Finite Group Databasebeta

← #17040838

Subgroups — #17040838

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763036C211yesmaximal, characteristic
2C2×C4×#63229C211yesmaximal, characteristic, complemented
3C2×#11762765C211yesmaximal, characteristic, complemented
4#14531466C211yesmaximal, characteristic, complemented
5#14529637C211yesmaximal, characteristic, complemented
6#14529509C211yesmaximal, characteristic, complemented
7#14529484C211yesmaximal, characteristic, complemented
8#14529473C211yesmaximal, characteristic, complemented
9#14529223C211yesmaximal, characteristic
10#14529215C211yesmaximal, characteristic, complemented
11#14528807C211yesmaximal, characteristic, complemented
12C4×#10587479C211yesmaximal, characteristic, complemented
13#14528617C211yesmaximal, characteristic, complemented
14#14528609C211yesmaximal, characteristic, complemented
15C17×#2324C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C22×#63223C2211yescharacteristic
18C23×#3622C2211yescharacteristic
19C2×#10587610C2211yescharacteristic
20C2×C4×Dic34C2211yescharacteristic
21#11763036C418yescomplemented
22C4×#63229—24no—
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32#11762765—24no—
33C2×#10587474C2211yescharacteristic
34C2×#10587474C2211yescharacteristic
35C2×C4×#3622C2211yescharacteristic
36C2×C4×#3622C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762704—21no—
40#11762703—21no—
41#11762697—21no—
42#11762690—21no—
43#11762669C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51C2×C17×#121C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C2×C17×#117C2211yescharacteristic
54C17×#390—21no—
55C17×#390C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#388—21no—
58C17×#386—21no—
59C17×#385—21no—
60C17×#380C2211yescharacteristic
61C4×C4⋊C4×C17C2211yescharacteristic
62C22×D4×C17—21no—
63C22×D4×C17C2311yescharacteristic
64C23×C4×C17—21no—
65C23×C4×C17C2311yescharacteristic
66C22×Dic34C2311yescharacteristic
67C17×#121—22no—
68C17×#121C2×C414yes—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C17×#117—22no—
72C17×#117—22no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C42×C17—21no—
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×C22⋊Dic17—21no—
90C2×C22⋊Dic17—21no—
91C2×#63229—22no—
92C2×#63229—22no—
93C2×#63229C2×C414yes—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97#10587610—22no—
98#10587610C2×C414yescomplemented
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×#63223—22no—
105C2×#63223—22no—
106C2×#63223—22no—
107C22×#3622—22no—
108C22×#3622—22no—
109C22×#3622C2×C414yescomplemented
110C22×#3622C2311yescharacteristic
111C22×#3622C2311yescharacteristic
112C22×#3622C2311yescharacteristic
113C22×#3622C2311yescharacteristic
114C4×Dic34—22no—
115C4×Dic34—22no—
116C17×#95—21no—
117C17×#95—21no—
118C17×#95—21no—
119#10587503—22no—
120#10587503—22no—
121#10587503C2×C414yes—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587481—22no—
128#10587481—22no—
129#10587481C2×C414yescomplemented
130#10587479—22no—
131#10587479—22no—
132#10587479C2×C414yescomplemented
133#10587474—22no—
134#10587474—22no—
135C4×#3622—22no—
136C4×#3622D414yescomplemented
137C24×C17—21no—
138C24×C17C2×D411yescharacteristic
139C2×D4×C17—24no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17—21no—
144C2×D4×C17—21no—
145C2×D4×C17—81no—
146C2×D4×C17—81no—
147C2×D4×C17C22×C412yes—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—42no—
156C22×C4×C17—81no—
157C22×C4×C17—81no—
158C22×C4×C17C22×C412yes—
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17#4112yes—
165C22×C4×C17C2411yescharacteristic
166C22×Dic17—21no—
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34C22×C412yes—
176C4⋊C4×C17—22no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—81no—
184C4⋊C4×C17C22×C412yes—
185C22⋊C4×C17—24no—
186C22⋊C4×C17—22no—
187C22⋊C4×C17—22no—
188C22⋊C4×C17—42no—
189C22⋊C4×C17—81no—
190C22⋊C4×C17—81no—
191C22⋊C4×C17—81no—
192C42×C17—21no—
193C42×C17—21no—
194C42×C17—81no—
195C42×C17C2×D412yes—
196C42×C17#4112yes—
197C22⋊Dic17—24no—
198C22⋊Dic17—81no—
199#63229—28no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17C22×C412yes—
206#63225—24no—
207#63225—42no—
208#63225—81no—
209C4×Dic17—42no—
210C4×Dic17—81no—
211#63223—28no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—28no—
219C2×#3622C22×C412yes—
220C2×#3622C22×C412yes—
221C2×#3622C2×D412yes—
222C2×#3622C2×D412yes—
223C2×#3622#4112yes—
224C2×#3622#4112yes—
225#21523—171nomaximal, Sylow
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—22no—
233C23×C17—41no—
234C23×C17—41no—
235C23×C17—41no—
236C23×C17—41no—
237C23×C17—41no—
238C23×C17—41no—
239C23×C17C4×D412yes—
240C23×C17C22×D411yescharacteristic
241D4×C17—22no—
242D4×C17—22no—
243D4×C17—41no—
244D4×C17—41no—
245D4×C17—84no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
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×C17—41no—
271C2×C4×C17—84no—
272C2×C4×C17C4×D412yes—
273C2×C4×C17#12112yes—
274C2×C4×C17#12112yes—
275C2×C4×C17C23×C411yescharacteristic
276C2×C4×C17C22×D411yescharacteristic
277C2×C4×C17C2×#4111yescharacteristic
278C2×C4×C17C2×#4111yescharacteristic
279C2×C4×C17C2×#4111yescharacteristic
280C2×C4×C17C2×#4111yescharacteristic
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291C2×Dic17—41no—
292C2×Dic17—41no—
293Dic34—22no—
294Dic34—22no—
295Dic34—41no—
296Dic34—41no—
297#3622—22no—
298#3622—22no—
299#3622—22no—
300#3622C4×D414yes—
301C2×#465—171no—
302C2×C4×SD16—171no—
303C2×#405—171no—
304#2324D1711yescharacteristic
305#2323—171no—
306#1968—171no—
307#1941—171no—
308#1936—171no—
309#1935—171no—
310#1895—171no—
311#1894—171no—
312#1807—171no—
313C4×#106—171no—
314#1763—171no—
315#1762—171no—
316C22×C17—22no—
317C22×C17—22no—
318C22×C17—22no—
319C22×C17—41no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17—42no—
324C22×C17—42no—
325C22×C17—42no—
326C22×C17—81no—
327C22×C17—81no—
328C22×C17—81no—
329C22×C17—81no—
330C22×C17C2×C4×D411yescharacteristic
331C22×C17C2×#12111yescharacteristic
332C22×C17#53211yescharacteristic
333C22×C17#57711yescharacteristic
334C22×C17#57711yescharacteristic
335C22×C17#57711yescharacteristic
336C22×C17#57711yescharacteristic
337C4×C17—22no—
338C4×C17—22no—
339C4×C17—22no—
340C4×C17—42no—
341C4×C17—41no—
342C4×C17—41no—
343C4×C17—81no—
344C4×C17—81no—
345C4×C17C2×C4×D411yescharacteristic
346C4×C17C4×#4111yescharacteristic
347C4×C17#52511yescharacteristic
348C4×C17#52911yescharacteristic
349Dic17—42no—
350Dic17—42no—
351Dic17—81no—
352Dic17—81no—
353C22×SD16—171no—
354C22×M16—171no—
355C23×C8—171no—
356C2×C22⋊Q8—171no—
357C2×#121C2×D1711yescharacteristic
358C2×C4×Q8—171no—
359C2×C4×D4C2×D1711yescharacteristic
360C2×#117C2×D1711yescharacteristic
361#465—178no—
362C4×SD16—344no—
363C2×#106—171no—
364C2×#106—171no—
365C2×#106—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#102—171no—
370C2×#102—171no—
371C2×#102—171no—
372#405—344no—
373C2×#97—171no—
374C2×#97—171no—
375C2×C4×C8—171no—
376C2×C4×C8—171no—
377#391—171no—
378#390—21no—
379#390C2×D1711yescharacteristic
380#389—171no—
381#389—341no—
382#389C2×D1711yescharacteristic
383#388—21no—
384#388—341no—
385#388—341no—
386#387—341no—
387#386—21no—
388#385—21no—
389#380—171no—
390#380C2×D1711yescharacteristic
391C4×C4⋊C4—171no—
392C4×C4⋊C4C2×D1711yescharacteristic
393#340—171no—
394#340—171no—
395#340—171no—
396#340—171no—
397#340—171no—
398#340—171no—
399C2×C17—42no—
400C2×C17—82no—
401C2×C17C4×#12111yescharacteristic
402C2×C17#294911yescharacteristic
403C2×C17#295211yescharacteristic
404C2×C17#305711yescharacteristic
405C2×C17#306011yescharacteristic
406C2×C17#317311yescharacteristic
407C2×C17#317411yescharacteristic
408C22×Q8—171no—
409C22×D4—21no—
410C22×D4C22×D1711yescharacteristic
411C23×C4—21no—
412C23×C4C22×D1711yescharacteristic
413C2×SD16—174no—
414C2×SD16—342no—
415C2×SD16—342no—
416C2×M16—342no—
417C2×M16—342no—
418C2×M16—342no—
419C22×C8—171no—
420C22×C8—174no—
421C22×C8—171no—
422C22×C8—171no—
423C22×C8—171no—
424C22×C8—342no—
425C22×C8—342no—
426C22⋊Q8—174no—
427C22⋊Q8—342no—
428#121—22no—
429#121C4×D1714yes—
430C4×Q8—342no—
431C4×Q8—342no—
432C4×D4—22no—
433C4×D4—22no—
434#117—22no—
435#117—22no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—171no—
438C2×C4⋊C4—171no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4—341no—
442C2×C4⋊C4—341no—
443C2×C4⋊C4C22⋊D1711yescharacteristic
444C2×C4⋊C4C22⋊D1711yescharacteristic
445C2×C4⋊C4C22×D1711yescharacteristic
446C2×C4⋊C4C22×D1711yescharacteristic
447C2×C4⋊C4C22×D1711yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—341no—
454C2×C22⋊C4—341no—
455C2×C22⋊C4C22⋊D1711yescharacteristic
456C2×C22⋊C4C22⋊D1711yescharacteristic
457C2×C42—21no—
458C2×C42—341no—
459C2×C42—341no—
460C2×C42C22×D1711yescharacteristic
461C2×C42C22×D1711yescharacteristic
462#106—174no—
463#106—342no—
464#106—342no—
465#103—174no—
466#103—342no—
467#103—342no—
468#102—174no—
469#102—342no—
470#102—342no—
471#97—342no—
472#97—342no—
473C4×C8—174no—
474C4×C8—342no—
475#95—21no—
476#95—21no—
477#95—21no—
478#95—341no—
479#95—341no—
480#95—341no—
481#95—341no—
482#95—341no—
483C17#2152311yescharacteristic, complemented, Sylow
484C24—21no—
485C24C2×C22⋊D1711yescharacteristic
486C2×Q8—172no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—81no—
499C2×D4—81no—
500C2×D4C2×C4×D1712yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—42no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—341no—
515C22×C4C2×C4×D1712yes—
516C22×C4#6325212yes—
517C22×C4C2×C22⋊D1711yescharacteristic
518C22×C4C2×C22⋊D1711yescharacteristic
519C22×C4C2×C22⋊D1711yescharacteristic
520C22×C4C2×C22⋊D1711yescharacteristic
521C22×C4C2×C22⋊D1711yescharacteristic
522C22×C4C23×D1711yescharacteristic
523SD16—348no—
524M16—348no—
525C2×C8—172no—
526C2×C8—172no—
527C2×C8—172no—
528C2×C8—172no—
529C2×C8—172no—
530C2×C8—172no—
531C2×C8—348no—
532C2×C8—341no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—81no—
546C4⋊C4—172no—
547C4⋊C4—341no—
548C4⋊C4—344no—
549C4⋊C4—341no—
550C4⋊C4—341no—
551C4⋊C4—341no—
552C4⋊C4—341no—
553C4⋊C4—682no—
554C4⋊C4—1361no—
555C4⋊C4C2×C4×D1712yes—
556C22⋊C4—24no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—42no—
560C22⋊C4—81no—
561C22⋊C4—81no—
562C22⋊C4—81no—
563C22⋊C4—344no—
564C22⋊C4—1361no—
565C42—21no—
566C42—21no—
567C42—81no—
568C42—682no—
569C42—1361no—
570C42D4×D1712yes—
571C42#6325412yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23C4×C22⋊D1712yes—
586C23C22×C22⋊D1711yescharacteristic
587Q8—342no—
588Q8—342no—
589Q8—681no—
590Q8—681no—
591D4—22no—
592D4—22no—
593D4—41no—
594D4—41no—
595D4—84no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
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—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—84no—
622C2×C4—342no—
623C2×C4—342no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4C4×C22⋊D1712yes—
635C2×C4#1058762512yes—
636C2×C4#1058762512yes—
637C2×C4C22×C4×D1711yescharacteristic
638C2×C4C2×#6325211yescharacteristic
639C2×C4C2×D4×D1711yescharacteristic
640C2×C4C2×#6325411yescharacteristic
641C2×C4#41×D1711yescharacteristic
642C2×C4#41×D1711yescharacteristic
643C8—174no—
644C8—342no—
645C8—342no—
646C8—342no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×C4×C22⋊D1711yescharacteristic
662C22C2×#1058762511yescharacteristic
663C22#1176335511yescharacteristic
664C22#1176361711yescharacteristic
665C22#1176362011yescharacteristic
666C22#1176362011yescharacteristic
667C22#1176362011yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—682no—
677C4—682no—
678C4—1361no—
679C4—1361no—
680C4C4×D4×D1711yescharacteristic
681C4C4×#6325411yescharacteristic
682C4#1176331811yescharacteristic
683C4#1176333911yescharacteristic
684C2—42no—
685C2—82no—
686C2C4×#1058762511yescharacteristic
687C2#1453764111yescharacteristic
688C2#1453766511yescharacteristic
689C2#1453842011yescharacteristic
690C2#1453844311yescharacteristic
691C2#1453939511yescharacteristic
692C2#1453940611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2324C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
cosocleC22×C4×C17C24
hypercenter#2324D17
hypocenterC17#21523
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
2. C2×C4×#63229C2semidirect
3. C2×#11762765C2semidirect
4. #14531466C2semidirect
5. #14529637C2semidirect
6. #14529509C2semidirect
7. #14529484C2semidirect
8. #14529473C2semidirect
10. #14529215C2semidirect
11. #14528807C2semidirect
12. C4×#10587479C2semidirect
13. #14528617C2semidirect
14. #14528609C2semidirect
21. #11763036C4semidirect
98. #10587610C2×C4semidirect
109. C22×#3622C2×C4semidirect
129. #10587481C2×C4semidirect
132. #10587479C2×C4semidirect
136. C4×#3622D4semidirect
483. C17#21523semidirect

Maximal subgroups

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

subgroupindex
C2×#117630362
C2×C4×#632292
C2×#117627652
#145314662
#145296372
#145295092
#145294842
#145294732
#145292232
#145292152
#145288072
C4×#105874792
#145286172
#145286092
C17×#23242
#2152317