Finite Group Databasebeta

← C17×#14506 #16872250

Subgroups — C17×#14506 #16872250

Order 4352.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C17×#123C211yesmaximal, characteristic
2C2×C17×#388C212yesmaximal, complemented
3C17×#1445C212yesmaximal, complemented
4C17×#1439C212yesmaximal, complemented
5C17×#1438C211yesmaximal, characteristic
6C17×#1438C211yesmaximal, characteristic
7C17×#1437C212yesmaximal, complemented
8C17×C4⋊#95C212yesmaximal, complemented
9C17×C22⋊#95C212yesmaximal, complemented
10C23×D4×C17C2211yescharacteristic
11C24×C4×C17C2211yescharacteristic
12C2×C17×#123—28no—
13C2×C17×#123C418yescomplemented
14C22×C4⋊C4×C17C2212yes—
15C22×C4⋊C4×C17C2212yes—
16C22×C22⋊C4×C17C2212yes—
17C22×C22⋊C4×C17C2212yes—
18C22×C22⋊C4×C17C2211yescharacteristic
19C22×C22⋊C4×C17C2212yes—
20C22×C22⋊C4×C17C2212yes—
21C22×C22⋊C4×C17C2211yescharacteristic
22C22×C22⋊C4×C17C2211yescharacteristic
23C22×C42×C17C2212yes—
24C17×#388—28no—
25C2×C17×#95C2212yes—
26C2×C17×#95C2212yes—
27C2×C17×#95C2212yes—
28C2×C17×#95C2212yes—
29C2×C17×#95C2212yes—
30C2×C17×#95C2212yes—
31C2×C17×#95C2211yescharacteristic
32C2×C17×#95C2212yes—
33C2×C17×#95C2211yescharacteristic
34C25×C17C2311yescharacteristic
35C25×C17C2311yescharacteristic
36C22×D4×C17—24no—
37C22×D4×C17—42no—
38C22×D4×C17—41no—
39C22×D4×C17—41no—
40C22×D4×C17C2×C414yes—
41C23×C4×C17—24no—
42C23×C4×C17—42no—
43C23×C4×C17—41no—
44C23×C4×C17—41no—
45C23×C4×C17C2×C414yes—
46C23×C4×C17C2312yes—
47C23×C4×C17C2312yes—
48C23×C4×C17C2312yes—
49C23×C4×C17C2311yescharacteristic
50C23×C4×C17C2312yes—
51C23×C4×C17C2312yes—
52C23×C4×C17C2312yes—
53C17×#123—232no—
54C2×C4⋊C4×C17—24no—
55C2×C4⋊C4×C17—24no—
56C2×C4⋊C4×C17—24no—
57C2×C4⋊C4×C17—24no—
58C2×C4⋊C4×C17—42no—
59C2×C4⋊C4×C17C2×C418yes—
60C2×C22⋊C4×C17—24no—
61C2×C22⋊C4×C17—24no—
62C2×C22⋊C4×C17—24no—
63C2×C22⋊C4×C17—24no—
64C2×C22⋊C4×C17—24no—
65C2×C22⋊C4×C17—24no—
66C2×C22⋊C4×C17—24no—
67C2×C22⋊C4×C17—24no—
68C2×C22⋊C4×C17—41no—
69C2×C22⋊C4×C17—42no—
70C2×C22⋊C4×C17—42no—
71C2×C22⋊C4×C17—42no—
72C2×C22⋊C4×C17—42no—
73C2×C22⋊C4×C17—41no—
74C2×C22⋊C4×C17—42no—
75C2×C22⋊C4×C17C2×C418yes—
76C2×C22⋊C4×C17C2×C414yes—
77C2×C22⋊C4×C17D418yes—
78C2×C42×C17—24no—
79C2×C42×C17—24no—
80C2×C42×C17—42no—
81C17×#95—24no—
82C17×#95—24no—
83C17×#95—42no—
84C17×#95—42no—
85C17×#95—42no—
86C17×#95—42no—
87C17×#95—41no—
88C17×#95—42no—
89C17×#95—41no—
90C24×C17—22no—
91C24×C17—22no—
92C24×C17—22no—
93C24×C17—21no—
94C24×C17—21no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17—22no—
100C24×C17—22no—
101C24×C17—21no—
102C24×C17—21no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17C22×C412yes—
106C24×C17C22×C412yes—
107C24×C17C2×D414yes—
108C24×C17C2411yescharacteristic
109C2×D4×C17—28no—
110C2×D4×C17—44no—
111C2×D4×C17—44no—
112C2×D4×C17—48no—
113C2×D4×C17—48no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—28no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17—22no—
144C22×C4×C17—22no—
145C22×C4×C17—22no—
146C22×C4×C17—22no—
147C22×C4×C17—22no—
148C22×C4×C17—22no—
149C22×C4×C17—22no—
150C22×C4×C17—22no—
151C22×C4×C17—22no—
152C22×C4×C17—22no—
153C22×C4×C17—22no—
154C22×C4×C17—22no—
155C22×C4×C17—44no—
156C22×C4×C17—44no—
157C22×C4×C17—48no—
158C22×C4×C17—48no—
159C22×C4×C17C22×C414yes—
160C22×C4×C17C22×C412yes—
161C22×C4×C17C22×C414yes—
162C22×C4×C17C2×D414yes—
163C22×C4×C17C2×D414yes—
164C22×C4×C17#4114yes—
165C22×C4×C17#4114yes—
166C4⋊C4×C17—216no—
167C4⋊C4×C17—48no—
168C22⋊C4×C17—216no—
169C22⋊C4×C17—44no—
170C22⋊C4×C17—48no—
171C22⋊C4×C17—48no—
172C22⋊C4×C17—48no—
173C22⋊C4×C17—44no—
174C22⋊C4×C17C22⋊C4116yescomplemented
175C42×C17—48no—
176#14506C1711yesmaximal, characteristic, direct product factor, Sylow
177C23×C17—24no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—24no—
181C23×C17—24no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—24no—
187C23×C17—22no—
188C23×C17—24no—
189C23×C17—22no—
190C23×C17—22no—
191C23×C17—24no—
192C23×C17—42no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—41no—
197C23×C17—42no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—42no—
201C23×C17—41no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17—42no—
212C23×C17—41no—
213C23×C17—42no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—42no—
217C23×C17—41no—
218C23×C17C2×C22⋊C414yes—
219C23×C17C23×C411yescharacteristic
220C23×C17C22×D412yes—
221C23×C17C2×#4112yes—
222C23×C17C2×#4112yes—
223C23×C17H(5,𝔽2)12yes—
224C23×C17H(5,𝔽2)11yescharacteristic
225C23×C17H(5,𝔽2)11yescharacteristic
226C23×C17H(5,𝔽2)12yes—
227C23×C172−1+412yes—
228D4×C17—416no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—24no—
244C2×C4×C17—24no—
245C2×C4×C17—24no—
246C2×C4×C17—24no—
247C2×C4×C17—416no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—42no—
258C2×C4×C17—42no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
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—42no—
270C2×C4×C17—42no—
271C2×C4×C17—42no—
272C2×C4×C17—42no—
273C2×C4×C17—42no—
274C2×C4×C17—42no—
275C2×C4×C17—42no—
276C2×C4×C17C2×C22⋊C418yes—
277C22×#123C2×C1711yescharacteristic
278C2×#388C2×C1712yescomplemented
279#1445C2×C1712yescomplemented
280#1439C2×C1712yescomplemented
281#1438C2×C1711yescharacteristic
282#1438C2×C1711yescharacteristic
283#1437C2×C1712yescomplemented
284C4⋊#95C2×C1712yescomplemented
285C22⋊#95C2×C1712yescomplemented
286C22×C17—24no—
287C22×C17—28no—
288C22×C17—44no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17—44no—
294C22×C17—44no—
295C22×C17—42no—
296C22×C17—44no—
297C22×C17—44no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—82no—
301C22×C17—82no—
302C22×C17—82no—
303C22×C17—82no—
304C22×C17C22×C22⋊C411yescharacteristic
305C22×C17C4×#4112yes—
306C22×C17#51811yescharacteristic
307C22×C17#51811yescharacteristic
308C22×C17#52012yes—
309C22×C17#53512yes—
310C22×C17#53512yes—
311C22×C17#53812yes—
312C22×C17#53912yes—
313C22×C17#54012yes—
314C22×C17#54212yes—
315C22×C17#54612yes—
316C22×C17#54612yes—
317C22×C17#54712yes—
318C22×C17#54712yes—
319C22×C17#55111yescharacteristic
320C22×C17#55312yes—
321C22×C17#55911yescharacteristic
322C22×C17#56011yescharacteristic
323C22×C17#56212yes—
324C22×C17#56311yescharacteristic
325C4×C17—28no—
326C4×C17—44no—
327C4×C17—42no—
328C4×C17—44no—
329C4×C17—44no—
330C4×C17—82no—
331C4×C17—82no—
332C23×D4C22×C1711yescharacteristic
333C24×C4C22×C1711yescharacteristic
334C2×#123—28no—
335C2×#123C4×C1718yescomplemented
336C22×C4⋊C4C22×C1712yes—
337C22×C4⋊C4C22×C1712yes—
338C22×C22⋊C4C22×C1712yes—
339C22×C22⋊C4C22×C1712yes—
340C22×C22⋊C4C22×C1711yescharacteristic
341C22×C22⋊C4C22×C1712yes—
342C22×C22⋊C4C22×C1712yes—
343C22×C22⋊C4C22×C1711yescharacteristic
344C22×C22⋊C4C22×C1711yescharacteristic
345C22×C42C22×C1712yes—
346#388—28no—
347C2×#95C22×C1712yes—
348C2×#95C22×C1712yes—
349C2×#95C22×C1712yes—
350C2×#95C22×C1712yes—
351C2×#95C22×C1712yes—
352C2×#95C22×C1712yes—
353C2×#95C22×C1711yescharacteristic
354C2×#95C22×C1712yes—
355C2×#95C22×C1711yescharacteristic
356C2×C17—44no—
357C2×C17—82no—
358C2×C17#230911yescharacteristic
359C2×C17#233512yes—
360C2×C17#233912yes—
361C2×C17#237411yescharacteristic
362C2×C17#237611yescharacteristic
363C2×C17#270312yes—
364C2×C17#270612yes—
365C2×C17#271012yes—
366C2×C17#271812yes—
367C25C23×C1711yescharacteristic
368C25C23×C1711yescharacteristic
369C22×D4—24no—
370C22×D4—42no—
371C22×D4—41no—
372C22×D4—41no—
373C22×D4C2×C4×C1714yes—
374C23×C4—24no—
375C23×C4—42no—
376C23×C4—41no—
377C23×C4—41no—
378C23×C4C2×C4×C1714yes—
379C23×C4C23×C1712yes—
380C23×C4C23×C1712yes—
381C23×C4C23×C1712yes—
382C23×C4C23×C1711yescharacteristic
383C23×C4C23×C1712yes—
384C23×C4C23×C1712yes—
385C23×C4C23×C1712yes—
386#123—232no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4C2×C4×C1718yes—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—41no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—41no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4C2×C4×C1718yes—
409C2×C22⋊C4C2×C4×C1714yes—
410C2×C22⋊C4D4×C1718yes—
411C2×C42—24no—
412C2×C42—24no—
413C2×C42—42no—
414#95—24no—
415#95—24no—
416#95—41no—
417#95—42no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—41no—
423C17#1450611yescharacteristic, direct product factor, Sylow
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C1712yes—
440C24C22×C4×C1712yes—
441C24C2×D4×C1714yes—
442C24C24×C1711yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C1714yes—
494C22×C4C22×C4×C1712yes—
495C22×C4C22×C4×C1714yes—
496C22×C4C2×D4×C1714yes—
497C22×C4C2×D4×C1714yes—
498C22×C4#41×C1714yes—
499C22×C4#41×C1714yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C17116yescomplemented
509C42—48no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C1714yes—
552C23C23×C4×C1711yescharacteristic
553C23C22×D4×C1712yes—
554C23C2×#41×C1712yes—
555C23C2×#41×C1712yes—
556C23C17×H(5,𝔽2)12yes—
557C23C17×H(5,𝔽2)11yescharacteristic
558C23C17×H(5,𝔽2)11yescharacteristic
559C23C17×H(5,𝔽2)12yes—
560C23C17×2−1+412yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C1718yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C1711yescharacteristic
629C22C4×#41×C1712yes—
630C22C17×#51811yescharacteristic
631C22C17×#51811yescharacteristic
632C22C17×#52012yes—
633C22C17×#53512yes—
634C22C17×#53512yes—
635C22C17×#53812yes—
636C22C17×#53912yes—
637C22C17×#54012yes—
638C22C17×#54212yes—
639C22C17×#54612yes—
640C22C17×#54612yes—
641C22C17×#54712yes—
642C22C17×#54712yes—
643C22C17×#55111yescharacteristic
644C22C17×#55312yes—
645C22C17×#55911yescharacteristic
646C22C17×#56011yescharacteristic
647C22C17×#56212yes—
648C22C17×#56311yescharacteristic
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C17×#230911yescharacteristic
659C2C17×#233512yes—
660C2C17×#233912yes—
661C2C17×#237411yescharacteristic
662C2C17×#237611yescharacteristic
663C2C17×#270312yes—
664C2C17×#270612yes—
665C2C17×#271012yes—
666C2C17×#271812yes—
subgroupG/H
center, socleC24×C17C24
commutatorC23C23×C4×C17
Frattini, cosocleC24C24×C17
prime radicalC23×D4×C17C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C17×#388C2semidirect
3. C17×#1445C2semidirect
4. C17×#1439C2semidirect
7. C17×#1437C2semidirect
8. C17×C4⋊#95C2semidirect
9. C17×C22⋊#95C2semidirect
13. C2×C17×#123C4semidirect
174. C22⋊C4×C17C22⋊C4semidirect
176. #14506C17direct (N × H)
278. C2×#388C2×C17semidirect
279. #1445C2×C17semidirect
280. #1439C2×C17semidirect
283. #1437C2×C17semidirect
284. C4⋊#95C2×C17semidirect
285. C22⋊#95C2×C17semidirect
335. C2×#123C4×C17semidirect
423. C17#14506direct (N × H)
508. C22⋊C4C22⋊C4×C17semidirect

Maximal subgroups

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

subgroupindex
C22×C17×#1232
C2×C17×#3882
C17×#14452
C17×#14392
C17×#14382
C17×#14372
C17×C4⋊#952
C17×C22⋊#952
#1450617