Finite Group Databasebeta

← C17×#18268 #16876012

Subgroups — C17×#18268 #16876012

Order 4352.

940 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3538C211yesmaximal, characteristic
2C17×#3092C211yesmaximal, characteristic
3C2×C17×#483C211yesmaximal, characteristic, complemented
4C17×#2502C211yesmaximal, characteristic, complemented
5C17×#1611C214yesmaximal, complemented
6C17×#1609C214yesmaximal, complemented
7C17×#1508C211yesmaximal, characteristic, complemented
8C17×#1501C211yesmaximal, characteristic
9C17×#1472C211yesmaximal, characteristic, complemented
10C17×#563—22no—
11C17×#552—21no—
12C17×#552—21no—
13C17×#551—22no—
14D4×Q8×C17—21no—
15D4×Q8×C17—21no—
16C17×#544—21no—
17C17×#544C2212yes—
18C17×#541—21no—
19C17×#539—21no—
20C17×#539C2212yes—
21C17×#536—21no—
22C17×#525C2211yescharacteristic
23C2×C17×C22⋊Q8C2211yescharacteristic
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#484—21no—
27C17×#483—22no—
28C17×#483C2214yes—
29C17×#483C2212yes—
30C17×#481—21no—
31C17×#480C2212yes—
32C17×#428—21no—
33C2×C17×#106—21no—
34C17×#417—21no—
35C2×C17×#103—21no—
36C17×#407C2211yescharacteristic
37C2×C17×C22⋊C8C2211yescharacteristic
38C17×#398C2214yes—
39C17×#391—21no—
40C17×#385—22no—
41C17×#384—21no—
42C4×C22⋊C4×C17C2211yescharacteristic
43C17×#328C2212yes—
44C17×#328C2214yes—
45C17×#327C2212yes—
46C17×#324C2214yes—
47C2×#41×C17D411yescharacteristic
48C22×Q8×C17—21no—
49C22×Q8×C17D411yescharacteristic
50C22×Q8×C17D411yescharacteristic
51C23×C4×C17—21no—
52C23×C4×C17C2311yescharacteristic
53C2×M16×C17—21no—
54C22×C8×C17—21no—
55C17×C4⋊Q8—21no—
56C17×C4⋊Q8—22no—
57C17×C4⋊Q8—22no—
58C17×C4⋊Q8—22no—
59C17×#126—22no—
60C17×#126—22no—
61C17×#126—22no—
62C17×#126—22no—
63C17×#125—21no—
64C17×#125—22no—
65C17×#125—22no—
66C17×#125—22no—
67C17×#125—22no—
68C17×#124—22no—
69C17×#123—22no—
70C17×#123—22no—
71C17×#123—22no—
72C17×#123—21no—
73C17×#123—22no—
74C17×C22⋊Q8—21no—
75C17×C22⋊Q8—21no—
76C17×C22⋊Q8—22no—
77C17×C22⋊Q8—22no—
78C17×C22⋊Q8—22no—
79C17×C22⋊Q8—21no—
80C17×C22⋊Q8—21no—
81C17×C22⋊Q8—21no—
82C17×C22⋊Q8—21no—
83C17×C22⋊Q8—21no—
84C17×C22⋊Q8—21no—
85C17×C22⋊Q8—22no—
86C17×C22⋊Q8D412yes—
87C17×C22⋊Q8D411yescharacteristic
88C17×C22⋊Q8C2312yes—
89C17×C22⋊Q8C2311yescharacteristic
90C17×#121D411yescharacteristic
91C17×#121C2311yescharacteristic
92C17×#120—21no—
93C4×Q8×C17—21no—
94C4×Q8×C17—21no—
95C4×D4×C17—22no—
96C4×D4×C17—21no—
97C4×D4×C17—21no—
98C4×D4×C17—22no—
99C17×#117—21no—
100C17×#117—22no—
101C17×#117—22no—
102C17×#117D411yescharacteristic
103C2×C4⋊C4×C17—21no—
104C2×C4⋊C4×C17—21no—
105C2×C4⋊C4×C17—21no—
106C2×C4⋊C4×C17—21no—
107C2×C4⋊C4×C17—21no—
108C2×C4⋊C4×C17—21no—
109C2×C4⋊C4×C17—24no—
110C2×C4⋊C4×C17D411yescharacteristic
111C2×C4⋊C4×C17D411yescharacteristic
112C2×C4⋊C4×C17C2312yes—
113C2×C4⋊C4×C17C2312yes—
114C2×C22⋊C4×C17—21no—
115C2×C22⋊C4×C17—21no—
116C2×C22⋊C4×C17D412yes—
117C2×C42×C17—21no—
118C17×#107—22no—
119C17×#106—24no—
120C17×#106—22no—
121C17×#103—24no—
122C17×#103—22no—
123C17×#102—22no—
124C17×C22⋊C8—21no—
125C17×C22⋊C8—21no—
126C17×C22⋊C8C2312yes—
127C17×C22⋊C8C2312yes—
128C17×#95—24no—
129C17×#95C2312yes—
130C24×C17C2×D411yescharacteristic
131#41×C17—42no—
132C2×Q8×C17—22no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—42no—
137C2×Q8×C17—42no—
138C2×Q8×C17—82no—
139C2×Q8×C17C2×D412yes—
140C2×Q8×C17C2×D411yescharacteristic
141C2×D4×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17—21no—
144C2×D4×C17C2×D411yescharacteristic
145C22×C4×C17—21no—
146C22×C4×C17—22no—
147C22×C4×C17—22no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—22no—
151C22×C4×C17—42no—
152C22×C4×C17—41no—
153C22×C4×C17—41no—
154C22×C4×C17—82no—
155C22×C4×C17C2×D412yes—
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D412yes—
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2411yescharacteristic
163M16×C17—42no—
164C2×C8×C17—22no—
165C2×C8×C17—22no—
166C2×C8×C17—42no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
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—21no—
184C4⋊C4×C17—22no—
185C4⋊C4×C17—22no—
186C4⋊C4×C17—41no—
187C4⋊C4×C17—41no—
188C4⋊C4×C17—84no—
189C4⋊C4×C17C2×D412yes—
190C4⋊C4×C17C2×D411yescharacteristic
191C4⋊C4×C17C2×D411yescharacteristic
192C22⋊C4×C17—22no—
193C22⋊C4×C17—22no—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—24no—
196C22⋊C4×C17—22no—
197C22⋊C4×C17—21no—
198C22⋊C4×C17—22no—
199C22⋊C4×C17—22no—
200C22⋊C4×C17—21no—
201C22⋊C4×C17—22no—
202C22⋊C4×C17—21no—
203C22⋊C4×C17—21no—
204C22⋊C4×C17—21no—
205C42×C17—22no—
206C42×C17—22no—
207C42×C17—22no—
208C42×C17—21no—
209C42×C17—42no—
210#18268C1711yesmaximal, characteristic, direct product factor, Sylow
211C23×C17—22no—
212C23×C17—22no—
213C23×C17—21no—
214C23×C17—41no—
215C23×C17#12011yescharacteristic
216C23×C17#12011yescharacteristic
217C23×C17C22×D411yescharacteristic
218Q8×C17—42no—
219Q8×C17—41no—
220Q8×C17—42no—
221Q8×C17—42no—
222Q8×C17—81no—
223Q8×C17—82no—
224D4×C17—41no—
225D4×C17—42no—
226D4×C17—42no—
227D4×C17—81no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—24no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—44no—
250C2×C4×C17—81no—
251C2×C4×C17—82no—
252C2×C4×C17—82no—
253C2×C4×C17—82no—
254C2×C4×C17#12011yescharacteristic
255C2×C4×C17#12011yescharacteristic
256C2×C4×C17C22×D411yescharacteristic
257C2×C4×C17C22×D411yescharacteristic
258C8×C17—42no—
259C8×C17—42no—
260#3538C2×C1711yescharacteristic
261#3092C2×C1711yescharacteristic
262C2×#483C2×C1711yescharacteristic, complemented
263#2502C2×C1711yescharacteristic, complemented
264#1611C2×C1714yescomplemented
265#1609C2×C1714yescomplemented
266#1508C2×C1711yescharacteristic, complemented
267#1501C2×C1711yescharacteristic
268#1472C2×C1711yescharacteristic, complemented
269C22×C17—21no—
270C22×C17—21no—
271C22×C17—21no—
272C22×C17—21no—
273C22×C17—21no—
274C22×C17—21no—
275C22×C17—21no—
276C22×C17—44no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17#45612yes—
281C22×C17C2×#12011yescharacteristic
282C22×C17#57712yes—
283C4×C17—21no—
284C4×C17—21no—
285C4×C17—21no—
286C4×C17—44no—
287C4×C17—42no—
288C4×C17—41no—
289C4×C17—42no—
290C4×C17—41no—
291C4×C17—41no—
292C4×C17—42no—
293C4×C17—42no—
294C4×C17—82no—
295C4×C17#45812yes—
296#563—22no—
297#552—21no—
298#552—21no—
299#551—22no—
300D4×Q8—21no—
301D4×Q8—21no—
302#544—21no—
303#544C22×C1712yes—
304#541—21no—
305#539—21no—
306#539C22×C1712yes—
307#536—21no—
308#525C22×C1711yescharacteristic
309C2×C22⋊Q8C22×C1711yescharacteristic
310C2×C22⋊Q8C22×C1711yescharacteristic
311C22×C4⋊C4C22×C1711yescharacteristic
312#484—21no—
313#483—22no—
314#483C22×C1714yes—
315#483C22×C1712yes—
316#481—21no—
317#480C22×C1712yes—
318#428—21no—
319C2×#106—21no—
320#417—21no—
321C2×#103—21no—
322#407C22×C1711yescharacteristic
323C2×C22⋊C8C22×C1711yescharacteristic
324#398C22×C1714yes—
325#391—21no—
326#385—22no—
327#384—21no—
328C4×C22⋊C4C22×C1711yescharacteristic
329#328C22×C1714yes—
330#328C22×C1712yes—
331#327C22×C1712yes—
332#324C22×C1714yes—
333C2×C17—22no—
334C2×C17—22no—
335C2×C17—41no—
336C2×C17—81no—
337C2×C17#301011yescharacteristic
338C2×C17C2×#45611yescharacteristic
339C2×C17C2×#45811yescharacteristic
340C2×#41D4×C1711yescharacteristic
341C22×Q8—21no—
342C22×Q8D4×C1711yescharacteristic
343C22×Q8D4×C1711yescharacteristic
344C23×C4—21no—
345C23×C4C23×C1711yescharacteristic
346C2×M16—21no—
347C22×C8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—22no—
350C4⋊Q8—22no—
351C4⋊Q8—22no—
352#126—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#125—21no—
357#125—22no—
358#125—22no—
359#125—22no—
360#125—22no—
361#124—22no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—21no—
366#123—22no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8D4×C1712yes—
380C22⋊Q8D4×C1711yescharacteristic
381C22⋊Q8C23×C1712yes—
382C22⋊Q8C23×C1711yescharacteristic
383#121D4×C1711yescharacteristic
384#121C23×C1711yescharacteristic
385#120—21no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×D4—22no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—22no—
392#117—21no—
393#117—22no—
394#117—22no—
395#117D4×C1711yescharacteristic
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4D4×C1711yescharacteristic
404C2×C4⋊C4D4×C1711yescharacteristic
405C2×C4⋊C4C23×C1712yes—
406C2×C4⋊C4C23×C1712yes—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4D4×C1712yes—
410C2×C42—21no—
411#107—22no—
412#106—24no—
413#106—22no—
414#103—24no—
415#103—22no—
416#102—22no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8C23×C1712yes—
420C22⋊C8C23×C1712yes—
421#95—24no—
422#95C23×C1712yes—
423C17#1826811yescharacteristic, direct product factor, Sylow
424C24C2×D4×C1711yescharacteristic
425#41—42no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—82no—
433C2×Q8C2×D4×C1712yes—
434C2×Q8C2×D4×C1711yescharacteristic
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C2×D4×C1711yescharacteristic
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—82no—
449C22×C4C2×D4×C1712yes—
450C22×C4C2×D4×C1711yescharacteristic
451C22×C4C2×D4×C1711yescharacteristic
452C22×C4C2×D4×C1711yescharacteristic
453C22×C4C2×D4×C1712yes—
454C22×C4C2×D4×C1711yescharacteristic
455C22×C4C2×D4×C1711yescharacteristic
456C22×C4C24×C1711yescharacteristic
457M16—42no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—42no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—84no—
483C4⋊C4C2×D4×C1712yes—
484C4⋊C4C2×D4×C1711yescharacteristic
485C4⋊C4C2×D4×C1711yescharacteristic
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—21no—
503C42—42no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—41no—
508C23C17×#12011yescharacteristic
509C23C17×#12011yescharacteristic
510C23C22×D4×C1711yescharacteristic
511Q8—42no—
512Q8—41no—
513Q8—42no—
514Q8—42no—
515Q8—81no—
516Q8—82no—
517D4—41no—
518D4—42no—
519D4—42no—
520D4—81no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4C17×#12011yescharacteristic
548C2×C4C17×#12011yescharacteristic
549C2×C4C22×D4×C1711yescharacteristic
550C2×C4C22×D4×C1711yescharacteristic
551C8—42no—
552C8—42no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—44no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22C17×#45612yes—
565C22C2×C17×#12011yescharacteristic
566C22C17×#57712yes—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4C17×#45812yes—
580C2—22no—
581C2—22no—
582C2—41no—
583C2—81no—
584C2C17×#301011yescharacteristic
585C2C2×C17×#45611yescharacteristic
586C2C2×C17×#45811yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#525C22
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
3. C2×C17×#483C2semidirect
4. C17×#2502C2semidirect
5. C17×#1611C2semidirect
6. C17×#1609C2semidirect
7. C17×#1508C2semidirect
9. C17×#1472C2semidirect
210. #18268C17direct (N × H)
262. C2×#483C2×C17semidirect
263. #2502C2×C17semidirect
264. #1611C2×C17semidirect
265. #1609C2×C17semidirect
266. #1508C2×C17semidirect
268. #1472C2×C17semidirect
423. C17#18268direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#35382
C17×#30922
C2×C17×#4832
C17×#25022
C17×#16112
C17×#16092
C17×#15082
C17×#15012
C17×#14722
#1826817