Finite Group Databasebeta

← C17×#18393 #16876137

Subgroups — C17×#18393 #16876137

Order 4352.

910 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3491C211yesmaximal, characteristic
2C17×#3232C212yesmaximal, complemented
3C17×#2611C211yesmaximal, characteristic, complemented
4C17×#1615C212yesmaximal, complemented
5C17×#1614C211yesmaximal, characteristic, complemented
6C17×#1613C211yesmaximal, characteristic
7C17×#1609C212yesmaximal, complemented
8C17×#1608C211yesmaximal, characteristic, complemented
9C17×#1607C211yesmaximal, characteristic
10C17×#1500C212yesmaximal, complemented
11C17×#1458C211yesmaximal, characteristic, complemented
12C17×#552—22no—
13C17×#552—22no—
14C17×#551—22no—
15C17×#551—22no—
16C17×#543C2211yescharacteristic
17C17×#539—21no—
18C17×#539C2212yes—
19C17×#534C2211yescharacteristic
20C17×#529—21no—
21C17×#529C2212yes—
22C2×C17×#125C2211yescharacteristic, complemented
23C2×C17×#123—21no—
24C17×#518—21no—
25C2×C17×#117C2211yescharacteristic
26C17×#499—22no—
27C17×#488—22no—
28C17×#484C2212yescomplemented
29C17×#483C2212yescomplemented
30C17×#481C2212yes—
31C17×#480C2212yes—
32C17×#479—22no—
33C17×#478—22no—
34C17×#431C2212yescomplemented
35C17×#400C2211yescharacteristic, complemented
36C17×#399C2211yescharacteristic
37C17×#398C2211yescharacteristic, complemented
38C17×#397C2211yescharacteristic
39C17×#389—21no—
40C17×#388—21no—
41C17×#387—21no—
42C17×#385—21no—
43C17×#376C2211yescharacteristic, complemented
44C17×#328C2212yescomplemented
45C17×#328C2212yescomplemented
46C17×#327C2212yes—
47C17×#327C2212yes—
48C17×#324C2212yescomplemented
49C17×#323C2211yescharacteristic
50C17×#323C2211yescharacteristic
51C22×D4×C17—21no—
52C23×C4×C17D411yescharacteristic
53C17×#126—22no—
54C17×#126—22no—
55C17×#126—22no—
56C17×#126—22no—
57C17×#125—21no—
58C17×#125—22no—
59C17×#125—22no—
60C17×#125—22no—
61C17×#125—22no—
62C17×#125D412yes—
63C17×#124—21no—
64C17×#124D412yes—
65C17×#123—22no—
66C17×#123—22no—
67C17×#123—22no—
68C17×#123—22no—
69C17×#123—22no—
70C17×#123—22no—
71C17×#123—22no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8—22no—
74C17×C22⋊Q8—22no—
75C17×C22⋊Q8—21no—
76C17×C22⋊Q8—22no—
77C17×C22⋊Q8C2312yes—
78C17×#121—21no—
79C17×#121C2312yes—
80C17×#120—22no—
81C4×Q8×C17—21no—
82C4×Q8×C17D412yes—
83C4×D4×C17—22no—
84C4×D4×C17—22no—
85C4×D4×C17—21no—
86C4×D4×C17—22no—
87C4×D4×C17—22no—
88C4×D4×C17D412yes—
89C17×#117—22no—
90C17×#117—22no—
91C17×#117—22no—
92C17×#117—22no—
93C17×#117—22no—
94C2×C4⋊C4×C17—21no—
95C2×C4⋊C4×C17—21no—
96C2×C4⋊C4×C17—21no—
97C2×C4⋊C4×C17—21no—
98C2×C4⋊C4×C17—21no—
99C2×C4⋊C4×C17—21no—
100C2×C4⋊C4×C17D411yescharacteristic
101C2×C4⋊C4×C17C2311yescharacteristic
102C2×C4⋊C4×C17C2311yescharacteristic
103C2×C4⋊C4×C17C2311yescharacteristic
104C2×C22⋊C4×C17—22no—
105C2×C22⋊C4×C17—21no—
106C2×C22⋊C4×C17—21no—
107C2×C22⋊C4×C17—21no—
108C2×C22⋊C4×C17—21no—
109C2×C22⋊C4×C17—21no—
110C2×C22⋊C4×C17D411yescharacteristic
111C2×C22⋊C4×C17C2311yescharacteristic
112C2×C42×C17—21no—
113C2×C42×C17D411yescharacteristic
114C2×C42×C17C2311yescharacteristic
115C17×#107—22no—
116C17×#107—22no—
117C17×#106—22no—
118C17×#106—22no—
119C17×C4⋊C8—22no—
120C17×#103—22no—
121C17×#103—22no—
122C17×#102—22no—
123C17×#102—22no—
124C17×C22⋊C8C2312yes—
125C17×C22⋊C8C2312yes—
126C4×C8×C17—22no—
127C17×#95—21no—
128C17×#95—21no—
129C17×#95—21no—
130C17×#95—21no—
131C17×#95—21no—
132C17×#95C2311yescharacteristic
133C17×#95C2311yescharacteristic
134C24×C17—21no—
135C24×C17C2×D411yescharacteristic
136C2×Q8×C17—21no—
137C2×Q8×C17C2×D412yes—
138C2×D4×C17—22no—
139C2×D4×C17—22no—
140C2×D4×C17—21no—
141C2×D4×C17—42no—
142C2×D4×C17—42no—
143C2×D4×C17C2×D412yes—
144C22×C4×C17—21no—
145C22×C4×C17—22no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—22no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—22no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—41no—
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2411yescharacteristic
163C2×C8×C17—22no—
164C2×C8×C17—22no—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17—22no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—22no—
179C4⋊C4×C17—42no—
180C4⋊C4×C17—42no—
181C4⋊C4×C17—41no—
182C4⋊C4×C17—42no—
183C4⋊C4×C17—41no—
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—22no—
198C22⋊C4×C17—42no—
199C22⋊C4×C17—42no—
200C42×C17—22no—
201C42×C17—21no—
202C42×C17—22no—
203C42×C17—21no—
204C42×C17—42no—
205C42×C17C2×D412yes—
206C42×C17C2×D412yes—
207#18393C1711yesmaximal, characteristic, direct product factor, Sylow
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—22no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—42no—
214C23×C17—81no—
215C23×C17—81no—
216C23×C17#13512yes—
217C23×C17C22×D411yescharacteristic
218Q8×C17—42no—
219Q8×C17—81no—
220D4×C17—42no—
221D4×C17—44no—
222D4×C17—42no—
223D4×C17—42no—
224D4×C17—81no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—41no—
241C2×C4×C17—42no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—82no—
255C2×C4×C17—81no—
256C2×C4×C17—82no—
257C2×C4×C17—81no—
258C2×C4×C17#12011yescharacteristic
259C2×C4×C17#12011yescharacteristic
260C2×C4×C17#12012yes—
261C2×C4×C17C22×D412yes—
262C8×C17—42no—
263C8×C17—42no—
264#3491C2×C1711yescharacteristic
265#3232C2×C1712yescomplemented
266#2611C2×C1711yescharacteristic, complemented
267#1615C2×C1712yescomplemented
268#1614C2×C1711yescharacteristic, complemented
269#1613C2×C1711yescharacteristic
270#1609C2×C1712yescomplemented
271#1608C2×C1711yescharacteristic, complemented
272#1607C2×C1711yescharacteristic
273#1500C2×C1712yescomplemented
274#1458C2×C1711yescharacteristic, complemented
275C22×C17—21no—
276C22×C17—21no—
277C22×C17—22no—
278C22×C17—42no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—81no—
285C22×C17—162no—
286C22×C17C2×#12011yescharacteristic
287C22×C17C2×#13511yescharacteristic
288C22×C17#57511yescharacteristic
289C4×C17—21no—
290C4×C17—21no—
291C4×C17—22no—
292C4×C17—22no—
293C4×C17—42no—
294C4×C17—41no—
295C4×C17—42no—
296C4×C17—41no—
297C4×C17—42no—
298C4×C17—41no—
299C4×C17—42no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17—81no—
303#552—22no—
304#552—22no—
305#551—22no—
306#551—22no—
307#543C22×C1711yescharacteristic
308#539—21no—
309#539C22×C1712yes—
310#534C22×C1711yescharacteristic
311#529—21no—
312#529C22×C1712yes—
313C2×#125C22×C1711yescharacteristic, complemented
314C2×#123—21no—
315#518—21no—
316C2×#117C22×C1711yescharacteristic
317#499—22no—
318#488—22no—
319#484C22×C1712yescomplemented
320#483C22×C1712yescomplemented
321#481C22×C1712yes—
322#480C22×C1712yes—
323#479—22no—
324#478—22no—
325#431C22×C1712yescomplemented
326#400C22×C1711yescharacteristic, complemented
327#399C22×C1711yescharacteristic
328#398C22×C1711yescharacteristic, complemented
329#397C22×C1711yescharacteristic
330#389—21no—
331#388—21no—
332#387—21no—
333#385—21no—
334#376C22×C1711yescharacteristic, complemented
335#328C22×C1712yescomplemented
336#328C22×C1712yescomplemented
337#327C22×C1712yes—
338#327C22×C1712yes—
339#324C22×C1712yescomplemented
340#323C22×C1711yescharacteristic
341#323C22×C1711yescharacteristic
342C2×C17—21no—
343C2×C17—21no—
344C2×C17—42no—
345C2×C17—82no—
346C2×C17#300211yescharacteristic
347C2×C17#301811yescharacteristic
348C2×C17#302811yescharacteristic
349C22×D4—21no—
350C23×C4D4×C1711yescharacteristic
351#126—22no—
352#126—22no—
353#126—22no—
354#126—22no—
355#125—21no—
356#125—22no—
357#125—22no—
358#125—22no—
359#125—22no—
360#125D4×C1712yes—
361#124—21no—
362#124D4×C1712yes—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8C23×C1712yes—
376#121—21no—
377#121C23×C1712yes—
378#120—22no—
379C4×Q8—21no—
380C4×Q8D4×C1712yes—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4D4×C1712yes—
387#117—22no—
388#117—22no—
389#117—22no—
390#117—22no—
391#117—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4D4×C1711yescharacteristic
399C2×C4⋊C4C23×C1711yescharacteristic
400C2×C4⋊C4C23×C1711yescharacteristic
401C2×C4⋊C4C23×C1711yescharacteristic
402C2×C22⋊C4—22no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4D4×C1711yescharacteristic
409C2×C22⋊C4C23×C1711yescharacteristic
410C2×C42—21no—
411C2×C42D4×C1711yescharacteristic
412C2×C42C23×C1711yescharacteristic
413#107—22no—
414#107—22no—
415#106—22no—
416#106—22no—
417C4⋊C8—22no—
418#103—22no—
419#103—22no—
420#102—22no—
421#102—22no—
422C22⋊C8C23×C1712yes—
423C22⋊C8C23×C1712yes—
424C4×C8—22no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95C23×C1711yescharacteristic
431#95C23×C1711yescharacteristic
432C17#1839311yescharacteristic, direct product factor, Sylow
433C24—21no—
434C24C2×D4×C1711yescharacteristic
435C2×Q8—21no—
436C2×Q8C2×D4×C1712yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4C2×D4×C1712yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4C2×D4×C1711yescharacteristic
457C22×C4C2×D4×C1711yescharacteristic
458C22×C4C2×D4×C1711yescharacteristic
459C22×C4C2×D4×C1711yescharacteristic
460C22×C4C2×D4×C1711yescharacteristic
461C22×C4C24×C1711yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4C2×D4×C1712yes—
484C4⋊C4C2×D4×C1712yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C42—22no—
500C42—21no—
501C42—21no—
502C42—22no—
503C42—42no—
504C42C2×D4×C1712yes—
505C42C2×D4×C1712yes—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—81no—
513C23—81no—
514C23C17×#13512yes—
515C23C22×D4×C1711yescharacteristic
516Q8—42no—
517Q8—81no—
518D4—42no—
519D4—44no—
520D4—42no—
521D4—42no—
522D4—81no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4C17×#12011yescharacteristic
557C2×C4C17×#12011yescharacteristic
558C2×C4C17×#12012yes—
559C2×C4C22×D4×C1712yes—
560C8—42no—
561C8—42no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—162no—
573C22C2×C17×#12011yescharacteristic
574C22C2×C17×#13511yescharacteristic
575C22C17×#57511yescharacteristic
576C4—21no—
577C4—21no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—81no—
590C2—21no—
591C2—21no—
592C2—42no—
593C2—82no—
594C2C17×#300211yescharacteristic
595C2C17×#301811yescharacteristic
596C2C17×#302811yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#534C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#3232C2semidirect
3. C17×#2611C2semidirect
4. C17×#1615C2semidirect
5. C17×#1614C2semidirect
7. C17×#1609C2semidirect
8. C17×#1608C2semidirect
10. C17×#1500C2semidirect
11. C17×#1458C2semidirect
22. C2×C17×#125C22semidirect
28. C17×#484C22semidirect
29. C17×#483C22semidirect
34. C17×#431C22semidirect
35. C17×#400C22semidirect
37. C17×#398C22semidirect
43. C17×#376C22semidirect
44. C17×#328C22semidirect
45. C17×#328C22semidirect
48. C17×#324C22semidirect
207. #18393C17direct (N × H)
265. #3232C2×C17semidirect
266. #2611C2×C17semidirect
267. #1615C2×C17semidirect
268. #1614C2×C17semidirect
270. #1609C2×C17semidirect
271. #1608C2×C17semidirect
273. #1500C2×C17semidirect
274. #1458C2×C17semidirect
313. C2×#125C22×C17semidirect
319. #484C22×C17semidirect
320. #483C22×C17semidirect
325. #431C22×C17semidirect
326. #400C22×C17semidirect
328. #398C22×C17semidirect
334. #376C22×C17semidirect
335. #328C22×C17semidirect
336. #328C22×C17semidirect
339. #324C22×C17semidirect
432. C17#18393direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34912
C17×#32322
C17×#26112
C17×#16152
C17×#16142
C17×#16132
C17×#16092
C17×#16082
C17×#16072
C17×#15002
C17×#14582
#1839317