
338Dic17—42no—
339Dic17—81no—
340C22×Q16C2×D1711yescharacteristic
341C22×SD16—171no—
342C2×#124—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8—171no—
345C2×C4×Q8C2×D1711yescharacteristic
346C2×C4×Q8C2×D1711yescharacteristic
347C2×C4×D4—171no—
348#464—178no—
349#441—24no—
350#440—344no—
351C2×#107—171no—
352C2×#106C2×D1711yescharacteristic
353C2×C4⋊C8—171no—
354C2×C4⋊C8—171no—
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×#97—171no—
364C2×#97C2×D1711yescharacteristic
365#391—171no—
366#390—171no—
367#388—341no—
368#388—341no—
369#382—171no—
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#340—171no—
383#340—171no—
384#336—171no—
385#336—171no—
386C2×C17C4×#12111yescharacteristic
387C2×C17#294311yescharacteristic
388C2×C17C4×#13711yescharacteristic
389C2×C17#303711yescharacteristic
390C2×C17#304011yescharacteristic
391C2×C17#311111yescharacteristic
392C2×C17#311711yescharacteristic
393D17—82no—
394C22×Q8C22×D1711yescharacteristic
395C22×Q8C22×D1711yescharacteristic
396C22×D4—171no—
397C23×C4—341no—
398C2×Q16—22no—
399C2×Q16—22no—
400C2×Q16C4×D1714yescomplemented
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—22no—
412C4×Q8—22no—
413C4×Q8—174no—
414C4×Q8—342no—
415C4×Q8—342no—
416C4×Q8—342no—
417C4×Q8C22⋊D1714yes—
418C4×D4—342no—
419C4×D4—342no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
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—21no—
436C2×C42—171no—
437C2×C42—171no—
438C2×C42—171no—
439C2×C42—171no—
440C2×C42—171no—
441C2×C42—171no—
442C2×C42—341no—
443C2×C42—341no—
444C2×C42—341no—
445C2×C42—341no—
446C2×C42—341no—
447C2×C42C22×D1711yescharacteristic
448#107—342no—
449#106—22no—
450C4⋊C8—174no—
451C4⋊C8—342no—
452#103—22no—
453#103—22no—
454#103—174no—
455#103—342no—
456#103—342no—
457#102—174no—
458#102—342no—
459#102—342no—
460#97—22no—
461#97—342no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469C17#2150811yescharacteristic, complemented, Sylow
470C24—341no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8C2×C4×D1712yes—
480C2×Q8C2×C4×D1712yes—
481C2×Q8#6325212yes—
482C2×Q8C2×C22⋊D1712yes—
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—28no—
507SD16—348no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—172no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8C2×C4×D1712yes—
520C4⋊C4—21no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—172no—
526C4⋊C4—172no—
527C4⋊C4—344no—
528C4⋊C4—344no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4C2×C22⋊D1712yes—
534C22⋊C4—344no—
535C22⋊C4—344no—
536C22⋊C4—682no—
537C22⋊C4—682no—
538C22⋊C4—682no—
539C42—24no—
540C42—42no—
541C42—172no—
542C42—172no—
543C42—342no—
544C42—344no—
545C42—344no—
546C42—341no—
547C42—341no—
548C42—681no—
549C42—681no—
550C42—681no—
551C42—682no—
552C42—682no—
553C42—682no—
554C42#6325412yes—
555C42C2×C22⋊D1712yes—
556C23—342no—
557C23—681no—
558C23—681no—
559C23—681no—
560C23—681no—
561C23—681no—
562C23—681no—
563C23C2×D4×D1711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8C4×C22⋊D1714yes—
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—22no—
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—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—342no—
602C2×C4—342no—
603C2×C4—341no—
604C2×C4—341no—
605C2×C4—341no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—1364no—
614C2×C4#1058762512yes—
615C2×C4#1058762512yes—
616C2×C4C22×C4×D1711yescharacteristic
617C2×C4C2×#6325211yescharacteristic
618C2×C4C2×#6325411yescharacteristic
619C2×C4#41×D1711yescharacteristic
620C2×C4#41×D1711yescharacteristic
621C2×C4C22×C22⋊D1711yescharacteristic
622C8—22no—
623C8—41no—
624C8—342no—
625C8—681no—
626C22—682no—
627C22—682no—
628C22—682no—
629C22—1361no—
630C22—1361no—
631C22—1361no—
632C22—1361no—
633C22C4×D4×D1711yescharacteristic
634C22C2×#1058762511yescharacteristic
635C22#1176331811yescharacteristic
636C22#137×D1711yescharacteristic
637C22#1176360811yescharacteristic
638C22#1176360811yescharacteristic
639C22#1176360811yescharacteristic
640C4—22no—
641C4—22no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—342no—
647C4—342no—
648C4—342no—
649C4—682no—
650C4—681no—
651C4—682no—
652C4—1361no—
653C4C2×C4×C22⋊D1711yescharacteristic
654C4C4×#6325411yescharacteristic
655C4#1176333911yescharacteristic
656C4#1176335511yescharacteristic
657C2—1362no—
658C2C4×#1058762511yescharacteristic
659C2#1453758311yescharacteristic
660C2#1453760011yescharacteristic
661C2#1453825311yescharacteristic
662C2#1453828011yescharacteristic
663C2#1453889911yescharacteristic
664C2#1453895911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#441C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenterC2×#441D17
hypocenterC17#21508
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
4. C4×Q8×Dic17C2semidirect
6. #14529553C2semidirect
7. #14529428C2semidirect
10. #14529127C2semidirect
11. #14528797C2semidirect
12. #14528638C2semidirect
13. #14528614C2semidirect
15. C2×C17×#441C2semidirect
21. #11763033C4semidirect
68. C2×Q16×C17C2×C4semidirect
87. Q8×Dic17C2×C4semidirect
122. #10587506C2×C4semidirect
133. #10587485C2×C4semidirect
295. C2×#441D17semidirect
400. C2×Q16C4×D17semidirect
469. C17#21508semidirect

Maximal subgroups

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

subgroupindex
C2×#117630332
C2×#117629072
C4×#105876332
C4×Q8×Dic172
#145297132
#145295532
#145294282
#145294102
#145291372
#145291272
#145287972
#145286382
#145286142
#145286102
C2×C17×#4412
#2150817