Finite Group Databasebeta

← #17091452

Subgroups — #17091452

Order 4352.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#103×D17C211yesmaximal, characteristic
2#14532482C211yesmaximal, characteristic
3Q8×#63227C211yesmaximal, characteristic
4#14530324C211yesmaximal, characteristic, complemented
5#14530164C211yesmaximal, characteristic, complemented
6#14530160C211yesmaximal, characteristic, complemented
7#14529922C211yesmaximal, characteristic, complemented
8#14529054C211yesmaximal, characteristic, complemented
9#14529052C211yesmaximal, characteristic, complemented
10#14528975C211yesmaximal, characteristic
11#14528960C211yesmaximal, characteristic
12#14528806C211yesmaximal, characteristic
13#14528787C211yesmaximal, characteristic, complemented
14#14528717C211yesmaximal, characteristic
15C2×C17×#475C211yesmaximal, characteristic, complemented
16C22×Q8×D17C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C2×#10587632—21no—
19C2×Q8×Dic17C2211yescharacteristic
20C2×C4⋊C4×D17C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30#103×D17—24no—
31C2×#10587491C2211yescharacteristic
32#11762742—21no—
33#11762728—21no—
34#11762715—21no—
35#11762715C2211yescharacteristic
36#11762714—21no—
37#11762714C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762680—21no—
41#11762678C2211yescharacteristic
42#11762666—21no—
43#11762666C2211yescharacteristic
44#11762666C2211yescharacteristic
45C4×#63227—21no—
46C4×#63227C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762564C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×C4⋊Q8C2211yescharacteristic
54C2×C4×Q8×C17C2211yescharacteristic
55C17×#475—24no—
56C2×C17×#106C2211yescharacteristic
57C2×C17×#106C2211yescharacteristic
58C2×C17×C4⋊C8C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C2×C17×#103C2211yescharacteristic
61C22×Q8×C17—21no—
62C22×Q8×C17C2311yescharacteristic
63C2×Q8×D17—22no—
64C2×Q8×D17—22no—
65C2×Q8×D17—22no—
66C2×Q8×D17—41no—
67C2×Q8×D17—41no—
68C2×Q8×D17Q814yescomplemented
69C22×C4×D17—21no—
70C22×C4×D17—21no—
71C22×C4×D17C2311yescharacteristic
72C22×Dic34—21no—
73C22×Dic34C2311yescharacteristic
74C22×C8×C17C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C17×C4⋊Q8—22no—
77C17×C4⋊Q8—41no—
78C4×Q8×C17—22no—
79C4×Q8×C17—41no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87#10587632—44no—
88Q8×Dic17—22no—
89Q8×Dic17—41no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×#63227D411yescharacteristic
93C2×#63227D411yescharacteristic
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C2×C8×D17—22no—
108C2×C8×D17—22no—
109C2×C8×D17—41no—
110C4⋊C4×D17—22no—
111C4⋊C4×D17—41no—
112C17×#106—22no—
113C17×#106—22no—
114C17×C4⋊C8—22no—
115C17×#103—22no—
116C17×#103D414yes—
117#10587506—22no—
118#10587506—22no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502D411yescharacteristic
128#10587502D411yescharacteristic
129#10587491—22no—
130#10587488—41no—
131#10587487—22no—
132#10587487—41no—
133#10587485—41no—
134#10587481—22no—
135#10587481—41no—
136#10587479—41no—
137C23×D17C2×D411yescharacteristic
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—42no—
142C2×Q8×C17—42no—
143C2×Q8×C17—81no—
144C2×Q8×C17C2×D412yes—
145C2×Q8×C17C2×Q812yes—
146C2×Q8×C17#4112yes—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2411yescharacteristic
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161Q8×D17—28no—
162Q8×D17—44no—
163Q8×D17—44no—
164Q8×D17—44no—
165C2×C4×D17—21no—
166C2×C4×D17—21no—
167C2×C4×D17—21no—
168C2×C4×D17—21no—
169C2×C4×D17—24no—
170C2×C4×D17—42no—
171C2×C4×D17—42no—
172C2×C4×D17—42no—
173C2×C4×D17—42no—
174C2×C4×D17—81no—
175C2×C4×D17C2×Q812yes—
176C2×C4×D17#4112yes—
177C2×Dic34—21no—
178C2×Dic34—21no—
179C2×Dic34—21no—
180C2×Dic34—21no—
181C2×Dic34—21no—
182C2×Dic34—24no—
183C2×Dic34—42no—
184C2×Dic34—42no—
185C2×Dic34C2×Q812yes—
186C2×C8×C17—21no—
187C2×C8×C17—21no—
188C2×C8×C17—21no—
189C2×C8×C17—21no—
190C2×C8×C17—21no—
191C2×C8×C17C2×D412yes—
192C4⋊C4×C17—21no—
193C4⋊C4×C17—21no—
194C4⋊C4×C17—21no—
195C4⋊C4×C17—21no—
196C4⋊C4×C17—21no—
197C4⋊C4×C17—42no—
198C4⋊C4×C17—42no—
199C4⋊C4×C17C2×D412yes—
200C42×C17—21no—
201C42×C17—21no—
202C42×C17—42no—
203#63227—22no—
204#63227—42no—
205#63227—42no—
206#63227SD1614yes—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—42no—
214C4⋊Dic17—42no—
215#63225—42no—
216#63225—42no—
217#63225—42no—
218#63225—81no—
219C4×Dic17—21no—
220C4×Dic17—21no—
221C4×Dic17—42no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228C8×D17—44no—
229#25955—171nomaximal, Sylow
230C23×C17C22×D411yescharacteristic
231C22×D17—21no—
232C22×D17—21no—
233C22×D17—21no—
234C22×D17—21no—
235C22×D17C22⋊Q812yes—
236C22×D17C2×SD1612yes—
237C22×D17#13712yes—
238Q8×C17—22no—
239Q8×C17—22no—
240Q8×C17—22no—
241Q8×C17—84no—
242Q8×C17C22⋊Q814yescomplemented
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17C2×SD1612yes—
261C2×C4×C17C22×D411yescharacteristic
262C2×C4×C17C22×Q811yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×C4×C17H(5,𝔽2)11yescharacteristic
266C2×C4×C17H(5,𝔽2)11yescharacteristic
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—22no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
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×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17C22⋊Q812yes—
287C2×Dic17C2×SD1612yes—
288C4×D17—22no—
289C4×D17—22no—
290C4×D17—22no—
291C4×D17—22no—
292C4×D17—48no—
293C4×D17—84no—
294Dic34—22no—
295Dic34—22no—
296Dic34—48no—
297Dic34—41no—
298Dic34—41no—
299C8×C17—22no—
300C8×C17—41no—
301#3622—41no—
302#3622—41no—
303C2×#475D1711yescharacteristic, complemented
304C22×#103—171no—
305#2440—171no—
306Q8×C22⋊C4—171no—
307#2096—171no—
308#2066—171no—
309#2066—171no—
310#2024—171no—
311#1865—171no—
312#1865—171no—
313#1853—171no—
314#1850—171no—
315#1807—171no—
316#1803—171no—
317#1787—171no—
318C22×C17C2×C22⋊Q811yescharacteristic
319C22×C17D4211yescharacteristic
320C22×C17#54611yescharacteristic
321C22×C17C22×SD1611yescharacteristic
322C22×C17C2×#13711yescharacteristic
323C22×C17#57711yescharacteristic
324C22×C17#57811yescharacteristic
325C2×D17—22no—
326C2×D17—22no—
327C2×D17—22no—
328C2×D17—22no—
329C2×D17—22no—
330C2×D17—22no—
331C2×D17#47514yescomplemented
332C4×C17—24no—
333C4×C17—22no—
334C4×C17—41no—
335C4×C17—42no—
336C4×C17—81no—
337C4×C17—81no—
338C4×C17C2×C22⋊Q811yescharacteristic
339C4×C17#54311yescharacteristic
340C4×C17#54611yescharacteristic
341C4×C17#55111yescharacteristic
342Dic17—22no—
343Dic17—22no—
344Dic17—42no—
345Dic17—41no—
346Dic17—81no—
347Dic17—81no—
348Dic17—81no—
349C23×Q8—171no—
350C23×C8—171no—
351C2×C4⋊Q8C2×D1711yescharacteristic
352C2×C22⋊Q8—341no—
353C2×C4×Q8—171no—
354C2×C4×Q8C2×D1711yescharacteristic
355C22×C4⋊C4—171no—
356#475—24no—
357C2×#106—171no—
358C2×#106—171no—
359C2×#106C2×D1711yescharacteristic
360C2×#106C2×D1711yescharacteristic
361C2×C4⋊C8—171no—
362C2×C4⋊C8C2×D1711yescharacteristic
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#103—344no—
370C2×#103C2×D1711yescharacteristic
371C2×#103C2×D1711yescharacteristic
372C2×C22⋊C8—171no—
373#398—341no—
374#395—341no—
375#391—171no—
376#391—171no—
377#391—341no—
378#391—341no—
379#389—171no—
380#389—171no—
381#385—341no—
382#384—171no—
383#380—171no—
384#380—171no—
385#380—341no—
386C4×C22⋊C4—171no—
387C4×C22⋊C4—341no—
388#340—171no—
389#340—171no—
390#340—171no—
391#340—171no—
392#336—171no—
393#336—171no—
394C2×C17#243711yescharacteristic
395C2×C17C2×#47511yescharacteristic
396C2×C17#308211yescharacteristic
397C2×C17#327011yescharacteristic
398C2×C17D4×SD1611yescharacteristic
399C2×C17#329811yescharacteristic
400C2×C17#331611yescharacteristic
401D17—24no—
402C22×Q8—21no—
403C22×Q8—174no—
404C22×Q8—171no—
405C22×Q8—341no—
406C22×Q8—342no—
407C22×Q8—342no—
408C22×Q8—342no—
409C22×Q8—681no—
410C22×Q8—681no—
411C22×Q8C22×D1711yescharacteristic
412C23×C4—171no—
413C23×C4—341no—
414C23×C4—341no—
415C22×C8—171no—
416C22×C8—171no—
417C22×C8—342no—
418C22×C8—342no—
419C22×C8—681no—
420C22×C8C22×D1711yescharacteristic
421C22×C8C22×D1711yescharacteristic
422C4⋊Q8—22no—
423C4⋊Q8—41no—
424C22⋊Q8—684no—
425C4×Q8—22no—
426C4×Q8—41no—
427C4×Q8—342no—
428C4×Q8—681no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—171no—
434C2×C4⋊C4—342no—
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—681no—
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C4⋊C4C22×D1711yescharacteristic
445C2×C22⋊C4—171no—
446C2×C22⋊C4—171no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C42—21no—
450C2×C42—171no—
451C2×C42—341no—
452C2×C42C22×D1711yescharacteristic
453#106—22no—
454#106—22no—
455#106—681no—
456#106—681no—
457C4⋊C8—22no—
458C4⋊C8—681no—
459#103—22no—
460#103—342no—
461#103—342no—
462#103—342no—
463#103—342no—
464#103—681no—
465#103—681no—
466#103C22⋊D1714yes—
467C22⋊C8—342no—
468#95—171no—
469#95—171no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478C17#2595511yescharacteristic, complemented, Sylow
479C24—171no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—81no—
486C2×Q8—172no—
487C2×Q8—344no—
488C2×Q8—341no—
489C2×Q8—348no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×Q8—341no—
494C2×Q8—682no—
495C2×Q8—682no—
496C2×Q8—684no—
497C2×Q8—684no—
498C2×Q8—684no—
499C2×Q8Q8×D1712yes—
500C2×Q8#6325612yes—
501C2×Q8C2×C22⋊D1712yes—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—171no—
507C22×C4—171no—
508C22×C4—172no—
509C22×C4—172no—
510C22×C4—171no—
511C22×C4—344no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—341no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—341no—
518C22×C4—341no—
519C22×C4—341no—
520C22×C4—682no—
521C22×C4—682no—
522C22×C4—682no—
523C22×C4—682no—
524C22×C4—1361no—
525C22×C4D4×D1711yescharacteristic
526C22×C4D4×D1711yescharacteristic
527C22×C4C23×D1711yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—684no—
540C2×C8C2×C22⋊D1712yes—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—341no—
549C4⋊C4—341no—
550C4⋊C4—341no—
551C4⋊C4—341no—
552C4⋊C4—341no—
553C4⋊C4—341no—
554C4⋊C4—682no—
555C4⋊C4—682no—
556C4⋊C4—682no—
557C4⋊C4—682no—
558C4⋊C4—682no—
559C4⋊C4—1361no—
560C4⋊C4C2×C22⋊D1712yes—
561C22⋊C4—174no—
562C22⋊C4—342no—
563C22⋊C4—682no—
564C22⋊C4—682no—
565C42—21no—
566C42—21no—
567C42—42no—
568C42—341no—
569C42—341no—
570C42—682no—
571C23—172no—
572C23—172no—
573C23—172no—
574C23—341no—
575C23—341no—
576C23—341no—
577C23—341no—
578C23C2×D4×D1711yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—84no—
583Q8—342no—
584Q8—342no—
585Q8—688no—
586Q8—681no—
587Q8—681no—
588Q8#1058763214yescomplemented
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—172no—
607C2×C4—172no—
608C2×C4—341no—
609C2×C4—341no—
610C2×C4—341no—
611C2×C4—341no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—341no—
618C2×C4—341no—
619C2×C4—341no—
620C2×C4—688no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
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—1364no—
633C2×C4SD16×D1712yes—
634C2×C4#1058769111yescharacteristic
635C2×C4C2×Q8×D1711yescharacteristic
636C2×C4C2×#6325611yescharacteristic
637C2×C4#41×D1711yescharacteristic
638C2×C4#1058769611yescharacteristic
639C2×C4C22×C22⋊D1711yescharacteristic
640C8—22no—
641C8—41no—
642C8—681no—
643C8—681no—
644C22—174no—
645C22—342no—
646C22—342no—
647C22—342no—
648C22—342no—
649C22—342no—
650C22—342no—
651C22C22⋊Q8×D1711yescharacteristic
652C22D4×C22⋊D1711yescharacteristic
653C22#1176345011yescharacteristic
654C22C2×SD16×D1711yescharacteristic
655C22#137×D1711yescharacteristic
656C22#1176361511yescharacteristic
657C22#1176362511yescharacteristic
658C4—24no—
659C4—22no—
660C4—41no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—342no—
665C4—342no—
666C4—681no—
667C4—682no—
668C4—1361no—
669C4—1361no—
670C4—1361no—
671C4C2×#1058763211yescharacteristic
672C4#1176343411yescharacteristic
673C4#1176344711yescharacteristic
674C4#1176348611yescharacteristic
675C2—344no—
676C2#1453244711yescharacteristic
677C2D17×#47511yescharacteristic
678C2#1453861911yescharacteristic
679C2#1454034811yescharacteristic
680C2SD16×C22⋊D1711yescharacteristic
681C2#1454064611yescharacteristic
682C2#1454089211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#475C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenterC2×#475D17
hypocenterC17#25955
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
4. #14530324C2semidirect
5. #14530164C2semidirect
6. #14530160C2semidirect
7. #14529922C2semidirect
8. #14529054C2semidirect
9. #14529052C2semidirect
13. #14528787C2semidirect
15. C2×C17×#475C2semidirect
68. C2×Q8×D17Q8semidirect
242. Q8×C17C22⋊Q8semidirect
303. C2×#475D17semidirect
331. C2×D17#475semidirect
478. C17#25955semidirect
588. Q8#10587632semidirect

Maximal subgroups

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

subgroupindex
C2×#103×D172
#145324822
Q8×#632272
#145303242
#145301642
#145301602
#145299222
#145290542
#145290522
#145289752
#145289602
#145288062
#145287872
#145287172
C2×C17×#4752
#2595517