Finite Group Databasebeta

← #17035548

Subgroups — #17035548

Order 4352.

2258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763316C211yesmaximal, characteristic
2C22×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3C2×#11762810C211yesmaximal, characteristic, complemented
4C2×#11762772C211yesmaximal, characteristic, complemented
5#14529098C214yesmaximal, complemented
6#14529088C214yesmaximal, complemented
7#14528504C211yesmaximal, characteristic, complemented
8#14528503C211yesmaximal, characteristic, complemented
9C2×C17×#417C211yesmaximal, characteristic, complemented
10C23×C4×D17—21no—
11C23×C4×D17C2211yescharacteristic
12C22×#63223C2211yescharacteristic
13#11763316—24no—
14#11763316C418yescomplemented
15C2×#10587621—21no—
16C2×#10587632C2212yes—
17C2×#10587624C2212yes—
18C2×#10587623—21no—
19C2×C4×C22⋊D17—22no—
20C2×#10587612C2211yescharacteristic
21C22×C4⋊Dic17C2211yescharacteristic
22C22×#63225—21no—
23C2×C4⋊C4×D17—24no—
24C2×C4⋊C4×D17—24no—
25C2×C4⋊C4×D17—22no—
26C2×C4⋊C4×D17C2214yes—
27#11762810—24no—
28C2×#10587506C2212yescomplemented
29C2×#10587503C2212yescomplemented
30#11762772—24no—
31C2×#10587491C2214yes—
32#11762572C2214yescomplemented
33#11762571C2214yescomplemented
34C22×#41×C17C2211yescharacteristic
35C22×M16×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C17×#417—24no—
38C2×C17×#103C2212yescomplemented
39C2×C17×#102C2212yescomplemented
40C24×D17D411yescharacteristic
41C2×#41×C17—22no—
42C2×#41×C17—22no—
43C2×#41×C17—22no—
44C2×#41×C17—22no—
45C2×#41×C17—22no—
46C2×#41×C17C2×C414yescomplemented
47C22×Q8×C17C2311yescharacteristic
48C22×D4×C17—21no—
49C22×D4×C17C2311yescharacteristic
50C23×C4×C17—21no—
51C23×C4×C17—21no—
52C23×C4×C17C2311yescharacteristic
53C22×C22⋊D17—22no—
54C23×Dic17—21no—
55C23×Dic17D411yescharacteristic
56C22×C4×D17—22no—
57C22×C4×D17—22no—
58C22×C4×D17—22no—
59C22×C4×D17—22no—
60C22×C4×D17—28no—
61C22×C4×D17—24no—
62C22×C4×D17—48no—
63C22×C4×D17C2×C414yes—
64C22×C4×D17D412yes—
65C22×C4×D17D414yes—
66C22×C4×D17C2312yes—
67C2×M16×C17—22no—
68C2×M16×C17—22no—
69C2×M16×C17—22no—
70C22×C8×C17C2312yes—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—22no—
73C2×C4⋊C4×C17D414yescomplemented
74C2×C4⋊C4×C17C2312yes—
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—22no—
77#10587632—24no—
78#10587632C2×C418yescomplemented
79#10587624—24no—
80#10587624C2×C418yescomplemented
81#10587623—28no—
82#10587621—28no—
83C4×C22⋊D17—216no—
84C2×#63227—22no—
85#10587612—22no—
86#10587612C2×C414yescomplemented
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17D414yescomplemented
90C2×C4⋊Dic17C2312yes—
91C2×C4⋊Dic17C2312yes—
92C2×#63225—28no—
93C2×#63225—24no—
94C2×#63225—22no—
95C2×C4×Dic17—21no—
96C2×#63223—22no—
97C2×#63223—22no—
98C2×#63223—22no—
99C22×#3622C2312yes—
100C4⋊C4×D17—216no—
101C4⋊C4×D17—28no—
102C4⋊C4×D17—28no—
103C17×#103—24no—
104C17×#102—24no—
105#10587506—24no—
106#10587503—24no—
107#10587491—28no—
108C24×C17—21no—
109C24×C17C2×D411yescharacteristic
110C23×D17—22no—
111C23×D17—22no—
112C23×D17—24no—
113C23×D17—24no—
114C23×D17C22⋊C414yes—
115C23×D17C2×D412yes—
116#41×C17—28no—
117#41×C17—28no—
118#41×C17—44no—
119#41×C17—44no—
120C2×Q8×C17—21no—
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17C22×C412yescomplemented
126C2×D4×C17—24no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—42no—
133C2×D4×C17—42no—
134C2×D4×C17C22×C412yescomplemented
135C22×C4×C17—24no—
136C22×C4×C17—22no—
137C22×C4×C17—21no—
138C22×C4×C17—24no—
139C22×C4×C17—44no—
140C22×C4×C17—42no—
141C22×C4×C17—42no—
142C22×C4×C17C22×C412yes—
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D412yes—
147C22×C4×C17#4112yes—
148C22×C4×C17#4112yes—
149C22×C4×C17C2411yescharacteristic
150C2×C22⋊D17—28no—
151C2×C22⋊D17—44no—
152C2×C22⋊D17—44no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—24no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—44no—
159C22×Dic17C22⋊C414yes—
160C22×Dic17C2×D412yes—
161C2×C4×D17—28no—
162C2×C4×D17—28no—
163C2×C4×D17—24no—
164C2×C4×D17—28no—
165C2×C4×D17—28no—
166C2×C4×D17—24no—
167C2×C4×D17—22no—
168C2×C4×D17—22no—
169C2×C4×D17—22no—
170C2×C4×D17—22no—
171C2×C4×D17—216no—
172C2×C4×D17—432no—
173C2×C4×D17—48no—
174C2×C4×D17—48no—
175C2×C4×D17C22⋊C418yes—
176C2×C4×D17C22×C414yes—
177C2×C4×D17C2×D414yes—
178M16×C17—44no—
179C2×C8×C17—22no—
180C2×C8×C17—22no—
181C2×C8×C17—22no—
182C4⋊C4×C17—24no—
183C4⋊C4×C17—22no—
184C4⋊C4×C17C2×D414yes—
185C22⋊Dic17—24no—
186C22⋊Dic17—28no—
187#63227—28no—
188C4⋊Dic17—24no—
189C4⋊Dic17—22no—
190C4⋊Dic17—22no—
191C4⋊Dic17C22×C414yescomplemented
192C4⋊Dic17C2×D414yes—
193#63225—216no—
194#63225—28no—
195C4×Dic17—24no—
196#63223—44no—
197C2×#3622—22no—
198C2×#3622—22no—
199C2×#3622—22no—
200#20909—171nomaximal, Sylow
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17C2×C22⋊C412yes—
209C23×C17#12012yes—
210C23×C17#12312yes—
211C23×C17C22×D411yescharacteristic
212C22×D17—28no—
213C22×D17—216no—
214C22×D17—28no—
215C22×D17—22no—
216C22×D17—22no—
217C22×D17—22no—
218C22×D17—22no—
219C22×D17—28no—
220C22×D17—216no—
221C22×D17C2×C22⋊C414yes—
222C22×D17AGL(1,ℤ/8)14yes—
223C22×D17#13714yes—
224Q8×C17—22no—
225Q8×C17—22no—
226Q8×C17—41no—
227Q8×C17—41no—
228D4×C17—22no—
229D4×C17—22no—
230D4×C17—48no—
231D4×C17—41no—
232D4×C17—41no—
233C2×C4×C17—22no—
234C2×C4×C17—24no—
235C2×C4×C17—48no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—44no—
239C2×C4×C17—48no—
240C2×C4×C17C2×C22⋊C412yes—
241C2×C4×C17C4×D414yes—
242C2×C4×C17C4×D414yes—
243C2×C4×C17#12012yes—
244C2×C4×C17#12114yes—
245C2×C4×C17#12114yes—
246C2×C4×C17#12312yes—
247C2×C4×C17C23×C411yescharacteristic
248C2×C4×C17C22×D411yescharacteristic
249C2×C4×C17C22×D411yescharacteristic
250C2×C4×C17C22×D411yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C22⋊D17—416no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—28no—
257C2×Dic17—24no—
258C2×Dic17—24no—
259C2×Dic17—48no—
260C2×Dic17—44no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17C2×C22⋊C414yes—
264C4×D17—24no—
265C4×D17—24no—
266C4×D17—24no—
267C4×D17—24no—
268C4×D17—216no—
269C4×D17—432no—
270C8×C17—42no—
271#3622—42no—
272C2×#525—171no—
273C23×C4⋊C4—171no—
274C2×#417—171no—
275C2×#417D1711yescharacteristic, complemented
276C2×#407—171no—
277#1874—174no—
278#1872—174no—
279#1736—171no—
280#1736—171no—
281C22×C17—24no—
282C22×C17—24no—
283C22×C17—22no—
284C22×C17—22no—
285C22×C17—44no—
286C22×C17—42no—
287C22×C17—82no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17#38614yes—
291C22×C17C22×C22⋊C411yescharacteristic
292C22×C17C2×#12011yescharacteristic
293C22×C17C2×#12311yescharacteristic
294C22×C17C2×AGL(1,ℤ/8)12yes—
295C22×C17C2×#13712yes—
296C2×D17—232no—
297C2×D17—24no—
298C2×D17—24no—
299C2×D17—24no—
300C2×D17—24no—
301C2×D17—24no—
302C2×D17—24no—
303C2×D17#41718yescomplemented
304C4×C17—44no—
305C4×C17—42no—
306C4×C17#38614yes—
307C4×C17C2×C4×D411yescharacteristic
308C4×C17C2×C4×D411yescharacteristic
309C4×C17C2×#12111yescharacteristic
310C4×C17C2×#12111yescharacteristic
311Dic17—24no—
312Dic17—44no—
313Dic17—44no—
314C22×#41C2×D1711yescharacteristic
315C24×C4—171no—
316C24×C4—341no—
317C22×M16—171no—
318C22×M16C2×D1711yescharacteristic
319#525—178no—
320#525—344no—
321C2×#123—341no—
322C2×C22⋊Q8—172no—
323C2×#121—172no—
324C2×#120—341no—
325C2×C4×D4—342no—
326C2×#117—171no—
327C22×C4⋊C4—171no—
328C22×C4⋊C4—174no—
329C22×C4⋊C4—344no—
330C22×C4⋊C4—342no—
331C22×C4⋊C4—341no—
332C22×C4⋊C4—344no—
333C22×C4⋊C4C2×D1711yescharacteristic
334#417—24no—
335#417—344no—
336C2×#103—172no—
337C2×#103C2×D1712yescomplemented
338C2×#102—172no—
339C2×#102C2×D1712yescomplemented
340#407—344no—
341C2×C22⋊C8—174no—
342#340—174no—
343#340—174no—
344C2×C17—24no—
345C2×C17—82no—
346C2×C17C2×#38611yescharacteristic
347C2×C17C2×#41712yes—
348C2×C17#300311yescharacteristic
349C2×C17#300411yescharacteristic
350C2×C17#309212yes—
351D17—28no—
352C25—171no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41C4×D1714yescomplemented
359C22×Q8C22×D1711yescharacteristic
360C22×D4—21no—
361C22×D4—342no—
362C22×D4C22×D1711yescharacteristic
363C23×C4—21no—
364C23×C4—21no—
365C23×C4—171no—
366C23×C4—174no—
367C23×C4—172no—
368C23×C4—174no—
369C23×C4—172no—
370C23×C4—344no—
371C23×C4—348no—
372C23×C4—342no—
373C23×C4—342no—
374C23×C4—342no—
375C23×C4—342no—
376C23×C4—341no—
377C23×C4—688no—
378C23×C4C22×D1711yescharacteristic
379C2×M16—22no—
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—342no—
383C2×M16—342no—
384C2×M16—342no—
385C22×C8—172no—
386C22×C8C22×D1712yes—
387#123—348no—
388C22⋊Q8—178no—
389C22⋊Q8—344no—
390#121—178no—
391#121—344no—
392#120—348no—
393C4×D4—3416no—
394#117—174no—
395#117—342no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—174no—
399C2×C4⋊C4—172no—
400C2×C4⋊C4—172no—
401C2×C4⋊C4—348no—
402C2×C4⋊C4—3416no—
403C2×C4⋊C4—344no—
404C2×C4⋊C4—342no—
405C2×C4⋊C4—348no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—342no—
408C2×C4⋊C4—348no—
409C2×C4⋊C4C22⋊D1714yescomplemented
410C2×C4⋊C4C22×D1712yes—
411C2×C22⋊C4—342no—
412C2×C22⋊C4—342no—
413C2×C22⋊C4—341no—
414C2×C42—341no—
415#103—24no—
416#103—344no—
417#102—24no—
418#102—344no—
419C22⋊C8—348no—
420C17#2090911yescharacteristic, complemented, Sylow
421C24—21no—
422C24—172no—
423C24—174no—
424C24—344no—
425C24—344no—
426C24—342no—
427C24—342no—
428C24D4×D1711yescharacteristic
429#41—28no—
430#41—28no—
431#41—44no—
432#41—44no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D1712yescomplemented
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—348no—
448C2×D4—684no—
449C2×D4—684no—
450C2×D4C2×C4×D1712yescomplemented
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—172no—
459C22×C4—174no—
460C22×C4—174no—
461C22×C4—174no—
462C22×C4—178no—
463C22×C4—3416no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—348no—
471C22×C4—344no—
472C22×C4—348no—
473C22×C4—348no—
474C22×C4—344no—
475C22×C4—348no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—344no—
479C22×C4—688no—
480C22×C4—6832no—
481C22×C4—688no—
482C22×C4—684no—
483C22×C4C2×C4×D1712yes—
484C22×C4#6325212yes—
485C22×C4D4×D1712yes—
486C22×C4D4×D1712yes—
487C22×C4D4×D1711yescharacteristic
488C22×C4#6325412yes—
489C22×C4C2×C22⋊D1712yes—
490C22×C4C23×D1711yescharacteristic
491M16—44no—
492M16—684no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—342no—
497C2×C8—342no—
498C2×C8—342no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—174no—
502C4⋊C4—174no—
503C4⋊C4—348no—
504C4⋊C4—3416no—
505C4⋊C4—342no—
506C4⋊C4—342no—
507C4⋊C4—344no—
508C4⋊C4C2×C22⋊D1714yes—
509C22⋊C4—348no—
510C22⋊C4—348no—
511C22⋊C4—344no—
512C42—344no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—174no—
521C23—174no—
522C23—174no—
523C23—3416no—
524C23—348no—
525C23—348no—
526C23—3416no—
527C23—342no—
528C23—342no—
529C23—342no—
530C23—342no—
531C23—348no—
532C23C22⋊C4×D1712yes—
533C23#1058756912yes—
534C23#1058762312yes—
535C23C2×D4×D1711yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545D4—6816no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—174no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—348no—
557C2×C4—344no—
558C2×C4—344no—
559C2×C4—344no—
560C2×C4—344no—
561C2×C4—3416no—
562C2×C4—344no—
563C2×C4—344no—
564C2×C4—6832no—
565C2×C4—688no—
566C2×C4—684no—
567C2×C4—682no—
568C2×C4—682no—
569C2×C4C22⋊C4×D1712yes—
570C2×C4#1058756714yes—
571C2×C4#1058756912yes—
572C2×C4#1058757014yes—
573C2×C4C4×C22⋊D1714yes—
574C2×C4#1058762312yes—
575C2×C4#1058762414yes—
576C2×C4C22×C4×D1711yescharacteristic
577C2×C4C2×#6325211yescharacteristic
578C2×C4C2×D4×D1711yescharacteristic
579C2×C4C2×D4×D1711yescharacteristic
580C2×C4C2×#6325411yescharacteristic
581C2×C4C22×C22⋊D1711yescharacteristic
582C8—42no—
583C8—682no—
584C22—22no—
585C22—22no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—82no—
593C22—178no—
594C22—3432no—
595C22—344no—
596C22—344no—
597C22—344no—
598C22—344no—
599C22—344no—
600C22—344no—
601C22#1176268514yes—
602C22C2×C22⋊C4×D1711yescharacteristic
603C22C2×#1058762311yescharacteristic
604C22C2×#1058756911yescharacteristic
605C22AGL(1,ℤ/8)×D1712yes—
606C22#137×D1712yes—
607C4—44no—
608C4—42no—
609C4—344no—
610C4—684no—
611C4—684no—
612C4#1176268514yes—
613C4C2×#1058756711yescharacteristic
614C4C2×C4×C22⋊D1711yescharacteristic
615C4C2×#1058762411yescharacteristic
616C4C2×#1058757011yescharacteristic
617C2—24no—
618C2—82no—
619C2—348no—
620C2C2×#1176268511yescharacteristic
621C2D17×#41712yes—
622C2#1453800911yescharacteristic
623C2#1453801611yescharacteristic
624C2#1453870512yes—
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#417C2
socleC23×C17C22×D4
prime radicalC2×#11763316C2
cosocleC22×C4×C17C24
hypercenterC2×#417D17
hypocenterC17#20909
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D17C2semidirect
3. C2×#11762810C2semidirect
4. C2×#11762772C2semidirect
5. #14529098C2semidirect
6. #14529088C2semidirect
7. #14528504C2semidirect
8. #14528503C2semidirect
9. C2×C17×#417C2semidirect
14. #11763316C4semidirect
28. C2×#10587506C22semidirect
29. C2×#10587503C22semidirect
32. #11762572C22semidirect
33. #11762571C22semidirect
38. C2×C17×#103C22semidirect
39. C2×C17×#102C22semidirect
46. C2×#41×C17C2×C4semidirect
73. C2×C4⋊C4×C17D4semidirect
78. #10587632C2×C4semidirect
80. #10587624C2×C4semidirect
86. #10587612C2×C4semidirect
89. C2×C4⋊Dic17D4semidirect
125. C2×Q8×C17C22×C4semidirect
134. C2×D4×C17C22×C4semidirect
191. C4⋊Dic17C22×C4semidirect
275. C2×#417D17semidirect
303. C2×D17#417semidirect
337. C2×#103C2×D17semidirect
339. C2×#102C2×D17semidirect
358. C2×#41C4×D17semidirect
409. C2×C4⋊C4C22⋊D17semidirect
420. C17#20909semidirect
438. C2×Q8C2×C4×D17semidirect
450. C2×D4C2×C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#117633162
C22×C4⋊C4×D172
C2×#117628102
C2×#117627722
#145290982
#145290882
#145285042
#145285032
C2×C17×#4172
#2090917