Finite Group Databasebeta

← C17×#18145 #16875889

Subgroups — C17×#18145 #16875889

Order 4352.

1382 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3460C211yesmaximal, characteristic, complemented
2C17×#3003C212yesmaximal, complemented
3C17×#2408C211yesmaximal, characteristic, complemented
4C17×#1600C214yesmaximal, complemented
5C17×#1599C212yesmaximal, complemented
6C17×#1598C212yesmaximal, complemented
7C17×#1513C212yesmaximal, complemented
8C17×#1465C211yesmaximal, characteristic, complemented
9C23×D4×C17C2211yescharacteristic
10C2×C17×AGL(1,ℤ/8)—22no—
11C17×#544C2211yescharacteristic, complemented
12C17×#542—22no—
13C17×#540—22no—
14C17×#538—21no—
15C17×#538C2212yescomplemented
16C17×#537C2211yescharacteristic, complemented
17C17×#535—21no—
18C17×#532—22no—
19C17×#529—22no—
20C17×#525—21no—
21C17×#525C2212yes—
22C2×C17×#121—21no—
23C2×C17×#120—21no—
24C4×#41×C17—21no—
25C2×C4×D4×C17C2211yescharacteristic
26C17×#450—22no—
27C17×#450C2214yes—
28C17×#447—22no—
29C17×#447C2214yes—
30C17×#418—22no—
31C17×#407C2212yes—
32C17×#394C2212yes—
33C17×#392C2212yes—
34C17×#390—21no—
35C17×#380C2211yescharacteristic
36C17×#379—21no—
37C17×#328C2214yescomplemented
38C17×#327C2214yescomplemented
39C17×#324C2212yescomplemented
40C17×#323C2212yescomplemented
41C25×C17—21no—
42C2×#41×C17—21no—
43C2×#41×C17—21no—
44C2×#41×C17D412yescomplemented
45C22×D4×C17—22no—
46C22×D4×C17—22no—
47C22×D4×C17—21no—
48C22×D4×C17—22no—
49C22×D4×C17—44no—
50C22×D4×C17D411yescharacteristic
51C22×D4×C17C2312yes—
52C23×C4×C17D411yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C17×AGL(1,ℤ/8)—48no—
55C2×SD16×C17—24no—
56C2×D8×C17—24no—
57C2×M16×C17—22no—
58C17×C4⋊Q8—22no—
59C17×#127—22no—
60C17×#126—22no—
61C17×#126—22no—
62C17×#125—22no—
63C17×#124—22no—
64C17×#124—22no—
65C17×#124—22no—
66C17×#123—22no—
67C17×#123—22no—
68C17×#123—22no—
69C17×#123—22no—
70C17×C22⋊Q8—21no—
71C17×C22⋊Q8—21no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8D412yescomplemented
74C17×C22⋊Q8C2312yes—
75C17×#121—21no—
76C17×#121—22no—
77C17×#121—22no—
78C17×#121—22no—
79C17×#121—22no—
80C17×#121—22no—
81C17×#121—21no—
82C17×#121—44no—
83C17×#121D412yescomplemented
84C17×#121C2312yes—
85C17×#120—22no—
86C17×#120—22no—
87C17×#120—44no—
88C4×Q8×C17—21no—
89C4×Q8×C17—21no—
90C4×D4×C17—21no—
91C4×D4×C17—21no—
92C4×D4×C17—22no—
93C4×D4×C17—22no—
94C4×D4×C17—22no—
95C4×D4×C17—22no—
96C4×D4×C17—22no—
97C4×D4×C17—22no—
98C17×#117—22no—
99C17×#117—21no—
100C17×#117—21no—
101C17×#117D412yescomplemented
102C2×C4⋊C4×C17—21no—
103C2×C4⋊C4×C17C2311yescharacteristic
104C2×C22⋊C4×C17—21no—
105C2×C22⋊C4×C17—22no—
106C2×C22⋊C4×C17—22no—
107C2×C22⋊C4×C17—22no—
108C2×C22⋊C4×C17D411yescharacteristic
109C2×C22⋊C4×C17C2311yescharacteristic
110C2×C42×C17—22no—
111C2×C42×C17D411yescharacteristic
112C17×#102—24no—
113C17×#102—24no—
114C17×C22⋊C8—22no—
115C17×C22⋊C8C2314yes—
116C17×#95C2312yes—
117C24×C17—22no—
118C24×C17—44no—
119C24×C17—42no—
120C24×C17—42no—
121C24×C17—42no—
122C24×C17—82no—
123C24×C17C2×D411yescharacteristic
124C24×C17C2×D411yescharacteristic
125#41×C17—42no—
126#41×C17—44no—
127#41×C17—42no—
128#41×C17—82no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17C2×D412yes—
132C2×D4×C17—28no—
133C2×D4×C17—21no—
134C2×D4×C17—24no—
135C2×D4×C17—22no—
136C2×D4×C17—22no—
137C2×D4×C17—22no—
138C2×D4×C17—22no—
139C2×D4×C17—21no—
140C2×D4×C17—22no—
141C2×D4×C17—22no—
142C2×D4×C17—22no—
143C2×D4×C17—24no—
144C2×D4×C17—41no—
145C2×D4×C17—42no—
146C2×D4×C17—44no—
147C2×D4×C17—44no—
148C2×D4×C17—48no—
149C2×D4×C17—44no—
150C2×D4×C17—44no—
151C2×D4×C17—41no—
152C2×D4×C17—41no—
153C2×D4×C17—82no—
154C2×D4×C17C2×D412yes—
155C22×C4×C17—22no—
156C22×C4×C17—22no—
157C22×C4×C17—21no—
158C22×C4×C17—22no—
159C22×C4×C17—21no—
160C22×C4×C17—22no—
161C22×C4×C17—42no—
162C22×C4×C17—42no—
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2×D412yes—
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×D412yes—
169C22×C4×C17C2411yescharacteristic
170SD16×C17—48no—
171D8×C17—48no—
172M16×C17—44no—
173C2×C8×C17—24no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—22no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—22no—
181C4⋊C4×C17—22no—
182C4⋊C4×C17—22no—
183C4⋊C4×C17—42no—
184C4⋊C4×C17C2×D412yes—
185C4⋊C4×C17C2×D412yes—
186C22⋊C4×C17—22no—
187C22⋊C4×C17—22no—
188C22⋊C4×C17—22no—
189C22⋊C4×C17—22no—
190C22⋊C4×C17—22no—
191C22⋊C4×C17—22no—
192C22⋊C4×C17—22no—
193C22⋊C4×C17—22no—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—22no—
196C22⋊C4×C17—22no—
197C22⋊C4×C17—42no—
198C22⋊C4×C17—44no—
199C22⋊C4×C17—44no—
200C22⋊C4×C17—82no—
201C42×C17—22no—
202C42×C17—22no—
203C42×C17—21no—
204C42×C17—22no—
205C42×C17—21no—
206#18145C1711yesmaximal, characteristic, direct product factor, Sylow
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—24no—
210C23×C17—21no—
211C23×C17—22no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—42no—
219C23×C17—48no—
220C23×C17—42no—
221C23×C17—42no—
222C23×C17—42no—
223C23×C17—42no—
224C23×C17—42no—
225C23×C17—84no—
226C23×C17—84no—
227C23×C17—88no—
228C23×C17—88no—
229C23×C17#12012yes—
230C23×C17C22×D411yescharacteristic
231Q8×C17—42no—
232Q8×C17—42no—
233Q8×C17—81no—
234D4×C17—28no—
235D4×C17—44no—
236D4×C17—42no—
237D4×C17—44no—
238D4×C17—44no—
239D4×C17—42no—
240D4×C17—42no—
241D4×C17—42no—
242D4×C17—44no—
243D4×C17—82no—
244D4×C17—82no—
245D4×C17—82no—
246D4×C17—81no—
247D4×C17—82no—
248D4×C17—88no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—22no—
252C2×C4×C17—24no—
253C2×C4×C17—21no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—21no—
258C2×C4×C17—42no—
259C2×C4×C17—41no—
260C2×C4×C17—44no—
261C2×C4×C17—42no—
262C2×C4×C17—44no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17—42no—
268C2×C4×C17—81no—
269C2×C4×C17—82no—
270C2×C4×C17#12011yescharacteristic
271C2×C4×C17#12011yescharacteristic
272C2×C4×C17AGL(1,ℤ/8)12yescomplemented
273C2×C4×C17C22×D412yes—
274C8×C17—44no—
275#3460C2×C1711yescharacteristic, complemented
276#3003C2×C1712yescomplemented
277#2408C2×C1711yescharacteristic, complemented
278#1600C2×C1714yescomplemented
279#1599C2×C1712yescomplemented
280#1598C2×C1712yescomplemented
281#1513C2×C1712yescomplemented
282#1465C2×C1711yescharacteristic, complemented
283C22×C17—21no—
284C22×C17—21no—
285C22×C17—22no—
286C22×C17—22no—
287C22×C17—22no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17—42no—
294C22×C17—41no—
295C22×C17—42no—
296C22×C17—41no—
297C22×C17—41no—
298C22×C17—42no—
299C22×C17—44no—
300C22×C17—81no—
301C22×C17—82no—
302C22×C17—82no—
303C22×C17—816no—
304C22×C17—82no—
305C22×C17—82no—
306C22×C17C2×#12011yescharacteristic
307C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C17#57511yescharacteristic
309C4×C17—22no—
310C4×C17—22no—
311C4×C17—21no—
312C4×C17—42no—
313C4×C17—41no—
314C4×C17—42no—
315C4×C17—41no—
316C4×C17—42no—
317C4×C17—82no—
318C4×C17B(4,𝔽2)12yes—
319C23×D4C22×C1711yescharacteristic
320C2×AGL(1,ℤ/8)—22no—
321#544C22×C1711yescharacteristic, complemented
322#542—22no—
323#540—22no—
324#538—21no—
325#538C22×C1712yescomplemented
326#537C22×C1711yescharacteristic, complemented
327#535—21no—
328#532—22no—
329#529—22no—
330#525—21no—
331#525C22×C1712yes—
332C2×#121—21no—
333C2×#120—21no—
334C4×#41—21no—
335C2×C4×D4C22×C1711yescharacteristic
336#450—22no—
337#450C22×C1714yes—
338#447—22no—
339#447C22×C1714yes—
340#418—22no—
341#407C22×C1712yes—
342#394C22×C1712yes—
343#392C22×C1712yes—
344#390—21no—
345#380C22×C1711yescharacteristic
346#379—21no—
347#328C22×C1714yescomplemented
348#327C22×C1714yescomplemented
349#324C22×C1712yescomplemented
350#323C22×C1712yescomplemented
351C2×C17—21no—
352C2×C17—21no—
353C2×C17—42no—
354C2×C17—82no—
355C2×C17—81no—
356C2×C17—84no—
357C2×C17#300611yescharacteristic
358C2×C17#302111yescharacteristic
359C2×C17C2×B(4,𝔽2)11yescharacteristic
360C25—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41D4×C1712yescomplemented
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—21no—
367C22×D4—22no—
368C22×D4—44no—
369C22×D4D4×C1711yescharacteristic
370C22×D4C23×C1712yes—
371C23×C4D4×C1711yescharacteristic
372C23×C4C23×C1711yescharacteristic
373AGL(1,ℤ/8)—48no—
374C2×SD16—24no—
375C2×D8—24no—
376C2×M16—22no—
377C4⋊Q8—22no—
378#127—22no—
379#126—22no—
380#126—22no—
381#125—22no—
382#124—22no—
383#124—22no—
384#124—22no—
385#123—22no—
386#123—22no—
387#123—22no—
388#123—22no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—22no—
392C22⋊Q8D4×C1712yescomplemented
393C22⋊Q8C23×C1712yes—
394#121—21no—
395#121—22no—
396#121—22no—
397#121—22no—
398#121—22no—
399#121—22no—
400#121—21no—
401#121—44no—
402#121D4×C1712yescomplemented
403#121C23×C1712yes—
404#120—22no—
405#120—22no—
406#120—44no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417#117—22no—
418#117—21no—
419#117—21no—
420#117D4×C1712yescomplemented
421C2×C4⋊C4—21no—
422C2×C4⋊C4C23×C1711yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4D4×C1711yescharacteristic
428C2×C22⋊C4C23×C1711yescharacteristic
429C2×C42—22no—
430C2×C42D4×C1711yescharacteristic
431#102—24no—
432#102—24no—
433C22⋊C8—22no—
434C22⋊C8C23×C1714yes—
435#95C23×C1712yes—
436C17#1814511yescharacteristic, direct product factor, Sylow
437C24—22no—
438C24—44no—
439C24—42no—
440C24—42no—
441C24—42no—
442C24—82no—
443C24C2×D4×C1711yescharacteristic
444C24C2×D4×C1711yescharacteristic
445#41—42no—
446#41—44no—
447#41—42no—
448#41—82no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×D4×C1712yes—
452C2×D4—28no—
453C2×D4—21no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—41no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—48no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—82no—
474C2×D4C2×D4×C1712yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4C2×D4×C1711yescharacteristic
484C22×C4C2×D4×C1712yes—
485C22×C4C2×D4×C1711yescharacteristic
486C22×C4C2×D4×C1711yescharacteristic
487C22×C4C2×D4×C1711yescharacteristic
488C22×C4C2×D4×C1712yes—
489C22×C4C24×C1711yescharacteristic
490SD16—48no—
491D8—48no—
492M16—44no—
493C2×C8—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4C2×D4×C1712yes—
505C4⋊C4C2×D4×C1712yes—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—22no—
525C42—21no—
526C23—21no—
527C23—21no—
528C23—24no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—48no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—84no—
545C23—84no—
546C23—88no—
547C23—88no—
548C23C17×#12012yes—
549C23C22×D4×C1711yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—81no—
553D4—28no—
554D4—44no—
555D4—42no—
556D4—44no—
557D4—44no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—44no—
562D4—82no—
563D4—82no—
564D4—81no—
565D4—82no—
566D4—82no—
567D4—88no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4C17×#12011yescharacteristic
590C2×C4C17×#12011yescharacteristic
591C2×C4C17×AGL(1,ℤ/8)12yescomplemented
592C2×C4C22×D4×C1712yes—
593C8—44no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—44no—
611C22—81no—
612C22—82no—
613C22—82no—
614C22—816no—
615C22—82no—
616C22—82no—
617C22C2×C17×#12011yescharacteristic
618C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
619C22C17×#57511yescharacteristic
620C4—22no—
621C4—22no—
622C4—21no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—82no—
629C4C17×B(4,𝔽2)12yes—
630C2—21no—
631C2—21no—
632C2—42no—
633C2—81no—
634C2—84no—
635C2—82no—
636C2C17×#300611yescharacteristic
637C2C17×#302111yescharacteristic
638C2C2×C17×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#3460C2semidirect
2. C17×#3003C2semidirect
3. C17×#2408C2semidirect
4. C17×#1600C2semidirect
5. C17×#1599C2semidirect
6. C17×#1598C2semidirect
7. C17×#1513C2semidirect
8. C17×#1465C2semidirect
11. C17×#544C22semidirect
15. C17×#538C22semidirect
16. C17×#537C22semidirect
37. C17×#328C22semidirect
38. C17×#327C22semidirect
39. C17×#324C22semidirect
40. C17×#323C22semidirect
44. C2×#41×C17D4semidirect
73. C17×C22⋊Q8D4semidirect
83. C17×#121D4semidirect
101. C17×#117D4semidirect
206. #18145C17direct (N × H)
272. C2×C4×C17AGL(1,ℤ/8)semidirect
275. #3460C2×C17semidirect
276. #3003C2×C17semidirect
277. #2408C2×C17semidirect
278. #1600C2×C17semidirect
279. #1599C2×C17semidirect
280. #1598C2×C17semidirect
281. #1513C2×C17semidirect
282. #1465C2×C17semidirect
321. #544C22×C17semidirect
325. #538C22×C17semidirect
326. #537C22×C17semidirect
347. #328C22×C17semidirect
348. #327C22×C17semidirect
349. #324C22×C17semidirect
350. #323C22×C17semidirect
363. C2×#41D4×C17semidirect
392. C22⋊Q8D4×C17semidirect
402. #121D4×C17semidirect
420. #117D4×C17semidirect
436. C17#18145direct (N × H)
591. C2×C4C17×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C17×#34602
C17×#30032
C17×#24082
C17×#16002
C17×#15992
C17×#15982
C17×#15132
C17×#14652
#1814517