Finite Group Databasebeta

← #17005482

Subgroups — #17005482

Order 4352.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537944C211yesmaximal, characteristic, complemented
2#14537819C211yesmaximal, characteristic, complemented
3#14527675C211yesmaximal, characteristic, complemented
4#14527669C211yesmaximal, characteristic, complemented
5#14527663C211yesmaximal, characteristic, complemented
6#14527649C211yesmaximal, characteristic, complemented
7#14527649C211yesmaximal, characteristic, complemented
8#14527642C211yesmaximal, characteristic, complemented
9SD16×#3622C211yesmaximal, characteristic, complemented
10D8×#3622C211yesmaximal, characteristic, complemented
11#14527569C211yesmaximal, characteristic, complemented
12#14527244C211yesmaximal, characteristic, complemented
13C2×D4⋊#3622C211yesmaximal, characteristic, complemented
14#14527068C211yesmaximal, characteristic, complemented
15C17×#3105C211yesmaximal, characteristic
16#11762940C2211yescharacteristic, complemented
17Q8×#3622C2211yescharacteristic, complemented
18#11762887C2211yescharacteristic
19#11762887C2211yescharacteristic
20#11762887C2211yescharacteristic, complemented
21#11762881—21no—
22#11762881—21no—
23#11762881—21no—
24D4×#3622—21no—
25D4×#3622C2211yescharacteristic
26D4×#3622C2211yescharacteristic
27D4×#3622C2211yescharacteristic, complemented
28#11762870—21no—
29#11762866C2211yescharacteristic
30#11762862C2211yescharacteristic
31#11762836—21no—
32C2×C4⋊#3622C2211yescharacteristic
33#11762773—21no—
34C2×C22⋊#3622—21no—
35C2×C4×#3622C2211yescharacteristic
36#11762562C2211yescharacteristic, complemented
37#11762562C2211yescharacteristic, complemented
38#11762560C2211yescharacteristic, complemented
39#11762560C2211yescharacteristic, complemented
40Q8⋊#3622C2211yescharacteristic, complemented
41Q8⋊#3622C2211yescharacteristic, complemented
42D4⋊#3622C2211yescharacteristic
43D4⋊#3622C2211yescharacteristic
44D4⋊#3622C2211yescharacteristic
45D4⋊#3622C2211yescharacteristic
46D4⋊#3622C2211yescharacteristic, complemented
47D4⋊#3622C2211yescharacteristic, complemented
48#11762531C2211yescharacteristic, complemented
49#11762531C2211yescharacteristic, complemented
50C8×#3622C2211yescharacteristic, complemented
51C8×#3622C2211yescharacteristic, complemented
52C17×#529C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C17×#484C411yescharacteristic
55C17×#481C411yescharacteristic
56C17×#478C411yescharacteristic
57C17×#478C411yescharacteristic
58C17×#463C411yescharacteristic
59C17×#463C411yescharacteristic
60C17×#450C411yescharacteristic
61C17×#447C411yescharacteristic
62C4×SD16×C17C2211yescharacteristic
63C4×SD16×C17C2211yescharacteristic
64C4×D8×C17C2211yescharacteristic
65C4×D8×C17C2211yescharacteristic
66C17×#431C2211yescharacteristic
67C22×D4×C17C2×C411yescharacteristic
68C23×C4×C17—21no—
69C2×SD16×C17C2×C411yescharacteristic
70C2×SD16×C17C2×C411yescharacteristic
71C2×D8×C17C2×C411yescharacteristic
72C2×D8×C17C2×C411yescharacteristic
73C17×#126—21no—
74C17×#125C2×C411yescharacteristic
75C17×#124C2×C411yescharacteristic
76C17×#123—21no—
77C17×C22⋊Q8C2×C411yescharacteristic
78C17×#121C2×C411yescharacteristic
79C4×Q8×C17C2311yescharacteristic
80C4×D4×C17—21no—
81C4×D4×C17—21no—
82C4×D4×C17—21no—
83C4×D4×C17C2×C411yescharacteristic
84C4×D4×C17C2×C411yescharacteristic
85C4×D4×C17C2311yescharacteristic
86C4×D4×C17C2311yescharacteristic
87C4×D4×C17C2311yescharacteristic
88C17×#117—21no—
89C2×C4⋊C4×C17C2×C411yescharacteristic
90C2×C22⋊C4×C17—21no—
91C2×C42×C17C2311yescharacteristic
92C2×#63223—21no—
93C2×#63223—21no—
94C2×#63223—21no—
95C22×#3622—21no—
96C22×#3622—21no—
97C22×#3622—21no—
98C22×#3622—21no—
99C22×#3622—21no—
100C17×#107C2×C411yescharacteristic
101C17×#107C2×C411yescharacteristic
102C17×#106C2×C411yescharacteristic
103C17×#106C2×C411yescharacteristic
104C17×C4⋊C8C2×C411yescharacteristic
105C17×C4⋊C8C2×C411yescharacteristic
106C17×#103C2×C411yescharacteristic
107C17×#103C2×C411yescharacteristic
108C17×#102C2×C411yescharacteristic
109C17×#102C2×C411yescharacteristic
110C17×#102C2×C411yescharacteristic
111C17×#102C2×C411yescharacteristic
112C17×#102C2×C411yescharacteristic
113C17×#102C2×C411yescharacteristic
114C17×C22⋊C8C2×C411yescharacteristic
115C17×C22⋊C8C2×C411yescharacteristic
116C4×C8×C17C2311yescharacteristic
117C4×C8×C17C2311yescharacteristic
118C22⋊#3622—21no—
119C22⋊#3622—21no—
120C22⋊#3622—21no—
121C22⋊#3622—21no—
122C22⋊#3622—21no—
123C22⋊#3622—21no—
124C22⋊#3622—21no—
125C22⋊#3622D411yescharacteristic
126C22⋊#3622D411yescharacteristic
127C4⋊#3622—21no—
128C4⋊#3622—21no—
129C4⋊#3622—21no—
130C4⋊#3622D411yescharacteristic
131C4⋊#3622D411yescharacteristic
132C4⋊#3622C2311yescharacteristic
133C4⋊#3622C2311yescharacteristic
134C4⋊#3622C2311yescharacteristic
135C4⋊#3622C2311yescharacteristic
136#10587474—21no—
137#10587474—21no—
138C4×#3622—21no—
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C24×C17—21no—
145C2×Q8×C17C22×C411yescharacteristic
146C2×D4×C17—22no—
147C2×D4×C17—21no—
148C2×D4×C17—21no—
149C2×D4×C17—41no—
150C2×D4×C17C22×C411yescharacteristic
151C2×D4×C17C22×C411yescharacteristic
152C2×D4×C17C22×C411yescharacteristic
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—81no—
160C22×C4×C17—81no—
161C22×C4×C17C22×C411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17#4111yescharacteristic
164SD16×C17—22no—
165SD16×C17—41no—
166D8×C17—22no—
167D8×C17—41no—
168C2×C8×C17C22×C411yescharacteristic
169C2×C8×C17C22×C411yescharacteristic
170C2×C8×C17C22×C411yescharacteristic
171C2×C8×C17C22×C411yescharacteristic
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17C22×C411yescharacteristic
178C4⋊C4×C17C22×C411yescharacteristic
179C4⋊C4×C17C22×C411yescharacteristic
180C4⋊C4×C17C22×C411yescharacteristic
181C22⋊C4×C17—21no—
182C22⋊C4×C17—21no—
183C22⋊C4×C17—21no—
184C22⋊C4×C17—21no—
185C22⋊C4×C17—21no—
186C22⋊C4×C17—21no—
187C22⋊C4×C17—21no—
188C22⋊C4×C17—21no—
189C22⋊C4×C17—21no—
190C42×C17—21no—
191C42×C17C22×C411yescharacteristic
192C42×C17C2×D411yescharacteristic
193C42×C17#4111yescharacteristic
194C42×C17C2411yescharacteristic
195#63223—81no—
196#63223—81no—
197#63223—81no—
198C2×#3622—24no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—81no—
206C2×#3622—81no—
207C2×#3622—81no—
208C2×#3622—81no—
209C2×#3622C2×D411yescharacteristic
210C2×#3622C2×D411yescharacteristic
211C2×#3622C2×D411yescharacteristic
212C2×#3622C2×D411yescharacteristic
213#18014—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—42no—
218C23×C17—42no—
219C23×C17—81no—
220C23×C17C4×D411yescharacteristic
221Q8×C17—21no—
222Q8×C17—21no—
223D4×C17—21no—
224D4×C17—21no—
225D4×C17—21no—
226D4×C17—21no—
227D4×C17—21no—
228D4×C17—42no—
229D4×C17—42no—
230D4×C17#13112yes—
231C2×C4×C17—22no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—81no—
248C2×C4×C17—81no—
249C2×C4×C17—81no—
250C2×C4×C17C4×D411yescharacteristic
251C2×C4×C17C4×D411yescharacteristic
252C2×C4×C17C4×D411yescharacteristic
253C2×C4×C17C23×C411yescharacteristic
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C8×C17—21no—
257C8×C17—21no—
258C8×C17—21no—
259C8×C17—21no—
260#3622—22no—
261#3622—22no—
262#3622—21no—
263#3622—21no—
264#3622—21no—
265#3622—21no—
266#3622—41no—
267#3622—41no—
268#3622—41no—
269#3622—41no—
270#3105D1711yescharacteristic
271#2994—171no—
272#2974—171no—
273#1590—171no—
274#1589—171no—
275#1588—171no—
276#1586—171no—
277#1586—171no—
278#1585—171no—
279C8×SD16—171no—
280C8×D8—171no—
281#1573—171no—
282#1494—171no—
283C2×D4⋊C8—171no—
284#1450—171no—
285C22×C17—21no—
286C22×C17—21no—
287C22×C17—22no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—84no—
297C22×C17#44312yes—
298C22×C17C2×C4×D411yescharacteristic
299C4×C17—21no—
300C4×C17—21no—
301C4×C17—21no—
302C4×C17—21no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17—41no—
307C4×C17—41no—
308C4×C17C2×#13111yescharacteristic
309C4×C17#56811yescharacteristic
310C4×C17#57611yescharacteristic
311C4×C17#57711yescharacteristic
312#529C2×D1711yescharacteristic
313C2×C4×D4C2×D1711yescharacteristic
314#484Dic1711yescharacteristic
315#481Dic1711yescharacteristic
316#478Dic1711yescharacteristic
317#478Dic1711yescharacteristic
318#463Dic1711yescharacteristic
319#463Dic1711yescharacteristic
320#450Dic1711yescharacteristic
321#447Dic1711yescharacteristic
322#446—171no—
323C8×Q8—171no—
324C4×SD16C2×D1711yescharacteristic
325C4×SD16C2×D1711yescharacteristic
326C4×D8C2×D1711yescharacteristic
327C4×D8C2×D1711yescharacteristic
328#436—171no—
329#436—171no—
330#436—171no—
331#435—341no—
332#435—341no—
333#435—341no—
334C8×D4—171no—
335C8×D4—171no—
336C8×D4—171no—
337C8×D4—341no—
338#433—341no—
339#432—171no—
340#431—171no—
341#431C2×D1711yescharacteristic
342#424—341no—
343C2×C4⋊C8—171no—
344#407—341no—
345C2×C22⋊C8—341no—
346C2×C4×C8—171no—
347#335—171no—
348#335—171no—
349#334—171no—
350#334—171no—
351Q8⋊C8—171no—
352Q8⋊C8—171no—
353D4⋊C8—171no—
354D4⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8—171no—
358D4⋊C8—171no—
359#322—171no—
360#322—171no—
361C82—171no—
362C82—171no—
363C2×C17—22no—
364C2×C17—42no—
365C2×C17—81no—
366C2×C17—81no—
367C2×C17#293311yescharacteristic
368C2×C17#294611yescharacteristic
369C2×C17C2×#44311yescharacteristic
370C22×D4C2×Dic1711yescharacteristic
371C23×C4—21no—
372C2×SD16C2×Dic1711yescharacteristic
373C2×SD16C2×Dic1711yescharacteristic
374C2×D8C2×Dic1711yescharacteristic
375C2×D8C2×Dic1711yescharacteristic
376C2×M16—341no—
377C2×M16—341no—
378C2×M16—341no—
379C22×C8—341no—
380C22×C8—341no—
381C22×C8—341no—
382C22×C8—341no—
383C22×C8—341no—
384#126—21no—
385#125C2×Dic1711yescharacteristic
386#124C2×Dic1711yescharacteristic
387#123—21no—
388C22⋊Q8C2×Dic1711yescharacteristic
389#121C2×Dic1711yescharacteristic
390C4×Q8C22×D1711yescharacteristic
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4C2×Dic1711yescharacteristic
395C4×D4C2×Dic1711yescharacteristic
396C4×D4C22×D1711yescharacteristic
397C4×D4C22×D1711yescharacteristic
398C4×D4C22×D1711yescharacteristic
399#117—21no—
400C2×C4⋊C4C2×Dic1711yescharacteristic
401C2×C22⋊C4—21no—
402C2×C42C22×D1711yescharacteristic
403#107C2×Dic1711yescharacteristic
404#107C2×Dic1711yescharacteristic
405#106C2×Dic1711yescharacteristic
406#106C2×Dic1711yescharacteristic
407C4⋊C8—171no—
408C4⋊C8—171no—
409C4⋊C8—171no—
410C4⋊C8—171no—
411C4⋊C8—171no—
412C4⋊C8—171no—
413C4⋊C8—341no—
414C4⋊C8—341no—
415C4⋊C8—341no—
416C4⋊C8C2×Dic1711yescharacteristic
417C4⋊C8C2×Dic1711yescharacteristic
418#103C2×Dic1711yescharacteristic
419#103C2×Dic1711yescharacteristic
420#102C2×Dic1711yescharacteristic
421#102C2×Dic1711yescharacteristic
422#102C2×Dic1711yescharacteristic
423#102C2×Dic1711yescharacteristic
424#102C2×Dic1711yescharacteristic
425#102C2×Dic1711yescharacteristic
426C22⋊C8—171no—
427C22⋊C8—171no—
428C22⋊C8—341no—
429C22⋊C8—341no—
430C22⋊C8—341no—
431C22⋊C8—341no—
432C22⋊C8—341no—
433C22⋊C8—341no—
434C22⋊C8—341no—
435C22⋊C8C2×Dic1711yescharacteristic
436C22⋊C8C2×Dic1711yescharacteristic
437#97—341no—
438#97—341no—
439C4×C8—171no—
440C4×C8—171no—
441C4×C8—171no—
442C4×C8—171no—
443C4×C8—341no—
444C4×C8—341no—
445C4×C8C22×D1711yescharacteristic
446C4×C8C22×D1711yescharacteristic
447C17#1801411yescharacteristic, complemented, Sylow
448C24—21no—
449C2×Q8C22×Dic1711yescharacteristic
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—41no—
454C2×D4C22×Dic1711yescharacteristic
455C2×D4C22×Dic1711yescharacteristic
456C2×D4C22×Dic1711yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4D4×D1711yescharacteristic
466C22×C4#6325411yescharacteristic
467C22×C4C22×Dic1711yescharacteristic
468SD16—22no—
469SD16—41no—
470D8—22no—
471D8—41no—
472M16—1361no—
473M16—1361no—
474M16—1361no—
475C2×C8—171no—
476C2×C8—171no—
477C2×C8—171no—
478C2×C8—171no—
479C2×C8—344no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—1361no—
487C2×C8—1361no—
488C2×C8—1361no—
489C2×C8—1361no—
490C2×C8C22×Dic1711yescharacteristic
491C2×C8C22×Dic1711yescharacteristic
492C2×C8C22×Dic1711yescharacteristic
493C2×C8C22×Dic1711yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4C22×Dic1711yescharacteristic
500C4⋊C4C22×Dic1711yescharacteristic
501C4⋊C4C22×Dic1711yescharacteristic
502C4⋊C4C22×Dic1711yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C42—21no—
513C42D4×D1711yescharacteristic
514C42#6325411yescharacteristic
515C42C22×Dic1711yescharacteristic
516C42C23×D1711yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—42no—
521C23—42no—
522C23—81no—
523C23D4×Dic1711yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—42no—
532D4—42no—
533D4#1058763412yes—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4D4×Dic1711yescharacteristic
554C2×C4D4×Dic1711yescharacteristic
555C2×C4D4×Dic1711yescharacteristic
556C2×C4C2×D4×D1711yescharacteristic
557C2×C4C2×#6325411yescharacteristic
558C2×C4C23×Dic1711yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—342no—
564C8—342no—
565C8—341no—
566C8—341no—
567C8—341no—
568C8—341no—
569C8—681no—
570C8—681no—
571C8—681no—
572C8—681no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—84no—
585C22#1176292612yes—
586C22C2×D4×Dic1711yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4C2×#1058763411yescharacteristic
597C4#1176357611yescharacteristic
598C4#1176361311yescharacteristic
599C4#1176361911yescharacteristic
600C2—22no—
601C2—42no—
602C2—81no—
603C2—81no—
604C2#1453751111yescharacteristic
605C2#1453761711yescharacteristic
606C2C2×#1176292611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#3105C2
socleC22×C17C2×C4×D4
prime radicalC17×#447C4
cosocleC42×C17C24
hypercenter#3105D17
hypocenterC17#18014
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3105

Complements

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

NH ≅ G/Ntype
1. #14537944C2semidirect
2. #14537819C2semidirect
3. #14527675C2semidirect
4. #14527669C2semidirect
5. #14527663C2semidirect
6. #14527649C2semidirect
7. #14527649C2semidirect
8. #14527642C2semidirect
9. SD16×#3622C2semidirect
10. D8×#3622C2semidirect
11. #14527569C2semidirect
12. #14527244C2semidirect
13. C2×D4⋊#3622C2semidirect
14. #14527068C2semidirect
16. #11762940C22semidirect
17. Q8×#3622C22semidirect
20. #11762887C22semidirect
27. D4×#3622C22semidirect
36. #11762562C22semidirect
37. #11762562C22semidirect
38. #11762560C22semidirect
39. #11762560C22semidirect
40. Q8⋊#3622C22semidirect
41. Q8⋊#3622C22semidirect
46. D4⋊#3622C22semidirect
47. D4⋊#3622C22semidirect
48. #11762531C22semidirect
49. #11762531C22semidirect
50. C8×#3622C22semidirect
51. C8×#3622C22semidirect
447. C17#18014semidirect

Maximal subgroups

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

subgroupindex
#145379442
#145378192
#145276752
#145276692
#145276632
#145276492
#145276422
SD16×#36222
D8×#36222
#145275692
#145272442
C2×D4⋊#36222
#145270682
C17×#31052
#1801417