Finite Group Databasebeta

← #17051614

Subgroups — #17051614

Order 4352.

1038 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762808C211yesmaximal, characteristic
2#14531615C211yesmaximal, characteristic
3#14529473C211yesmaximal, characteristic, complemented
4#14529459C211yesmaximal, characteristic, complemented
5#14529425C211yesmaximal, characteristic, complemented
6#14529418C211yesmaximal, characteristic, complemented
7#14529044C211yesmaximal, characteristic, complemented
8#14529030C211yesmaximal, characteristic, complemented
9#14528874C211yesmaximal, characteristic, complemented
10#14528815C211yesmaximal, characteristic, complemented
11#14528722C211yesmaximal, characteristic
12#14528612C211yesmaximal, characteristic, complemented
13C4×#10587480C211yesmaximal, characteristic, complemented
14#14528568C211yesmaximal, characteristic, complemented
15C17×#2344C211yesmaximal, characteristic, complemented
16C22×#63252C2211yescharacteristic
17C22×#63223C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22#11762808C418yescomplemented
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×#10587480C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762715C2211yescharacteristic
36#11762685—21no—
37#11762685—21no—
38#11762671—21no—
39#11762669C2211yescharacteristic
40#95×D17—21no—
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C17×#117—21no—
50C22×C4⋊C4×C17C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#387—21no—
54C17×#387—21no—
55C17×#381—21no—
56C17×#380C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C22⋊C4×C17C2211yescharacteristic
60C23×C4×C17—21no—
61C23×C4×C17C2311yescharacteristic
62C22×C22⋊D17—21no—
63C2×#63252—22no—
64C2×#63252—22no—
65C2×#63252—22no—
66C2×#63252—41no—
67C2×#63252—41no—
68C2×#63252C2×C414yes—
69C22×D68C2311yescharacteristic
70C22×C4×D17—21no—
71C22×C4×D17—21no—
72C22×C4×D17—21no—
73C22×Dic34C2311yescharacteristic
74C17×#117—44no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—22no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17C2×C414yescomplemented
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C22⋊C4×C17—21no—
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C22⋊C4×C17D411yescharacteristic
87C2×C42×C17—21no—
88C2×C42×C17—21no—
89C2×C42×C17C2311yescharacteristic
90C2×C42×C17C2311yescharacteristic
91C2×C22⋊Dic17—21no—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63223—41no—
97C2×#63223—41no—
98C2×#63223C2×C414yescomplemented
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C4×D68—22no—
104C4×D68—22no—
105C4×D68—41no—
106C4×D68—41no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95D411yescharacteristic
113C17×#95D411yescharacteristic
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119C22⋊#3622—22no—
120#10587481—41no—
121#10587481—41no—
122#10587481C2×C414yescomplemented
123#10587481C2×C414yescomplemented
124#10587480—22no—
125#10587480—22no—
126#10587480C2×C414yes—
127#10587480C2×C414yes—
128#10587479—41no—
129#10587478—41no—
130#10587474—41no—
131C4×#3622—41no—
132C24×C17C2×D411yescharacteristic
133C23×D17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—24no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17C22×C412yes—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17#4112yes—
152C22×C4×C17C2411yescharacteristic
153C2×C22⋊D17—24no—
154C2×C22⋊D17—42no—
155C2×C22⋊D17—42no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159#63252—28no—
160#63252—44no—
161#63252—44no—
162#63252—44no—
163C2×D68—21no—
164C2×D68—21no—
165C2×D68—21no—
166C2×D68C22×C412yes—
167C2×D68#4112yes—
168C2×D68#4112yes—
169C2×C4×D17—24no—
170C2×C4×D17—42no—
171C2×C4×D17—42no—
172C2×C4×D17—42no—
173C2×C4×D17—42no—
174C2×C4×D17—42no—
175C2×C4×D17—42no—
176C2×C4×D17—81no—
177C2×C4×D17—81no—
178C2×Dic34—21no—
179C2×Dic34—21no—
180C2×Dic34—21no—
181C2×Dic34—21no—
182C2×Dic34—21no—
183C2×Dic34C22×C412yes—
184C4⋊C4×C17—24no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—42no—
192C4⋊C4×C17—42no—
193C4⋊C4×C17C22×C412yes—
194C4⋊C4×C17C22×C412yes—
195C22⋊C4×C17—22no—
196C22⋊C4×C17—41no—
197C22⋊C4×C17—42no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17—21no—
202C42×C17—42no—
203C42×C17—42no—
204C22⋊Dic17—81no—
205#63227—42no—
206#63227—42no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211#63223—44no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622C22×C412yes—
223C2×#3622C22×C412yes—
224#22582—171nomaximal, Sylow
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17C4×D412yes—
232C23×C17C22×D411yescharacteristic
233C22×D17—22no—
234C22×D17—22no—
235C22×D17—22no—
236C22×D17—41no—
237C22×D17—41no—
238C22×D17—41no—
239C22×D17—41no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—48no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17C4×D412yes—
271C2×C4×C17AGL(1,ℤ/8)12yes—
272C2×C4×C17C23×C411yescharacteristic
273C2×C4×C17C2×#4111yescharacteristic
274C2×C4×C17C2×#4111yescharacteristic
275C2×C4×C17C2×#4111yescharacteristic
276C2×C4×C17H(5,𝔽2)11yescharacteristic
277C2×C4×C17H(5,𝔽2)11yescharacteristic
278C22⋊D17—48no—
279C2×Dic17—22no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288D68—22no—
289D68—22no—
290D68—22no—
291D68#11714yescomplemented
292C4×D17—48no—
293C4×D17—84no—
294C4×D17—84no—
295Dic34—22no—
296Dic34—22no—
297Dic34—41no—
298Dic34—41no—
299#3622—22no—
300#3622—22no—
301#3622—41no—
302#3622—41no—
303C2×#417—171no—
304#2344D1711yescharacteristic, complemented
305#2339—171no—
306#1935—171no—
307#1933—171no—
308#1928—171no—
309#1927—171no—
310#1864—171no—
311#1862—171no—
312#1824—171no—
313#1809—171no—
314#1788—171no—
315#1762—171no—
316C4×#102—171no—
317#1753—171no—
318C22×C17—22no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17—41no—
325C22×C17C2×C4×D411yescharacteristic
326C22×C17#54611yescharacteristic
327C22×C17#54611yescharacteristic
328C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
329C22×C17#57511yescharacteristic
330C22×C17#57711yescharacteristic
331C22×C17#57811yescharacteristic
332C2×D17—24no—
333C2×D17—42no—
334C2×D17—42no—
335C2×D17—42no—
336C2×D17—42no—
337C2×D17—42no—
338C2×D17—42no—
339C4×C17—22no—
340C4×C17—22no—
341C4×C17—41no—
342C4×C17—42no—
343C4×C17—42no—
344C4×C17—41no—
345C4×C17—41no—
346C4×C17—81no—
347C4×C17—81no—
348C4×C17C2×#11711yescharacteristic
349C4×C17#51811yescharacteristic
350C4×C17#55111yescharacteristic
351C4×C17#55111yescharacteristic
352Dic17—42no—
353Dic17—81no—
354Dic17—81no—
355C22×#41—171no—
356C22×M16—171no—
357C2×C4×D4—171no—
358C2×C4×D4—171no—
359C2×#117—21no—
360C22×C4⋊C4C2×D1711yescharacteristic
361C2×#107—171no—
362C2×#106—171no—
363#417—178no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#102—171no—
369C2×#102—171no—
370C2×#102—171no—
371C2×#102—171no—
372C2×C22⋊C8—171no—
373C2×#97—171no—
374C2×C4×C8—171no—
375#391—171no—
376#391—171no—
377#389C2×D1711yescharacteristic
378#389C2×D1711yescharacteristic
379#387—21no—
380#387—21no—
381#386—341no—
382#386—341no—
383#381—21no—
384#381—341no—
385#380—171no—
386#380C2×D1711yescharacteristic
387C4×C4⋊C4C2×D1711yescharacteristic
388C4×C4⋊C4C2×D1711yescharacteristic
389C4×C22⋊C4C2×D1711yescharacteristic
390C2×#95—341no—
391#340—171no—
392#340—171no—
393#340—171no—
394#340—171no—
395#340—171no—
396#340—171no—
397#336—171no—
398#336—171no—
399C2×C17—42no—
400C2×C17#234211yescharacteristic
401C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
402C2×C17#294711yescharacteristic
403C2×C17#330611yescharacteristic
404C2×C17#331511yescharacteristic
405C2×C17#332311yescharacteristic
406C2×C17#333011yescharacteristic
407D17—44no—
408C2×#41—174no—
409C2×#41—342no—
410C2×#41—342no—
411C2×#41—342no—
412C2×#41—681no—
413C2×#41—681no—
414C22×Q8—171no—
415C22×D4—171no—
416C22×D4—341no—
417C23×C4—21no—
418C23×C4—341no—
419C23×C4—341no—
420C23×C4—341no—
421C23×C4C22×D1711yescharacteristic
422C2×M16—174no—
423C2×M16—681no—
424C2×M16—681no—
425C22×C8—171no—
426C22×C8—171no—
427C22×C8—171no—
428C22×C8—171no—
429C4×D4—342no—
430C4×D4—342no—
431C4×D4—681no—
432C4×D4—681no—
433#117—44no—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—171no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4C4×D1714yescomplemented
441C2×C4⋊C4C22×D1711yescharacteristic
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C4⋊C4C22×D1711yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—341no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4C22⋊D1711yescharacteristic
450C2×C22⋊C4C22⋊D1711yescharacteristic
451C2×C42—21no—
452C2×C42—21no—
453C2×C42C22×D1711yescharacteristic
454C2×C42C22×D1711yescharacteristic
455#107—681no—
456#106—681no—
457#103—174no—
458#103—174no—
459#103—681no—
460#103—681no—
461#102—174no—
462#102—174no—
463#102—342no—
464#102—342no—
465C22⋊C8—342no—
466#97—681no—
467C4×C8—681no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478#95C22⋊D1711yescharacteristic
479#95C22⋊D1711yescharacteristic
480C17#2258211yescharacteristic, complemented, Sylow
481C24—341no—
482C24C2×C22⋊D1711yescharacteristic
483#41—348no—
484#41—684no—
485#41—684no—
486#41—684no—
487C2×Q8—172no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×D4—172no—
494C2×D4—172no—
495C2×D4—172no—
496C2×D4—344no—
497C2×D4—341no—
498C2×D4—341no—
499C2×D4—341no—
500C2×D4—682no—
501C2×D4—682no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—341no—
514C22×C4—344no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—682no—
518C22×C4—682no—
519C22×C4—682no—
520C22×C4—682no—
521C22×C4—682no—
522C22×C4—682no—
523C22×C4—1361no—
524C22×C4—1361no—
525C22×C4C2×C4×D1712yes—
526C22×C4#6325212yes—
527C22×C4C2×C22⋊D1711yescharacteristic
528C22×C4C2×C22⋊D1711yescharacteristic
529C22×C4C2×C22⋊D1711yescharacteristic
530C22×C4C2×C22⋊D1711yescharacteristic
531C22×C4C2×C22⋊D1711yescharacteristic
532C22×C4C23×D1711yescharacteristic
533M16—684no—
534C2×C8—172no—
535C2×C8—172no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8—341no—
541C2×C8—341no—
542C2×C8—341no—
543C2×C8—341no—
544C2×C8—341no—
545C2×C8—341no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—24no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—341no—
556C4⋊C4—341no—
557C4⋊C4—341no—
558C4⋊C4—341no—
559C4⋊C4C2×C4×D1712yes—
560C4⋊C4C2×C4×D1712yes—
561C22⋊C4—22no—
562C22⋊C4—41no—
563C22⋊C4—42no—
564C22⋊C4—682no—
565C22⋊C4—682no—
566C22⋊C4—1361no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—42no—
572C42—42no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—342no—
580C23—342no—
581C23—342no—
582C23—681no—
583C23—681no—
584C23—681no—
585C23—681no—
586C23C4×C22⋊D1712yes—
587C23C22×C22⋊D1711yescharacteristic
588Q8—342no—
589Q8—342no—
590Q8—681no—
591Q8—681no—
592D4—174no—
593D4—342no—
594D4—342no—
595D4—342no—
596D4—688no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—342no—
628C2×C4—688no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4—681no—
635C2×C4—681no—
636C2×C4—681no—
637C2×C4—1364no—
638C2×C4—1364no—
639C2×C4C4×C22⋊D1712yes—
640C2×C4#1058761812yes—
641C2×C4C22×C4×D1711yescharacteristic
642C2×C4C2×#6325211yescharacteristic
643C2×C4#41×D1711yescharacteristic
644C2×C4#41×D1711yescharacteristic
645C2×C4#1058769611yescharacteristic
646C2×C4#1058769611yescharacteristic
647C8—342no—
648C8—342no—
649C8—681no—
650C8—681no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—344no—
659C22—682no—
660C22—682no—
661C22—682no—
662C22—682no—
663C22—682no—
664C22—682no—
665C22C2×C4×C22⋊D1711yescharacteristic
666C22#1176344711yescharacteristic
667C22#1176344711yescharacteristic
668C22C2×#1058761811yescharacteristic
669C22#1176360711yescharacteristic
670C22#1176361711yescharacteristic
671C22#1176362411yescharacteristic
672C4—22no—
673C4—22no—
674C4—41no—
675C4—41no—
676C4—42no—
677C4—42no—
678C4—41no—
679C4—81no—
680C4—81no—
681C4—682no—
682C4—1361no—
683C4—1361no—
684C4#117×D1711yescharacteristic
685C4#1176328311yescharacteristic
686C4#1176348311yescharacteristic
687C4#1176348311yescharacteristic
688C2—42no—
689C2—684no—
690C2#1453163411yescharacteristic
691C2C4×#1058761811yescharacteristic
692C2#1453762711yescharacteristic
693C2#1454073911yescharacteristic
694C2#1454087411yescharacteristic
695C2#1454099411yescharacteristic
696C2#1454108311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2344C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#2344D17
hypocenterC17#22582
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #14529473C2semidirect
4. #14529459C2semidirect
5. #14529425C2semidirect
6. #14529418C2semidirect
7. #14529044C2semidirect
8. #14529030C2semidirect
9. #14528874C2semidirect
10. #14528815C2semidirect
12. #14528612C2semidirect
13. C4×#10587480C2semidirect
14. #14528568C2semidirect
15. C17×#2344C2semidirect
22. #11762808C4semidirect
79. C2×C4⋊C4×C17C2×C4semidirect
98. C2×#63223C2×C4semidirect
122. #10587481C2×C4semidirect
123. #10587481C2×C4semidirect
291. D68#117semidirect
304. #2344D17semidirect
440. C2×C4⋊C4C4×D17semidirect
480. C17#22582semidirect

Maximal subgroups

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

subgroupindex
C2×#117628082
#145316152
#145294732
#145294592
#145294252
#145294182
#145290442
#145290302
#145288742
#145288152
#145287222
#145286122
C4×#105874802
#145285682
C17×#23442
#2258217