Finite Group Databasebeta

← C17×#13086 #16870830

Subgroups — C17×#13086 #16870830

Order 4352.

1118 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C17×#99C211yesmaximal, characteristic, complemented
2C17×#2403C211yesmaximal, characteristic
3C2×C17×AΣL(1,𝔽16)C212yesmaximal, complemented
4C17×#1783C211yesmaximal, characteristic, complemented
5C17×#1390C212yesmaximal, complemented
6C23×D4×C17C411yescharacteristic, complemented
7C23×D4×C17C2211yescharacteristic
8C2×C17×#120—21no—
9C2×C17×#120C411yescharacteristic, complemented
10C2×C17×#120C2211yescharacteristic
11C22×C22⋊C4×C17—21no—
12C2×C17×#100—21no—
13C2×C17×#100C2212yes—
14C2×C17×#99—24no—
15C2×C17×#99—21no—
16C2×C17×#99C2212yes—
17C17×#407—21no—
18C17×#392—21no—
19C17×#392—21no—
20C17×#386—21no—
21C17×#386C411yescharacteristic, complemented
22C17×#386C2211yescharacteristic
23C17×#379C411yescharacteristic, complemented
24C17×AΣL(1,𝔽16)—24no—
25C25×C17C2×C411yescharacteristic
26C25×C17C2×C411yescharacteristic, complemented
27C25×C17D411yescharacteristic
28C22×D4×C17—21no—
29C22×D4×C17—22no—
30C22×D4×C17—22no—
31C22×D4×C17—21no—
32C22×D4×C17—21no—
33C22×D4×C17—21no—
34C22×D4×C17—41no—
35C22×D4×C17—41no—
36C22×D4×C17—42no—
37C22×D4×C17—44no—
38C22×D4×C17—41no—
39C22×D4×C17—41no—
40C22×D4×C17C2×C411yescharacteristic
41C22×D4×C17D411yescharacteristic
42C22×D4×C17D411yescharacteristic
43C22×D4×C17C2311yescharacteristic
44C23×C4×C17—41no—
45C23×C4×C17C2×C411yescharacteristic, complemented
46C23×C4×C17D411yescharacteristic
47C2×M16×C17—22no—
48C17×#120—22no—
49C17×#120—22no—
50C17×#120—44no—
51C17×#120—42no—
52C2×C22⋊C4×C17—24no—
53C2×C22⋊C4×C17—22no—
54C2×C22⋊C4×C17—21no—
55C2×C22⋊C4×C17—21no—
56C2×C22⋊C4×C17—21no—
57C2×C22⋊C4×C17—21no—
58C2×C22⋊C4×C17—21no—
59C2×C22⋊C4×C17—21no—
60C2×C22⋊C4×C17—41no—
61C2×C22⋊C4×C17—42no—
62C2×C22⋊C4×C17C2×C411yescharacteristic, complemented
63C2×C22⋊C4×C17C2×C411yescharacteristic, complemented
64C17×#100—22no—
65C17×#100—22no—
66C17×#100—42no—
67C17×#99—24no—
68C17×#99—22no—
69C17×#99D414yescomplemented
70C17×C22⋊C8—41no—
71C17×C22⋊C8—41no—
72C17×#95—21no—
73C17×#95—21no—
74C24×C17—21no—
75C24×C17—22no—
76C24×C17—21no—
77C24×C17—21no—
78C24×C17—21no—
79C24×C17—21no—
80C24×C17—21no—
81C24×C17—21no—
82C24×C17—22no—
83C24×C17—22no—
84C24×C17—41no—
85C24×C17—42no—
86C24×C17—41no—
87C24×C17—42no—
88C24×C17—42no—
89C24×C17—41no—
90C24×C17—81no—
91C24×C17—81no—
92C24×C17—81no—
93C24×C17C22⋊C411yescharacteristic
94C24×C17C22⋊C411yescharacteristic
95C24×C17C22⋊C411yescharacteristic
96C24×C17C22×C411yescharacteristic
97C24×C17C2×D411yescharacteristic
98C2×D4×C17—24no—
99C2×D4×C17—22no—
100C2×D4×C17—22no—
101C2×D4×C17—22no—
102C2×D4×C17—21no—
103C2×D4×C17—21no—
104C2×D4×C17—22no—
105C2×D4×C17—41no—
106C2×D4×C17—41no—
107C2×D4×C17—41no—
108C2×D4×C17—41no—
109C2×D4×C17—41no—
110C2×D4×C17—41no—
111C2×D4×C17—42no—
112C2×D4×C17—44no—
113C2×D4×C17—44no—
114C2×D4×C17—44no—
115C2×D4×C17—44no—
116C2×D4×C17—42no—
117C2×D4×C17—42no—
118C2×D4×C17—42no—
119C2×D4×C17—44no—
120C2×D4×C17—44no—
121C2×D4×C17—44no—
122C2×D4×C17—41no—
123C2×D4×C17—41no—
124C2×D4×C17—44no—
125C2×D4×C17—42no—
126C2×D4×C17—42no—
127C2×D4×C17—42no—
128C2×D4×C17—41no—
129C2×D4×C17—41no—
130C2×D4×C17—81no—
131C2×D4×C17—81no—
132C2×D4×C17C2×D412yes—
133C2×D4×C17#4112yes—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—22no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—41no—
141C22×C4×C17—42no—
142C22×C4×C17—41no—
143C22×C4×C17—44no—
144C22×C4×C17—42no—
145C22×C4×C17—82no—
146C22×C4×C17—82no—
147C22×C4×C17C22⋊C411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149M16×C17—42no—
150M16×C17—42no—
151C2×C8×C17—42no—
152C22⋊C4×C17—22no—
153C22⋊C4×C17—24no—
154C22⋊C4×C17—42no—
155C22⋊C4×C17—42no—
156C22⋊C4×C17—42no—
157C22⋊C4×C17—44no—
158C22⋊C4×C17—42no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—42no—
161C22⋊C4×C17—41no—
162C22⋊C4×C17—81no—
163#13086C1711yesmaximal, characteristic, direct product factor, Sylow
164C23×C17—22no—
165C23×C17—22no—
166C23×C17—22no—
167C23×C17—22no—
168C23×C17—22no—
169C23×C17—21no—
170C23×C17—21no—
171C23×C17—21no—
172C23×C17—21no—
173C23×C17—21no—
174C23×C17—21no—
175C23×C17—21no—
176C23×C17—21no—
177C23×C17—22no—
178C23×C17—21no—
179C23×C17—21no—
180C23×C17—21no—
181C23×C17—41no—
182C23×C17—42no—
183C23×C17—42no—
184C23×C17—41no—
185C23×C17—41no—
186C23×C17—42no—
187C23×C17—41no—
188C23×C17—41no—
189C23×C17—41no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—42no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—42no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—81no—
205C23×C17—81no—
206C23×C17—81no—
207C23×C17—81no—
208C23×C17—82no—
209C23×C17—84no—
210C23×C17—82no—
211C23×C17—82no—
212C23×C17—84no—
213C23×C17—84no—
214C23×C17—84no—
215C23×C17—84no—
216C23×C17—82no—
217C23×C17#9911yescharacteristic
218C23×C17#9911yescharacteristic
219C23×C17#9912yes—
220C23×C17C2×C22⋊C411yescharacteristic
221C23×C17C4×D412yes—
222C23×C17#12011yescharacteristic
223C23×C17#12311yescharacteristic
224D4×C17—42no—
225D4×C17—42no—
226D4×C17—44no—
227D4×C17—44no—
228D4×C17—44no—
229D4×C17—42no—
230D4×C17—42no—
231D4×C17—48no—
232D4×C17—44no—
233D4×C17—84no—
234D4×C17—82no—
235D4×C17—82no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—22no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—41no—
245C2×C4×C17—42no—
246C2×C4×C17—41no—
247C2×C4×C17—42no—
248C2×C4×C17—44no—
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—88no—
256C2×C4×C17—82no—
257C2×C4×C17—82no—
258C2×C4×C17#12112yes—
259C8×C17—82no—
260C22×#99C2×C1711yescharacteristic, complemented
261#2403C2×C1711yescharacteristic
262C2×AΣL(1,𝔽16)C2×C1712yescomplemented
263#1783C2×C1711yescharacteristic, complemented
264#1390C2×C1712yescomplemented
265C22×C17—21no—
266C22×C17—21no—
267C22×C17—21no—
268C22×C17—21no—
269C22×C17—21no—
270C22×C17—21no—
271C22×C17—21no—
272C22×C17—21no—
273C22×C17—22no—
274C22×C17—21no—
275C22×C17—22no—
276C22×C17—42no—
277C22×C17—42no—
278C22×C17—42no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—42no—
282C22×C17—44no—
283C22×C17—44no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—42no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—81no—
298C22×C17—81no—
299C22×C17—84no—
300C22×C17—81no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—82no—
305C22×C17—82no—
306C22×C17—84no—
307C22×C17—84no—
308C22×C17—84no—
309C22×C17—81no—
310C22×C17—81no—
311C22×C17AΣL(1,𝔽16)12yescomplemented
312C22×C17#38611yescharacteristic
313C22×C17C2×#9911yescharacteristic
314C22×C17C2×#9911yescharacteristic
315C4×C17—22no—
316C4×C17—42no—
317C4×C17—41no—
318C4×C17—81no—
319C4×C17—81no—
320C4×C17—84no—
321C23×D4C4×C1711yescharacteristic, complemented
322C23×D4C22×C1711yescharacteristic
323C2×#120—21no—
324C2×#120C4×C1711yescharacteristic, complemented
325C2×#120C22×C1711yescharacteristic
326C22×C22⋊C4—21no—
327C2×#100—21no—
328C2×#100C22×C1712yes—
329C2×#99—21no—
330C2×#99—24no—
331C2×#99C22×C1712yes—
332#407—21no—
333#392—21no—
334#392—21no—
335#386—21no—
336#386C4×C1711yescharacteristic, complemented
337#386C22×C1711yescharacteristic
338#379C4×C1711yescharacteristic, complemented
339AΣL(1,𝔽16)—24no—
340C2×C17—21no—
341C2×C17—21no—
342C2×C17—22no—
343C2×C17—41no—
344C2×C17—41no—
345C2×C17—42no—
346C2×C17—41no—
347C2×C17—42no—
348C2×C17—82no—
349C2×C17—81no—
350C2×C17#189811yescharacteristic
351C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
352C2×C17#212111yescharacteristic
353C25C2×C4×C1711yescharacteristic
354C25C2×C4×C1711yescharacteristic, complemented
355C25D4×C1711yescharacteristic
356C22×D4—21no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4—41no—
363C22×D4—41no—
364C22×D4—42no—
365C22×D4—44no—
366C22×D4—41no—
367C22×D4—41no—
368C22×D4C2×C4×C1711yescharacteristic
369C22×D4D4×C1711yescharacteristic
370C22×D4D4×C1711yescharacteristic
371C22×D4C23×C1711yescharacteristic
372C23×C4—41no—
373C23×C4C2×C4×C1711yescharacteristic, complemented
374C23×C4D4×C1711yescharacteristic
375C2×M16—22no—
376#120—22no—
377#120—22no—
378#120—44no—
379#120—42no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—41no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4C2×C4×C1711yescharacteristic, complemented
391C2×C22⋊C4C2×C4×C1711yescharacteristic, complemented
392#100—22no—
393#100—22no—
394#100—42no—
395#99—24no—
396#99—22no—
397#99D4×C1714yescomplemented
398C22⋊C8—41no—
399C22⋊C8—41no—
400#95—21no—
401#95—21no—
402C17#1308611yescharacteristic, direct product factor, Sylow
403C24—21no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—22no—
413C24—41no—
414C24—42no—
415C24—41no—
416C24—42no—
417C24—42no—
418C24—41no—
419C24—81no—
420C24—81no—
421C24—81no—
422C24C22⋊C4×C1711yescharacteristic
423C24C22⋊C4×C1711yescharacteristic
424C24C22⋊C4×C1711yescharacteristic
425C24C22×C4×C1711yescharacteristic
426C24C2×D4×C1711yescharacteristic
427C2×D4—24no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—41no—
459C2×D4—81no—
460C2×D4—81no—
461C2×D4C2×D4×C1712yes—
462C2×D4#41×C1712yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4C22⋊C4×C1711yescharacteristic
477C22×C4C2×D4×C1711yescharacteristic
478M16—42no—
479M16—42no—
480C2×C8—42no—
481C22⋊C4—22no—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—41no—
491C22⋊C4—81no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—82no—
537C23—82no—
538C23—84no—
539C23—82no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—82no—
545C23C17×#9911yescharacteristic
546C23C17×#9911yescharacteristic
547C23C17×#9912yes—
548C23C2×C22⋊C4×C1711yescharacteristic
549C23C4×D4×C1712yes—
550C23C17×#12011yescharacteristic
551C23C17×#12311yescharacteristic
552D4—42no—
553D4—42no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—48no—
560D4—44no—
561D4—82no—
562D4—84no—
563D4—82no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—88no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4C17×#12112yes—
587C8—82no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—81no—
621C22—81no—
622C22—84no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—81no—
633C22—81no—
634C22C17×AΣL(1,𝔽16)12yescomplemented
635C22C17×#38611yescharacteristic
636C22C2×C17×#9911yescharacteristic
637C22C2×C17×#9911yescharacteristic
638C4—22no—
639C4—42no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—84no—
644C2—21no—
645C2—21no—
646C2—22no—
647C2—41no—
648C2—41no—
649C2—41no—
650C2—42no—
651C2—42no—
652C2—82no—
653C2—81no—
654C2C17×#189811yescharacteristic
655C2C2×C17×AΣL(1,𝔽16)11yescharacteristic
656C2C17×#212111yescharacteristic
subgroupG/H
center, socleC22×C17C2×#99
commutatorC24C22×C4×C17
Frattini, cosocleC22×D4C23×C17
prime radicalC17×#2403C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C24×C17 ◁ C23×D4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C17×#99C2semidirect
3. C2×C17×AΣL(1,𝔽16)C2semidirect
4. C17×#1783C2semidirect
5. C17×#1390C2semidirect
6. C23×D4×C17C4semidirect
9. C2×C17×#120C4semidirect
21. C17×#386C4semidirect
23. C17×#379C4semidirect
26. C25×C17C2×C4semidirect
45. C23×C4×C17C2×C4semidirect
62. C2×C22⋊C4×C17C2×C4semidirect
63. C2×C22⋊C4×C17C2×C4semidirect
69. C17×#99D4semidirect
163. #13086C17direct (N × H)
260. C22×#99C2×C17semidirect
262. C2×AΣL(1,𝔽16)C2×C17semidirect
263. #1783C2×C17semidirect
264. #1390C2×C17semidirect
311. C22×C17AΣL(1,𝔽16)semidirect
321. C23×D4C4×C17semidirect
324. C2×#120C4×C17semidirect
336. #386C4×C17semidirect
338. #379C4×C17semidirect
354. C25C2×C4×C17semidirect
373. C23×C4C2×C4×C17semidirect
390. C2×C22⋊C4C2×C4×C17semidirect
391. C2×C22⋊C4C2×C4×C17semidirect
397. #99D4×C17semidirect
402. C17#13086direct (N × H)
634. C22C17×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C22×C17×#992
C17×#24032
C2×C17×AΣL(1,𝔽16)2
C17×#17832
C17×#13902
#1308617