Finite Group Databasebeta

← #17039477

Subgroups — #17039477

Order 4352.

1086 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763417C211yesmaximal, characteristic
2C2×#11762868C211yesmaximal, characteristic
3#14531138C211yesmaximal, characteristic
4#14529740C211yesmaximal, characteristic, complemented
5#14529738C211yesmaximal, characteristic, complemented
6#14529705C211yesmaximal, characteristic, complemented
7#14529705C211yesmaximal, characteristic, complemented
8#14529095C211yesmaximal, characteristic
9#14529093C211yesmaximal, characteristic
10#14529090C211yesmaximal, characteristic
11#14529088C211yesmaximal, characteristic
12#14528617C211yesmaximal, characteristic, complemented
13#14528614C211yesmaximal, characteristic, complemented
14C4×#10587487C211yesmaximal, characteristic, complemented
15C4×C17×#103C211yesmaximal, characteristic, complemented
16#11763417—24no—
17#11763417C418yescomplemented
18C2×#10587630C2211yescharacteristic
19C2×#10587584—21no—
20C2×#10587562—21no—
21C2×#10587556—21no—
22C2×#10587576C2211yescharacteristic
23C2×#10587575—21no—
24C2×#10587583C2211yescharacteristic
25C2×#10587632C2211yescharacteristic
26C2×C4×Dic34C2211yescharacteristic
27C2×#10587578C2211yescharacteristic
28C2×#10587558C2211yescharacteristic
29C2×C4⋊C4×D17C2211yescharacteristic
30#11762868—24no—
31C2×#10587485C2211yescharacteristic
32C2×#10587485C2211yescharacteristic
33C2×#10587487C2211yescharacteristic
34C2×#10587487C2211yescharacteristic
35C2×#10587506C2211yescharacteristic
36C2×#10587506C2211yescharacteristic
37C2×#10587481C2211yescharacteristic
38C2×#10587481C2211yescharacteristic
39C2×#10587491C2211yescharacteristic
40C2×#10587491C2211yescharacteristic
41C2×#10587474C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4×#63225—21no—
45C4×C4⋊Dic17C2211yescharacteristic
46#11762660—21no—
47C4×#63227—21no—
48#11762656—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C4×C8×C17C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×C4×D17—21no—
64C22×C4×D17C2311yescharacteristic
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×C17—21no—
76C2×C42×C17C2311yescharacteristic
77#10587632—22no—
78#10587632C2×C414yes—
79#10587630—22no—
80#10587630C2×C414yescomplemented
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227D411yescharacteristic
85C2×#63227D411yescharacteristic
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225D411yescharacteristic
98C2×#63225D411yescharacteristic
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107#10587584—28no—
108#10587583—22no—
109#10587583C2×C414yes—
110#10587578—22no—
111#10587578—22no—
112C4⋊C4×D17—22no—
113C4⋊C4×D17—22no—
114#10587576—22no—
115#10587576C2×C414yescomplemented
116#10587575—28no—
117#10587562—28no—
118#10587558C2×C414yes—
119#10587558D414yes—
120#10587556—28no—
121C4×Dic34—22no—
122C4×Dic34C2×C414yescomplemented
123C17×#103—22no—
124C17×#103—22no—
125C4×C8×C17—22no—
126C17×#95—21no—
127#10587506—22no—
128#10587506—22no—
129#10587502—21no—
130#10587502—21no—
131#10587502—21no—
132#10587491—22no—
133#10587491—22no—
134#10587487—22no—
135#10587487—22no—
136#10587485—22no—
137#10587485—22no—
138#10587481—22no—
139#10587481—22no—
140#10587474—22no—
141C23×D17C2×D411yescharacteristic
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17C22×C412yes—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2411yescharacteristic
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162C2×C4×D17—21no—
163C2×C4×D17—21no—
164C2×C4×D17—21no—
165C2×C4×D17—21no—
166C2×C4×D17—24no—
167C2×C4×D17—81no—
168C2×C4×D17—81no—
169C2×C4×D17C22×C412yes—
170C2×C4×D17C2×D412yes—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34C22×C412yes—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17C22×C412yes—
188C4⋊C4×C17#4112yes—
189C4⋊C4×C17#4112yes—
190C42×C17—24no—
191C42×C17—81no—
192C42×C17C22×C412yes—
193C42×C17C2×D412yes—
194#63227—24no—
195#63227—24no—
196#63227—24no—
197#63227C22⋊C414yes—
198#63227C22⋊C414yes—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—24no—
203C4⋊Dic17—24no—
204C4⋊Dic17C22×C412yes—
205C4⋊Dic17#4112yes—
206C4⋊Dic17#4112yes—
207#63225—24no—
208#63225—24no—
209#63225—24no—
210#63225—24no—
211#63225—24no—
212#63225—24no—
213#63225C22⋊C414yes—
214#63225C22⋊C414yes—
215C4×Dic17—24no—
216C4×Dic17—24no—
217C4×Dic17—81no—
218C4×Dic17—81no—
219C4×Dic17—81no—
220C4×Dic17C22×C412yes—
221C4×Dic17C2×D412yes—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228#21398—171nomaximal, Sylow
229C23×C17C22×D411yescharacteristic
230C22×D17—21no—
231C22×D17—21no—
232C22×D17—21no—
233C22×D17—21no—
234C22×D17—21no—
235C22×D17—21no—
236C22×D17C2×C22⋊C412yes—
237Q8×C17—22no—
238Q8×C17—22no—
239Q8×C17—41no—
240Q8×C17—41no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
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—41no—
257C2×C4×C17C2×C22⋊C412yes—
258C2×C4×C17C2×C22⋊C412yes—
259C2×C4×C17C23×C411yescharacteristic
260C2×C4×C17C22×D411yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17C2×C22⋊C412yes—
284C2×Dic17C2×C22⋊C412yes—
285C2×Dic17C2×C22⋊C412yes—
286C4×D17—22no—
287C4×D17—22no—
288C4×D17—22no—
289C4×D17—22no—
290C4×D17—84no—
291Dic34—22no—
292Dic34—22no—
293Dic34—41no—
294Dic34—41no—
295C8×C17—41no—
296C8×C17—41no—
297#3622—41no—
298#3622—41no—
299C2×#541—171no—
300C2×#433—171no—
301#2274—171no—
302#1989—171no—
303#1989—171no—
304#1981—171no—
305#1981—171no—
306#1873—171no—
307#1873—171no—
308#1872—171no—
309#1872—171no—
310#1763—171no—
311#1763—171no—
312C4×#103—171no—
313C4×#103D1711yescharacteristic, complemented
314C22×C17C22×C22⋊C411yescharacteristic
315C22×C17#52511yescharacteristic
316C22×C17#52511yescharacteristic
317C22×C17#57711yescharacteristic
318C22×C17#57711yescharacteristic
319C22×C17#57811yescharacteristic
320C22×C17#57811yescharacteristic
321C2×D17—22no—
322C2×D17—22no—
323C2×D17—22no—
324C2×D17—41no—
325C2×D17—41no—
326C2×D17—41no—
327C2×D17—41no—
328C4×C17—22no—
329C4×C17—22no—
330C4×C17—42no—
331C4×C17—42no—
332C4×C17—42no—
333C4×C17—42no—
334C4×C17C4×#4111yescharacteristic
335C4×C17C4×#4111yescharacteristic
336C4×C17#53211yescharacteristic
337C4×C17#53211yescharacteristic
338Dic17—22no—
339Dic17—22no—
340Dic17—22no—
341Dic17—42no—
342Dic17—42no—
343Dic17—42no—
344Dic17—42no—
345#541—178no—
346#541—344no—
347C2×C4⋊Q8—171no—
348C2×#126—341no—
349C2×#126—341no—
350C2×#125—171no—
351C2×#125—341no—
352C2×#125—341no—
353C2×C22⋊Q8—171no—
354C2×C22⋊Q8—171no—
355C2×C4×Q8—171no—
356C2×C4×Q8C2×D1711yescharacteristic
357C2×#117—171no—
358C2×#117—171no—
359C22×C4⋊C4—171no—
360#433—344no—
361C2×C4⋊C8—171no—
362C2×C4⋊C8—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#103C2×D1711yescharacteristic
370C2×#103C2×D1711yescharacteristic
371C2×C22⋊C8—171no—
372C2×C22⋊C8—171no—
373C2×#97—171no—
374C2×C4×C8C2×D1711yescharacteristic
375C4×C4⋊C4—171no—
376C4×C4⋊C4—171no—
377C4×C4⋊C4—171no—
378C4×C4⋊C4—341no—
379C4×C4⋊C4—341no—
380C4×C4⋊C4C2×D1711yescharacteristic
381C4×C22⋊C4—341no—
382C4×C22⋊C4—341no—
383#340—171no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388#340—171no—
389#340C2×D1711yescharacteristic
390#340C2×D1711yescharacteristic
391C2×C17#229711yescharacteristic
392C2×C17#289811yescharacteristic
393C2×C17#289811yescharacteristic
394C2×C17#312011yescharacteristic
395C2×C17#312011yescharacteristic
396C2×C17#312311yescharacteristic
397C2×C17#312711yescharacteristic
398D17—42no—
399C22×Q8—171no—
400C22×Q8C22×D1711yescharacteristic
401C23×C4—171no—
402C23×C4—341no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8C22×D1711yescharacteristic
406C22×C8C22×D1711yescharacteristic
407C4⋊Q8—174no—
408C4⋊Q8—342no—
409#126—348no—
410#126—348no—
411#125—174no—
412#125—348no—
413#125—342no—
414#125—348no—
415C22⋊Q8—174no—
416C22⋊Q8—174no—
417C22⋊Q8—342no—
418C22⋊Q8—342no—
419C4×Q8—22no—
420C4×Q8—174no—
421C4×Q8—342no—
422C4×Q8C4×D1714yescomplemented
423#117—174no—
424#117—174no—
425#117—342no—
426#117—342no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—342no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4—341no—
442C2×C4⋊C4—342no—
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C4⋊C4C22×D1711yescharacteristic
445C2×C4⋊C4C22×D1711yescharacteristic
446C2×C22⋊C4—171no—
447C2×C22⋊C4—171no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4—341no—
450C2×C22⋊C4—341no—
451C2×C42—21no—
452C2×C42—21no—
453C2×C42—171no—
454C2×C42—341no—
455C2×C42—341no—
456C2×C42—341no—
457C2×C42—341no—
458C2×C42—341no—
459C2×C42C22×D1711yescharacteristic
460C4⋊C8—342no—
461C4⋊C8—342no—
462#103—22no—
463#103—22no—
464#103—342no—
465#103—342no—
466#103—342no—
467#103—342no—
468#103—342no—
469#103—342no—
470C22⋊C8—342no—
471C22⋊C8—342no—
472#97—342no—
473C4×C8—22no—
474#95—21no—
475#95—341no—
476#95—341no—
477#95—341no—
478C17#2139811yescharacteristic, complemented, Sylow
479C24—171no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—172no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8C2×C4×D1712yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—171no—
497C22×C4—171no—
498C22×C4—171no—
499C22×C4—172no—
500C22×C4—172no—
501C22×C4—344no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—341no—
510C22×C4—1361no—
511C22×C4—1361no—
512C22×C4D4×D1711yescharacteristic
513C22×C4D4×D1711yescharacteristic
514C22×C4C23×D1711yescharacteristic
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—24no—
531C4⋊C4—174no—
532C4⋊C4—174no—
533C4⋊C4—172no—
534C4⋊C4—172no—
535C4⋊C4—172no—
536C4⋊C4—344no—
537C4⋊C4—344no—
538C4⋊C4—344no—
539C4⋊C4—344no—
540C4⋊C4—344no—
541C4⋊C4—344no—
542C4⋊C4—344no—
543C4⋊C4—341no—
544C4⋊C4—341no—
545C4⋊C4—341no—
546C4⋊C4—344no—
547C4⋊C4C2×C4×D1712yes—
548C4⋊C4#6325212yes—
549C4⋊C4#6325212yes—
550C22⋊C4—174no—
551C22⋊C4—174no—
552C22⋊C4—344no—
553C22⋊C4—344no—
554C22⋊C4—344no—
555C42—24no—
556C42—81no—
557C42—172no—
558C42—172no—
559C42—344no—
560C42—344no—
561C42—1361no—
562C42—1361no—
563C42—1361no—
564C42C2×C4×D1712yes—
565C42D4×D1712yes—
566C23—172no—
567C23—341no—
568C23—341no—
569C23—341no—
570C23—341no—
571C23—341no—
572C23—341no—
573C23C2×D4×D1711yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—41no—
577Q8—41no—
578Q8—342no—
579Q8—342no—
580Q8—681no—
581Q8—681no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—172no—
599C2×C4—172no—
600C2×C4—172no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—341no—
605C2×C4—341no—
606C2×C4—341no—
607C2×C4—342no—
608C2×C4—342no—
609C2×C4—342no—
610C2×C4—342no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—1364no—
624C2×C4C22⋊C4×D1712yes—
625C2×C4C22⋊C4×D1712yes—
626C2×C4C22×C4×D1711yescharacteristic
627C2×C4C2×#6325211yescharacteristic
628C2×C4C2×#6325211yescharacteristic
629C2×C4C2×D4×D1711yescharacteristic
630C2×C4#41×D1711yescharacteristic
631C2×C4#41×D1711yescharacteristic
632C8—41no—
633C8—41no—
634C8—681no—
635C8—681no—
636C22—342no—
637C22—342no—
638C22—342no—
639C22—681no—
640C22—681no—
641C22—681no—
642C22—681no—
643C22C2×C22⋊C4×D1711yescharacteristic
644C22#1176331811yescharacteristic
645C22#1176331811yescharacteristic
646C22#1176361511yescharacteristic
647C22#1176361611yescharacteristic
648C22#1176362211yescharacteristic
649C22#1176362311yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—342no—
657C4—342no—
658C4—342no—
659C4—682no—
660C4—682no—
661C4—682no—
662C4—682no—
663C4#1176327711yescharacteristic
664C4C4×#6325211yescharacteristic
665C4#1176335611yescharacteristic
666C4#1176335611yescharacteristic
667C2—682no—
668C2#1453127011yescharacteristic
669C2#1453731211yescharacteristic
670C2#1453731311yescharacteristic
671C2#1453899411yescharacteristic
672C2#1453900411yescharacteristic
673C2#1453903311yescharacteristic
674C2#1453908111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#103C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenterC4×#103D17
hypocenterC17#21398
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
4. #14529740C2semidirect
5. #14529738C2semidirect
6. #14529705C2semidirect
7. #14529705C2semidirect
12. #14528617C2semidirect
13. #14528614C2semidirect
14. C4×#10587487C2semidirect
15. C4×C17×#103C2semidirect
17. #11763417C4semidirect
69. C4×Q8×C17C2×C4semidirect
80. #10587630C2×C4semidirect
115. #10587576C2×C4semidirect
122. C4×Dic34C2×C4semidirect
313. C4×#103D17semidirect
422. C4×Q8C4×D17semidirect
478. C17#21398semidirect

Maximal subgroups

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

subgroupindex
C2×#117634172
C2×#117628682
#145311382
#145297402
#145297382
#145297052
#145290952
#145290932
#145290902
#145290882
#145286172
#145286142
C4×#105874872
C4×C17×#1032
#2139817