Finite Group Databasebeta

← #17070534

Subgroups — #17070534

Order 4352.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763437C211yesmaximal, characteristic
2C2×#11763116C211yesmaximal, characteristic, complemented
3C2×#11763104C211yesmaximal, characteristic, complemented
4#14530122C211yesmaximal, characteristic, complemented
5#14530112C211yesmaximal, characteristic, complemented
6#14529935C211yesmaximal, characteristic, complemented
7#14529925C211yesmaximal, characteristic, complemented
8#14529741C211yesmaximal, characteristic, complemented
9#14529736C211yesmaximal, characteristic, complemented
10#14529097C211yesmaximal, characteristic
11#14529092C211yesmaximal, characteristic
12#14528981C211yesmaximal, characteristic
13#14528971C211yesmaximal, characteristic
14#14528768C211yesmaximal, characteristic
15C17×#2323C211yesmaximal, characteristic
16#11763437—24no—
17#11763437—24no—
18C2×#10587555C2211yescharacteristic
19C2×#10587555C2211yescharacteristic
20C2×#10587574—21no—
21C2×#10587556C2211yescharacteristic
22C2×#10587556C2211yescharacteristic
23C2×#10587576—21no—
24C2×#10587564—21no—
25C2×#10587564—21no—
26C2×#10587612C2211yescharacteristic
27C22×C4⋊Dic17C2211yescharacteristic
28#11763116—24no—
29#11763104—24no—
30C2×#10587478C2211yescharacteristic
31C2×#10587478C2211yescharacteristic
32C2×#10587478C2211yescharacteristic
33C2×#10587479C2211yescharacteristic
34C2×C4⋊#3622C2211yescharacteristic
35C2×C4⋊#3622C2211yescharacteristic
36C2×#10587506C2211yescharacteristic
37C2×#10587506C2211yescharacteristic
38C2×#10587506C2211yescharacteristic
39C2×#10587506C2211yescharacteristic
40C2×C22⋊#3622C2211yescharacteristic
41C2×C22⋊#3622C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762564C2211yescharacteristic
52C2×C17×C22⋊Q8C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C2×C17×#117C2211yescharacteristic
55C17×#391C2211yescharacteristic
56C17×#389—21no—
57C17×#389C2211yescharacteristic
58C17×#388—21no—
59C17×#388—21no—
60C17×#387—21no—
61C17×#380C2211yescharacteristic
62C4×C4⋊C4×C17C2211yescharacteristic
63C22×Q8×C17C2311yescharacteristic
64C23×C4×C17C2311yescharacteristic
65C23×Dic17—21no—
66C22×Dic34—21no—
67C22×Dic34—21no—
68C17×C22⋊Q8—22no—
69C17×C22⋊Q8Q814yescomplemented
70C4×Q8×C17—22no—
71C4×Q8×C17—22no—
72C17×#117—22no—
73C17×#117—22no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C42×C17—21no—
87C2×C42×C17—21no—
88C2×C42×C17C2311yescharacteristic
89C2×C42×C17C2311yescharacteristic
90C2×C22⋊Dic17—21no—
91#10587612—22no—
92#10587612—22no—
93C2×C4⋊Dic17—22no—
94C2×C4⋊Dic17—22no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—21no—
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4⋊Dic17C2311yescharacteristic
102C2×#63225—21no—
103C2×#63225—21no—
104C2×#63225—21no—
105C2×#63225—21no—
106C2×C4×Dic17—21no—
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C22×#3622C2311yescharacteristic
111#10587576—28no—
112#10587574—28no—
113#10587564—28no—
114#10587564—28no—
115#10587556—22no—
116#10587556—22no—
117#10587556—22no—
118#10587556D414yes—
119#10587555—22no—
120#10587555—22no—
121#10587555—22no—
122#10587555D414yes—
123C17×#95—21no—
124C17×#95—21no—
125C17×#95—21no—
126C17×#95—21no—
127C17×#95—21no—
128#10587506—22no—
129#10587506—22no—
130#10587506—22no—
131#10587506—22no—
132C22⋊#3622—22no—
133C22⋊#3622—22no—
134#10587479—22no—
135#10587478—22no—
136#10587478—22no—
137#10587478—22no—
138C4⋊#3622—22no—
139C4⋊#3622—22no—
140C24×C17C2×D411yescharacteristic
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17C2×Q812yes—
147C22×C4×C17—21no—
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×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×Q812yes—
161C22×C4×C17#4112yes—
162C22×C4×C17C2411yescharacteristic
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17—24no—
168C22×Dic17—81no—
169C22×Dic17—81no—
170C2×Dic34—24no—
171C2×Dic34—24no—
172C2×Dic34—81no—
173C2×Dic34—81no—
174C2×Dic34—81no—
175C2×Dic34—81no—
176C4⋊C4×C17—22no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—24no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—42no—
185C4⋊C4×C17—81no—
186C4⋊C4×C17C2×Q812yes—
187C22⋊C4×C17—24no—
188C22⋊C4×C17—22no—
189C22⋊C4×C17—22no—
190C22⋊C4×C17—81no—
191C42×C17—21no—
192C42×C17—21no—
193C42×C17—42no—
194C42×C17—81no—
195C42×C17C2×D412yes—
196C42×C17C2×D412yes—
197C22⋊Dic17—24no—
198C4⋊Dic17—24no—
199C4⋊Dic17—24no—
200C4⋊Dic17—24no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17C2×D412yes—
206C4⋊Dic17C2×D412yes—
207C4⋊Dic17C2×D412yes—
208C4⋊Dic17C2×D412yes—
209#63225—24no—
210#63225—24no—
211#63225—24no—
212#63225—24no—
213C4×Dic17—24no—
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×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226#24311—171nomaximal, Sylow
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17C22⋊Q812yes—
234C23×C17C22×D411yescharacteristic
235Q8×C17—22no—
236Q8×C17—22no—
237Q8×C17—41no—
238Q8×C17—41no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
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—21no—
250C2×C4×C17—21no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
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#12012yes—
268C2×C4×C17#12012yes—
269C2×C4×C17C22⋊Q812yes—
270C2×C4×C17C22×D411yescharacteristic
271C2×C4×C17C22×D411yescharacteristic
272C2×C4×C17C22×Q811yescharacteristic
273C2×C4×C17C2×#4111yescharacteristic
274C2×C4×C172−1+411yescharacteristic
275C2×C4×C172−1+411yescharacteristic
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—22no—
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—
288C2×Dic17—84no—
289Dic34—84no—
290Dic34—84no—
291#3622—41no—
292#3622—41no—
293#3622—41no—
294#3622—41no—
295C2×#544—171no—
296C2×#479—171no—
297C2×#477—171no—
298#2323D1711yescharacteristic
299#2059—171no—
300#2057—171no—
301#2026—171no—
302#2024—171no—
303#1989—171no—
304#1988—171no—
305#1873—171no—
306#1872—171no—
307#1854—171no—
308#1852—171no—
309#1798—171no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—41no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17C2×#12011yescharacteristic
318C22×C17C2×C22⋊Q811yescharacteristic
319C22×C17#53611yescharacteristic
320C22×C17#57611yescharacteristic
321C22×C17#57711yescharacteristic
322C22×C17#57811yescharacteristic
323C22×C17#57811yescharacteristic
324C4×C17—22no—
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—42no—
328C4×C17—42no—
329C4×C17—41no—
330C4×C17—41no—
331C4×C17—81no—
332C4×C17—81no—
333C4×C17#54811yescharacteristic
334C4×C17#54811yescharacteristic
335C4×C17D4×Q811yescharacteristic
336C4×C17D4×Q811yescharacteristic
337Dic17—42no—
338Dic17—42no—
339Dic17—42no—
340Dic17—42no—
341#544—344no—
342#544—344no—
343C2×C4⋊Q8—171no—
344C2×C4⋊Q8—171no—
345C2×C4⋊Q8—341no—
346C2×#125—171no—
347C2×#125—171no—
348C2×#125—341no—
349C2×C22⋊Q8—341no—
350C2×C22⋊Q8—341no—
351C2×C22⋊Q8C2×D1711yescharacteristic
352C2×C4×Q8C2×D1711yescharacteristic
353C2×#117—171no—
354C2×#117C2×D1711yescharacteristic
355C22×C4⋊C4—171no—
356#479—344no—
357#477—344no—
358C2×#107—171no—
359C2×#107—171no—
360C2×#107—171no—
361C2×#106—171no—
362C2×C4⋊C8—171no—
363C2×C4⋊C8—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×C22⋊C8—171no—
369C2×C22⋊C8—171no—
370#391C2×D1711yescharacteristic
371#389—21no—
372#389C2×D1711yescharacteristic
373#388—21no—
374#388—21no—
375#387—21no—
376#380C2×D1711yescharacteristic
377C4×C4⋊C4C2×D1711yescharacteristic
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#336—171no—
387#336—171no—
388C2×C17—42no—
389C2×C17#240911yescharacteristic
390C2×C17#301111yescharacteristic
391C2×C17#301211yescharacteristic
392C2×C17#308211yescharacteristic
393C2×C17#308311yescharacteristic
394C2×C17#324211yescharacteristic
395C2×C17#324311yescharacteristic
396C22×Q8—341no—
397C22×Q8—341no—
398C22×Q8C22×D1711yescharacteristic
399C23×C4—341no—
400C23×C4C22×D1711yescharacteristic
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8—171no—
405C4⋊Q8—174no—
406C4⋊Q8—342no—
407C4⋊Q8—342no—
408C4⋊Q8—348no—
409C4⋊Q8—342no—
410#125—174no—
411#125—342no—
412#125—342no—
413#125—348no—
414#125—342no—
415C22⋊Q8—22no—
416C22⋊Q8—348no—
417C22⋊Q8—348no—
418C22⋊Q8Dic3414yescomplemented
419C4×Q8—22no—
420C4×Q8—22no—
421#117—22no—
422#117—22no—
423#117—342no—
424#117—342no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—342no—
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⋊C4C22⋊D1711yescharacteristic
442C2×C4⋊C4C22⋊D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C4⋊C4C22×D1711yescharacteristic
445C2×C4⋊C4C22×D1711yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4C22⋊D1711yescharacteristic
450C2×C22⋊C4C22⋊D1711yescharacteristic
451C2×C42—21no—
452C2×C42—21no—
453C2×C42—341no—
454C2×C42C22×D1711yescharacteristic
455C2×C42C22×D1711yescharacteristic
456#107—342no—
457#107—342no—
458#107—342no—
459#106—342no—
460C4⋊C8—342no—
461C4⋊C8—342no—
462#103—342no—
463#103—342no—
464#103—342no—
465#103—342no—
466C22⋊C8—342no—
467C22⋊C8—342no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473C17#2431111yescharacteristic, complemented, Sylow
474C24C2×C22⋊D1711yescharacteristic
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—344no—
481C2×Q8—344no—
482C2×Q8—1361no—
483C2×Q8—1361no—
484C2×Q8—1361no—
485C2×Q8—1361no—
486C2×Q8C2×Dic3412yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—344no—
500C22×C4—1361no—
501C22×C4—1361no—
502C22×C4C2×Dic3412yes—
503C22×C4#6325212yes—
504C22×C4C2×C22⋊D1711yescharacteristic
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C2×C22⋊D1711yescharacteristic
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C23×D1711yescharacteristic
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C4⋊C4—21no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—172no—
533C4⋊C4—172no—
534C4⋊C4—172no—
535C4⋊C4—172no—
536C4⋊C4—344no—
537C4⋊C4—344no—
538C4⋊C4—344no—
539C4⋊C4—344no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—344no—
545C4⋊C4—344no—
546C4⋊C4—344no—
547C4⋊C4C2×Dic3412yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—81no—
552C22⋊C4—344no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—81no—
557C42—344no—
558C42D4×D1712yes—
559C42D4×D1712yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1058761012yes—
567C23C22×C22⋊D1711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—1364no—
573Q8—1364no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—342no—
603C2×C4—342no—
604C2×C4—342no—
605C2×C4—342no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—1364no—
615C2×C4#1058761012yes—
616C2×C4#1058762312yes—
617C2×C4#1058762312yes—
618C2×C4C22×Dic3411yescharacteristic
619C2×C4C2×#6325211yescharacteristic
620C2×C4C2×D4×D1711yescharacteristic
621C2×C4C2×D4×D1711yescharacteristic
622C2×C4#1058769711yescharacteristic
623C2×C4#1058769711yescharacteristic
624C8—681no—
625C8—681no—
626C8—681no—
627C8—681no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1058762311yescharacteristic
636C22C2×#1058761011yescharacteristic
637C22#1176338011yescharacteristic
638C22#1176361211yescharacteristic
639C22#1176361711yescharacteristic
640C22#1176362411yescharacteristic
641C22#1176362411yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—682no—
652C4—682no—
653C4—682no—
654C4—682no—
655C4#1176346811yescharacteristic
656C4#1176346811yescharacteristic
657C4D4×Dic3411yescharacteristic
658C4D4×Dic3411yescharacteristic
659C2—42no—
660C2#1453214911yescharacteristic
661C2#1453807711yescharacteristic
662C2#1453808511yescharacteristic
663C2#1453862611yescharacteristic
664C2#1453863511yescharacteristic
665C2#1454004611yescharacteristic
666C2#1454006111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2323C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2323D17
hypocenterC17#24311
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×#11763116C2semidirect
3. C2×#11763104C2semidirect
4. #14530122C2semidirect
5. #14530112C2semidirect
6. #14529935C2semidirect
7. #14529925C2semidirect
8. #14529741C2semidirect
9. #14529736C2semidirect
69. C17×C22⋊Q8Q8semidirect
418. C22⋊Q8Dic34semidirect
473. C17#24311semidirect

Maximal subgroups

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

subgroupindex
C2×#117634372
C2×#117631162
C2×#117631042
#145301222
#145301122
#145299352
#145299252
#145297412
#145297362
#145290972
#145290922
#145289812
#145289712
#145287682
C17×#23232
#2431117