Finite Group Databasebeta

← C17×#18206 #16875950

Subgroups — C17×#18206 #16875950

Order 4352.

1028 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3505C211yesmaximal, characteristic
2C2×C17×#483C211yesmaximal, characteristic, complemented
3C2×C17×#480C211yesmaximal, characteristic, complemented
4C17×#2486C211yesmaximal, characteristic, complemented
5C17×#1612C214yesmaximal, complemented
6C17×#1609C214yesmaximal, complemented
7C17×#1520C211yesmaximal, characteristic, complemented
8C17×#1519C211yesmaximal, characteristic, complemented
9C17×#1459C211yesmaximal, characteristic, complemented
10C17×#553—22no—
11C17×#553—21no—
12C17×#552—21no—
13C17×#551—22no—
14C17×#548—21no—
15C17×#546—21no—
16C17×#539C2212yes—
17C17×#538C2212yes—
18C17×#534—21no—
19C17×#529—21no—
20C17×#525—21no—
21C2×C17×#123—21no—
22C2×C17×C22⋊Q8C2211yescharacteristic
23C2×C17×#121C2211yescharacteristic
24C17×#518—21no—
25C2×C4×D4×C17C2211yescharacteristic
26C22×C4⋊C4×C17C2211yescharacteristic
27C17×#483—22no—
28C17×#483C2214yescomplemented
29C17×#480—22no—
30C17×#480C2214yes—
31C2×C17×#107—21no—
32C2×C17×#106—21no—
33C2×C17×#103—21no—
34C2×C17×#102—21no—
35C2×C17×C22⋊C8C2211yescharacteristic
36C2×C17×C22⋊C8C2211yescharacteristic
37C17×#398C2212yescomplemented
38C17×#397C2212yescomplemented
39C17×#389—21no—
40C17×#381—21no—
41C17×#380—21no—
42C17×#380C2211yescharacteristic
43C17×#328C2214yescomplemented
44C17×#327C2214yes—
45C17×#324C2212yescomplemented
46C17×#323C2212yescomplemented
47C2×#41×C17—21no—
48C22×Q8×C17D411yescharacteristic
49C22×D4×C17—21no—
50C22×D4×C17D411yescharacteristic
51C22×D4×C17D411yescharacteristic
52C23×C4×C17—21no—
53C23×C4×C17D411yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C22×C8×C17—21no—
56C22×C8×C17—21no—
57C17×#126—22no—
58C17×#126—22no—
59C17×#126—22no—
60C17×#125—21no—
61C17×#125—22no—
62C17×#124—21no—
63C17×#124—22no—
64C17×#123—22no—
65C17×#123—22no—
66C17×#123—22no—
67C17×#123—22no—
68C17×#123—22no—
69C17×#123—44no—
70C17×C22⋊Q8—21no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—21no—
73C17×C22⋊Q8—21no—
74C17×C22⋊Q8D412yes—
75C17×C22⋊Q8C2312yes—
76C17×#121—21no—
77C17×#121—22no—
78C17×#121—22no—
79C17×#121—22no—
80C17×#121—21no—
81C17×#121—21no—
82C17×#121D412yes—
83C17×#121C2312yes—
84C17×#120—22no—
85C17×#120—22no—
86C4×Q8×C17—21no—
87C4×D4×C17—22no—
88C4×D4×C17—22no—
89C4×D4×C17—22no—
90C4×D4×C17—21no—
91C4×D4×C17—21no—
92C4×D4×C17—21no—
93C4×D4×C17—21no—
94C4×D4×C17—21no—
95C4×D4×C17—22no—
96C4×D4×C17—22no—
97C17×#117—22no—
98C17×#117—21no—
99C17×#117—22no—
100C2×C4⋊C4×C17—21no—
101C2×C4⋊C4×C17—21no—
102C2×C4⋊C4×C17—21no—
103C2×C4⋊C4×C17—21no—
104C2×C4⋊C4×C17—21no—
105C2×C4⋊C4×C17—21no—
106C2×C4⋊C4×C17—22no—
107C2×C4⋊C4×C17—21no—
108C2×C4⋊C4×C17D411yescharacteristic
109C2×C4⋊C4×C17D411yescharacteristic
110C2×C4⋊C4×C17C2312yes—
111C2×C4⋊C4×C17C2311yescharacteristic
112C2×C22⋊C4×C17—22no—
113C2×C22⋊C4×C17—21no—
114C2×C22⋊C4×C17—21no—
115C2×C22⋊C4×C17—22no—
116C2×C22⋊C4×C17—21no—
117C2×C22⋊C4×C17D411yescharacteristic
118C2×C22⋊C4×C17C2311yescharacteristic
119C2×C42×C17D411yescharacteristic
120C17×#107—24no—
121C17×#106—24no—
122C17×#103—24no—
123C17×#102—24no—
124C17×C22⋊C8—21no—
125C17×C22⋊C8—21no—
126C17×C22⋊C8C2312yes—
127C17×C22⋊C8C2312yes—
128C17×#95—22no—
129C17×#95—22no—
130C17×#95C2312yes—
131C24×C17—21no—
132C24×C17C2×D411yescharacteristic
133C24×C17C2×D411yescharacteristic
134#41×C17—82no—
135C2×Q8×C17—21no—
136C2×Q8×C17—42no—
137C2×Q8×C17C2×D412yes—
138C2×D4×C17—21no—
139C2×D4×C17—22no—
140C2×D4×C17—21no—
141C2×D4×C17—22no—
142C2×D4×C17—22no—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—22no—
146C2×D4×C17—21no—
147C2×D4×C17—42no—
148C2×D4×C17—42no—
149C2×D4×C17—42no—
150C2×D4×C17C2×D412yes—
151C22×C4×C17—21no—
152C22×C4×C17—22no—
153C22×C4×C17—22no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—22no—
159C22×C4×C17—22no—
160C22×C4×C17—42no—
161C22×C4×C17—42no—
162C22×C4×C17—41no—
163C22×C4×C17—41no—
164C22×C4×C17—42no—
165C22×C4×C17—42no—
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2×D411yescharacteristic
171C22×C4×C17C2×D411yescharacteristic
172C22×C4×C17C2×D411yescharacteristic
173C22×C4×C17C2×D411yescharacteristic
174C22×C4×C17C2411yescharacteristic
175C2×C8×C17—22no—
176C2×C8×C17—22no—
177C2×C8×C17—42no—
178C2×C8×C17—42no—
179C4⋊C4×C17—22no—
180C4⋊C4×C17—22no—
181C4⋊C4×C17—22no—
182C4⋊C4×C17—22no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—22no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—22no—
190C4⋊C4×C17—22no—
191C4⋊C4×C17—44no—
192C4⋊C4×C17—42no—
193C4⋊C4×C17—41no—
194C4⋊C4×C17—41no—
195C4⋊C4×C17C2×D412yes—
196C4⋊C4×C17C2×D412yes—
197C22⋊C4×C17—22no—
198C22⋊C4×C17—22no—
199C22⋊C4×C17—22no—
200C22⋊C4×C17—22no—
201C22⋊C4×C17—22no—
202C22⋊C4×C17—22no—
203C22⋊C4×C17—22no—
204C22⋊C4×C17—21no—
205C22⋊C4×C17—21no—
206C22⋊C4×C17—22no—
207C22⋊C4×C17—22no—
208C22⋊C4×C17—22no—
209C22⋊C4×C17—21no—
210C22⋊C4×C17—21no—
211C22⋊C4×C17—44no—
212C22⋊C4×C17—42no—
213C42×C17—22no—
214C42×C17—22no—
215C42×C17—21no—
216C42×C17—21no—
217#18206C1711yesmaximal, characteristic, direct product factor, Sylow
218C23×C17—21no—
219C23×C17—22no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—42no—
227C23×C17—81no—
228C23×C17—81no—
229C23×C17#12011yescharacteristic
230C23×C17#12011yescharacteristic
231C23×C17#13512yes—
232C23×C17AGL(1,ℤ/8)12yes—
233C23×C17C22×D411yescharacteristic
234Q8×C17—42no—
235Q8×C17—42no—
236D4×C17—42no—
237D4×C17—42no—
238D4×C17—42no—
239D4×C17—42no—
240D4×C17—42no—
241D4×C17—44no—
242D4×C17—42no—
243D4×C17—82no—
244D4×C17—82no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—22no—
251C2×C4×C17—21no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17—42no—
268C2×C4×C17—42no—
269C2×C4×C17—81no—
270C2×C4×C17—81no—
271C2×C4×C17—81no—
272C2×C4×C17—82no—
273C2×C4×C17—82no—
274C2×C4×C17—82no—
275C2×C4×C17—82no—
276C2×C4×C17#12011yescharacteristic
277C2×C4×C17#12011yescharacteristic
278C2×C4×C17C22×D411yescharacteristic
279C2×C4×C17C22×D411yescharacteristic
280C8×C17—42no—
281C8×C17—42no—
282#3505C2×C1711yescharacteristic
283C2×#483C2×C1711yescharacteristic, complemented
284C2×#480C2×C1711yescharacteristic, complemented
285#2486C2×C1711yescharacteristic, complemented
286#1612C2×C1714yescomplemented
287#1609C2×C1714yescomplemented
288#1520C2×C1711yescharacteristic, complemented
289#1519C2×C1711yescharacteristic, complemented
290#1459C2×C1711yescharacteristic, complemented
291C22×C17—21no—
292C22×C17—21no—
293C22×C17—21no—
294C22×C17—21no—
295C22×C17—21no—
296C22×C17—21no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—41no—
301C22×C17—42no—
302C22×C17—42no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17—42no—
306C22×C17—81no—
307C22×C17—162no—
308C22×C17#44912yes—
309C22×C17#44912yes—
310C22×C17C2×#12011yescharacteristic
311C22×C17C2×#13511yescharacteristic
312C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
313C4×C17—21no—
314C4×C17—21no—
315C4×C17—21no—
316C4×C17—41no—
317C4×C17—42no—
318C4×C17—41no—
319C4×C17—42no—
320C4×C17—41no—
321C4×C17—42no—
322C4×C17—82no—
323C4×C17—82no—
324C4×C17#45812yes—
325#553—22no—
326#553—21no—
327#552—21no—
328#551—22no—
329#548—21no—
330#546—21no—
331#539C22×C1712yes—
332#538C22×C1712yes—
333#534—21no—
334#529—21no—
335#525—21no—
336C2×#123—21no—
337C2×C22⋊Q8C22×C1711yescharacteristic
338C2×#121C22×C1711yescharacteristic
339#518—21no—
340C2×C4×D4C22×C1711yescharacteristic
341C22×C4⋊C4C22×C1711yescharacteristic
342#483—22no—
343#483C22×C1714yescomplemented
344#480—22no—
345#480C22×C1714yes—
346C2×#107—21no—
347C2×#106—21no—
348C2×#103—21no—
349C2×#102—21no—
350C2×C22⋊C8C22×C1711yescharacteristic
351C2×C22⋊C8C22×C1711yescharacteristic
352#398C22×C1712yescomplemented
353#397C22×C1712yescomplemented
354#389—21no—
355#381—21no—
356#380—21no—
357#380C22×C1711yescharacteristic
358#328C22×C1714yescomplemented
359#327C22×C1714yes—
360#324C22×C1712yescomplemented
361#323C22×C1712yescomplemented
362C2×C17—21no—
363C2×C17—21no—
364C2×C17—22no—
365C2×C17—22no—
366C2×C17—81no—
367C2×C17—82no—
368C2×C17C2×#44911yescharacteristic
369C2×C17#302211yescharacteristic
370C2×C17C2×#45811yescharacteristic
371C2×#41—21no—
372C22×Q8D4×C1711yescharacteristic
373C22×D4—21no—
374C22×D4D4×C1711yescharacteristic
375C22×D4D4×C1711yescharacteristic
376C23×C4—21no—
377C23×C4D4×C1711yescharacteristic
378C23×C4C23×C1711yescharacteristic
379C22×C8—21no—
380C22×C8—21no—
381#126—22no—
382#126—22no—
383#126—22no—
384#125—21no—
385#125—22no—
386#124—21no—
387#124—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—44no—
394C22⋊Q8—21no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398C22⋊Q8D4×C1712yes—
399C22⋊Q8C23×C1712yes—
400#121—21no—
401#121—22no—
402#121—22no—
403#121—22no—
404#121—21no—
405#121—21no—
406#121D4×C1712yes—
407#121C23×C1712yes—
408#120—22no—
409#120—22no—
410C4×Q8—21no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—22no—
420C4×D4—22no—
421#117—22no—
422#117—21no—
423#117—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4D4×C1711yescharacteristic
433C2×C4⋊C4D4×C1711yescharacteristic
434C2×C4⋊C4C23×C1712yes—
435C2×C4⋊C4C23×C1711yescharacteristic
436C2×C22⋊C4—22no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4D4×C1711yescharacteristic
442C2×C22⋊C4C23×C1711yescharacteristic
443C2×C42D4×C1711yescharacteristic
444#107—24no—
445#106—24no—
446#103—24no—
447#102—24no—
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8C23×C1712yes—
451C22⋊C8C23×C1712yes—
452#95—22no—
453#95—22no—
454#95C23×C1712yes—
455C17#1820611yescharacteristic, direct product factor, Sylow
456C24—21no—
457C24C2×D4×C1711yescharacteristic
458C24C2×D4×C1711yescharacteristic
459#41—82no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8C2×D4×C1712yes—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×D4×C1712yes—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4C2×D4×C1711yescharacteristic
492C22×C4C2×D4×C1711yescharacteristic
493C22×C4C2×D4×C1711yescharacteristic
494C22×C4C2×D4×C1711yescharacteristic
495C22×C4C2×D4×C1711yescharacteristic
496C22×C4C2×D4×C1711yescharacteristic
497C22×C4C2×D4×C1711yescharacteristic
498C22×C4C2×D4×C1711yescharacteristic
499C22×C4C24×C1711yescharacteristic
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—42no—
503C2×C8—42no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4C2×D4×C1712yes—
521C4⋊C4C2×D4×C1712yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—21no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23C17×#12011yescharacteristic
554C23C17×#12011yescharacteristic
555C23C17×#13512yes—
556C23C17×AGL(1,ℤ/8)12yes—
557C23C22×D4×C1711yescharacteristic
558Q8—42no—
559Q8—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—44no—
566D4—42no—
567D4—82no—
568D4—82no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4C17×#12011yescharacteristic
601C2×C4C17×#12011yescharacteristic
602C2×C4C22×D4×C1711yescharacteristic
603C2×C4C22×D4×C1711yescharacteristic
604C8—42no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—81no—
622C22—162no—
623C22C17×#44912yes—
624C22C17×#44912yes—
625C22C2×C17×#12011yescharacteristic
626C22C2×C17×#13511yescharacteristic
627C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4C17×#45812yes—
640C2—21no—
641C2—21no—
642C2—22no—
643C2—22no—
644C2—81no—
645C2—82no—
646C2C2×C17×#44911yescharacteristic
647C2C17×#302211yescharacteristic
648C2C2×C17×#45811yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#3505C2
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
2. C2×C17×#483C2semidirect
3. C2×C17×#480C2semidirect
4. C17×#2486C2semidirect
5. C17×#1612C2semidirect
6. C17×#1609C2semidirect
7. C17×#1520C2semidirect
8. C17×#1519C2semidirect
9. C17×#1459C2semidirect
28. C17×#483C22semidirect
37. C17×#398C22semidirect
38. C17×#397C22semidirect
43. C17×#328C22semidirect
45. C17×#324C22semidirect
46. C17×#323C22semidirect
217. #18206C17direct (N × H)
283. C2×#483C2×C17semidirect
284. C2×#480C2×C17semidirect
285. #2486C2×C17semidirect
286. #1612C2×C17semidirect
287. #1609C2×C17semidirect
288. #1520C2×C17semidirect
289. #1519C2×C17semidirect
290. #1459C2×C17semidirect
343. #483C22×C17semidirect
352. #398C22×C17semidirect
353. #397C22×C17semidirect
358. #328C22×C17semidirect
360. #324C22×C17semidirect
361. #323C22×C17semidirect
455. C17#18206direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#35052
C2×C17×#4832
C2×C17×#4802
C17×#24862
C17×#16122
C17×#16092
C17×#15202
C17×#15192
C17×#14592
#1820617