Finite Group Databasebeta

← C17×#18587 #16876331

Subgroups — C17×#18587 #16876331

Order 4352.

944 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3520C211yesmaximal, characteristic
2C17×#3218C211yesmaximal, characteristic, complemented
3C17×#3165C211yesmaximal, characteristic, complemented
4C2×C17×#474C211yesmaximal, characteristic, complemented
5C17×#1654C212yesmaximal, complemented
6C17×#1652C212yesmaximal, complemented
7C17×#1649C212yesmaximal, complemented
8C17×#1647C212yesmaximal, complemented
9C17×#1552C211yesmaximal, characteristic, complemented
10C17×#1490C211yesmaximal, characteristic, complemented
11C17×#1483C211yesmaximal, characteristic, complemented
12C17×#556C2211yescharacteristic
13C17×#554—21no—
14C17×#552—21no—
15C17×#551—22no—
16D4×Q8×C17C2211yescharacteristic
17C17×#548—21no—
18C17×#547C2211yescharacteristic
19C17×#547C2211yescharacteristic
20C17×#546—21no—
21C17×#542—21no—
22C17×#539—22no—
23C17×#533C2211yescharacteristic
24C2×C17×C4⋊Q8C2211yescharacteristic
25C17×#529C2211yescharacteristic
26C17×#525—21no—
27C2×C17×C22⋊Q8—21no—
28C17×#520—21no—
29C2×C4×D4×C17C2211yescharacteristic
30C17×#501—22no—
31C17×#500C2212yescomplemented
32C17×#498C2212yescomplemented
33C17×#497—21no—
34C17×#491—21no—
35C17×#474—21no—
36C17×#474—21no—
37C17×#474C2212yes—
38C17×#474C2212yes—
39C17×#462C2212yescomplemented
40C17×#461C2212yescomplemented
41C17×#451—21no—
42C17×#448—21no—
43C17×#432C2211yescharacteristic
44C17×#428—22no—
45C2×C17×#107—21no—
46C2×C17×C4⋊C8C2211yescharacteristic
47C2×C17×#102—21no—
48C4×M16×C17C2211yescharacteristic
49C17×#332C2212yescomplemented
50C17×#332C2212yescomplemented
51C17×Q8⋊C8C2212yescomplemented
52C17×D4⋊C8C2212yes—
53C17×D4⋊C8C2212yes—
54C17×D4⋊C8C2212yescomplemented
55C2×#41×C17—21no—
56C2×#41×C17—21no—
57C22×Q8×C17—21no—
58C22×Q8×C17—21no—
59C22×D4×C17D411yescharacteristic
60C23×C4×C17—21no—
61C2×Q16×C17—22no—
62C2×SD16×C17—22no—
63C2×M16×C17—21no—
64C22×C8×C17—21no—
65C17×C4⋊Q8—21no—
66C17×C4⋊Q8—21no—
67C17×C4⋊Q8—21no—
68C17×C4⋊Q8—21no—
69C17×C4⋊Q8C2312yes—
70C17×C4⋊Q8C2311yescharacteristic
71C17×C4⋊Q8C2311yescharacteristic
72C17×#126—22no—
73C17×#126—22no—
74C17×#126—21no—
75C17×#125—22no—
76C17×#125D411yescharacteristic
77C17×#125D412yes—
78C17×#124—22no—
79C17×#124—21no—
80C17×#124D411yescharacteristic
81C17×#123—22no—
82C17×#123—22no—
83C17×#123—21no—
84C17×#123—22no—
85C17×#123—22no—
86C17×C22⋊Q8—22no—
87C17×C22⋊Q8—22no—
88C17×C22⋊Q8—22no—
89C17×C22⋊Q8—21no—
90C17×C22⋊Q8—21no—
91C17×C22⋊Q8—21no—
92C17×C22⋊Q8—21no—
93C17×C22⋊Q8—21no—
94C17×C22⋊Q8—22no—
95C17×C22⋊Q8—21no—
96C17×C22⋊Q8D411yescharacteristic
97C17×#121—22no—
98C17×#121—21no—
99C17×#121—21no—
100C17×#121D411yescharacteristic
101C17×#120—22no—
102C4×Q8×C17—21no—
103C4×Q8×C17—21no—
104C4×Q8×C17—21no—
105C4×Q8×C17C2311yescharacteristic
106C4×D4×C17—22no—
107C4×D4×C17—22no—
108C4×D4×C17—22no—
109C4×D4×C17—21no—
110C4×D4×C17—21no—
111C4×D4×C17—21no—
112C4×D4×C17—21no—
113C4×D4×C17D411yescharacteristic
114C4×D4×C17D411yescharacteristic
115C4×D4×C17C2311yescharacteristic
116C4×D4×C17C2311yescharacteristic
117C4×D4×C17C2311yescharacteristic
118C17×#117—22no—
119C17×#117—21no—
120C17×#117—21no—
121C17×#117D412yes—
122C2×C4⋊C4×C17—22no—
123C2×C4⋊C4×C17—21no—
124C2×C4⋊C4×C17—21no—
125C2×C4⋊C4×C17—21no—
126C2×C4⋊C4×C17—21no—
127C2×C4⋊C4×C17D411yescharacteristic
128C2×C22⋊C4×C17—22no—
129C2×C22⋊C4×C17—21no—
130C2×C42×C17C2311yescharacteristic
131C17×#107—24no—
132C17×#107—22no—
133C17×#107—22no—
134C17×#106—24no—
135C17×C4⋊C8—21no—
136C17×C4⋊C8C2312yes—
137C17×C4⋊C8C2312yes—
138C17×#103—22no—
139C17×#103—22no—
140C17×#102—22no—
141C17×#102—22no—
142C17×C22⋊C8—21no—
143C17×#97—21no—
144C17×#97—21no—
145C4×C8×C17C2312yes—
146C24×C17—21no—
147#41×C17—42no—
148#41×C17—42no—
149#41×C17—82no—
150C2×Q8×C17—22no—
151C2×Q8×C17—21no—
152C2×Q8×C17—21no—
153C2×Q8×C17—22no—
154C2×Q8×C17—41no—
155C2×Q8×C17—82no—
156C2×Q8×C17—82no—
157C2×Q8×C17C2×D411yescharacteristic
158C2×D4×C17—22no—
159C2×D4×C17—22no—
160C2×D4×C17—21no—
161C2×D4×C17—22no—
162C2×D4×C17—21no—
163C2×D4×C17—41no—
164C2×D4×C17C2×D411yescharacteristic
165C2×D4×C17C2×D411yescharacteristic
166C2×D4×C17C2×D411yescharacteristic
167C22×C4×C17—22no—
168C22×C4×C17—22no—
169C22×C4×C17—22no—
170C22×C4×C17—21no—
171C22×C4×C17—21no—
172C22×C4×C17—21no—
173C22×C4×C17—21no—
174C22×C4×C17—21no—
175C22×C4×C17—21no—
176C22×C4×C17—41no—
177C22×C4×C17—82no—
178C22×C4×C17C2×D411yescharacteristic
179C22×C4×C17C2×D411yescharacteristic
180C22×C4×C17C2×D411yescharacteristic
181Q16×C17—82no—
182SD16×C17—82no—
183M16×C17—24no—
184C2×C8×C17—22no—
185C2×C8×C17—22no—
186C2×C8×C17—22no—
187C2×C8×C17—42no—
188C4⋊C4×C17—22no—
189C4⋊C4×C17—22no—
190C4⋊C4×C17—22no—
191C4⋊C4×C17—22no—
192C4⋊C4×C17—21no—
193C4⋊C4×C17—22no—
194C4⋊C4×C17—22no—
195C4⋊C4×C17—21no—
196C4⋊C4×C17—21no—
197C4⋊C4×C17—21no—
198C4⋊C4×C17—21no—
199C4⋊C4×C17—21no—
200C4⋊C4×C17—21no—
201C4⋊C4×C17—22no—
202C4⋊C4×C17—22no—
203C4⋊C4×C17—41no—
204C4⋊C4×C17—41no—
205C4⋊C4×C17C2×D411yescharacteristic
206C4⋊C4×C17C2×D412yes—
207C4⋊C4×C17C2×D411yescharacteristic
208C4⋊C4×C17C2×D411yescharacteristic
209C4⋊C4×C17C2×D412yes—
210C4⋊C4×C17C2×D411yescharacteristic
211C22⋊C4×C17—22no—
212C22⋊C4×C17—22no—
213C22⋊C4×C17—22no—
214C22⋊C4×C17—22no—
215C22⋊C4×C17—22no—
216C22⋊C4×C17—22no—
217C22⋊C4×C17—21no—
218C22⋊C4×C17—21no—
219C22⋊C4×C17—21no—
220C22⋊C4×C17—21no—
221C22⋊C4×C17—21no—
222C22⋊C4×C17—21no—
223C22⋊C4×C17—21no—
224C22⋊C4×C17—21no—
225C22⋊C4×C17—21no—
226C42×C17—22no—
227C42×C17—22no—
228C42×C17—21no—
229C42×C17C2×D411yescharacteristic
230C42×C17C2×D411yescharacteristic
231C42×C17C2×D411yescharacteristic
232C42×C17C2411yescharacteristic
233#18587C1711yesmaximal, characteristic, direct product factor, Sylow
234C23×C17—21no—
235C23×C17—21no—
236C23×C17—21no—
237C23×C17—42no—
238C23×C17—42no—
239C23×C17—81no—
240C23×C17#12011yescharacteristic
241Q8×C17—22no—
242Q8×C17—41no—
243Q8×C17—41no—
244Q8×C17—42no—
245Q8×C17—42no—
246Q8×C17—42no—
247Q8×C17—82no—
248Q8×C17—82no—
249D4×C17—21no—
250D4×C17—21no—
251D4×C17—21no—
252D4×C17—22no—
253D4×C17—42no—
254D4×C17—44no—
255D4×C17—42no—
256D4×C17—82no—
257D4×C17#13712yes—
258C2×C4×C17—22no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—22no—
262C2×C4×C17—21no—
263C2×C4×C17—22no—
264C2×C4×C17—21no—
265C2×C4×C17—21no—
266C2×C4×C17—22no—
267C2×C4×C17—21no—
268C2×C4×C17—21no—
269C2×C4×C17—44no—
270C2×C4×C17—41no—
271C2×C4×C17—41no—
272C2×C4×C17—42no—
273C2×C4×C17—42no—
274C2×C4×C17—42no—
275C2×C4×C17—81no—
276C2×C4×C17—81no—
277C2×C4×C17—81no—
278C2×C4×C17—82no—
279C2×C4×C17—82no—
280C2×C4×C17—82no—
281C2×C4×C17#12011yescharacteristic
282C2×C4×C17#12011yescharacteristic
283C2×C4×C17#12011yescharacteristic
284C2×C4×C17C22×D411yescharacteristic
285C2×C4×C17C22×D411yescharacteristic
286C2×C4×C17C22×D411yescharacteristic
287C8×C17—24no—
288C8×C17—42no—
289C8×C17—42no—
290#3520C2×C1711yescharacteristic
291#3218C2×C1711yescharacteristic, complemented
292#3165C2×C1711yescharacteristic, complemented
293C2×#474C2×C1711yescharacteristic, complemented
294#1654C2×C1712yescomplemented
295#1652C2×C1712yescomplemented
296#1649C2×C1712yescomplemented
297#1647C2×C1712yescomplemented
298#1552C2×C1711yescharacteristic, complemented
299#1490C2×C1711yescharacteristic, complemented
300#1483C2×C1711yescharacteristic, complemented
301C22×C17—21no—
302C22×C17—21no—
303C22×C17—22no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—41no—
312C22×C17—84no—
313C22×C17#45612yes—
314C22×C17C2×#12011yescharacteristic
315C4×C17—21no—
316C4×C17—21no—
317C4×C17—21no—
318C4×C17—21no—
319C4×C17—41no—
320C4×C17—42no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17—42no—
324C4×C17—41no—
325C4×C17—41no—
326C4×C17—82no—
327C4×C17C2×#13711yescharacteristic
328C4×C17#57511yescharacteristic
329C4×C17#57612yes—
330#556C22×C1711yescharacteristic
331#554—21no—
332#552—21no—
333#551—22no—
334D4×Q8C22×C1711yescharacteristic
335#548—21no—
336#547C22×C1711yescharacteristic
337#547C22×C1711yescharacteristic
338#546—21no—
339#542—21no—
340#539—22no—
341#533C22×C1711yescharacteristic
342C2×C4⋊Q8C22×C1711yescharacteristic
343#529C22×C1711yescharacteristic
344#525—21no—
345C2×C22⋊Q8—21no—
346#520—21no—
347C2×C4×D4C22×C1711yescharacteristic
348#501—22no—
349#500C22×C1712yescomplemented
350#498C22×C1712yescomplemented
351#497—21no—
352#491—21no—
353#474—21no—
354#474—21no—
355#474C22×C1712yes—
356#474C22×C1712yes—
357#462C22×C1712yescomplemented
358#461C22×C1712yescomplemented
359#451—21no—
360#448—21no—
361#432C22×C1711yescharacteristic
362#428—22no—
363C2×#107—21no—
364C2×C4⋊C8C22×C1711yescharacteristic
365C2×#102—21no—
366C4×M16C22×C1711yescharacteristic
367#332C22×C1712yescomplemented
368#332C22×C1712yescomplemented
369Q8⋊C8C22×C1712yescomplemented
370D4⋊C8C22×C1712yes—
371D4⋊C8C22×C1712yes—
372D4⋊C8C22×C1712yescomplemented
373C2×C17—22no—
374C2×C17—42no—
375C2×C17—81no—
376C2×C17—81no—
377C2×C17#300711yescharacteristic
378C2×C17#300811yescharacteristic
379C2×C17C2×#45611yescharacteristic
380C2×#41—21no—
381C2×#41—21no—
382C22×Q8—21no—
383C22×Q8—21no—
384C22×D4D4×C1711yescharacteristic
385C23×C4—21no—
386C2×Q16—22no—
387C2×SD16—22no—
388C2×M16—21no—
389C22×C8—21no—
390C4⋊Q8—21no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393C4⋊Q8—21no—
394C4⋊Q8C23×C1712yes—
395C4⋊Q8C23×C1711yescharacteristic
396C4⋊Q8C23×C1711yescharacteristic
397#126—22no—
398#126—22no—
399#126—21no—
400#125—22no—
401#125D4×C1711yescharacteristic
402#125D4×C1712yes—
403#124—22no—
404#124—21no—
405#124D4×C1711yescharacteristic
406#123—22no—
407#123—22no—
408#123—21no—
409#123—22no—
410#123—22no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—22no—
420C22⋊Q8—21no—
421C22⋊Q8D4×C1711yescharacteristic
422#121—22no—
423#121—21no—
424#121—21no—
425#121D4×C1711yescharacteristic
426#120—22no—
427C4×Q8—21no—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8C23×C1711yescharacteristic
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4D4×C1711yescharacteristic
439C4×D4D4×C1711yescharacteristic
440C4×D4C23×C1711yescharacteristic
441C4×D4C23×C1711yescharacteristic
442C4×D4C23×C1711yescharacteristic
443#117—22no—
444#117—21no—
445#117—21no—
446#117D4×C1712yes—
447C2×C4⋊C4—22no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—21no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4D4×C1711yescharacteristic
453C2×C22⋊C4—22no—
454C2×C22⋊C4—21no—
455C2×C42C23×C1711yescharacteristic
456#107—22no—
457#107—22no—
458#107—24no—
459#106—24no—
460C4⋊C8—21no—
461C4⋊C8C23×C1712yes—
462C4⋊C8C23×C1712yes—
463#103—22no—
464#103—22no—
465#102—22no—
466#102—22no—
467C22⋊C8—21no—
468#97—21no—
469#97—21no—
470C4×C8C23×C1712yes—
471C17#1858711yescharacteristic, direct product factor, Sylow
472C24—21no—
473#41—42no—
474#41—42no—
475#41—82no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—41no—
481C2×Q8—82no—
482C2×Q8—82no—
483C2×Q8C2×D4×C1711yescharacteristic
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—41no—
490C2×D4C2×D4×C1711yescharacteristic
491C2×D4C2×D4×C1711yescharacteristic
492C2×D4C2×D4×C1711yescharacteristic
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—41no—
503C22×C4—82no—
504C22×C4C2×D4×C1711yescharacteristic
505C22×C4C2×D4×C1711yescharacteristic
506C22×C4C2×D4×C1711yescharacteristic
507Q16—82no—
508SD16—82no—
509M16—24no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—42no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4C2×D4×C1711yescharacteristic
532C4⋊C4C2×D4×C1711yescharacteristic
533C4⋊C4C2×D4×C1712yes—
534C4⋊C4C2×D4×C1711yescharacteristic
535C4⋊C4C2×D4×C1711yescharacteristic
536C4⋊C4C2×D4×C1712yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C42—22no—
553C42—21no—
554C42—22no—
555C42C2×D4×C1711yescharacteristic
556C42C2×D4×C1711yescharacteristic
557C42C2×D4×C1711yescharacteristic
558C42C24×C1711yescharacteristic
559C23—21no—
560C23—21no—
561C23—21no—
562C23—42no—
563C23—42no—
564C23—81no—
565C23C17×#12011yescharacteristic
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—82no—
573Q8—82no—
574D4—21no—
575D4—21no—
576D4—21no—
577D4—22no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—82no—
582D4C17×#13712yes—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4C17×#12011yescharacteristic
607C2×C4C17×#12011yescharacteristic
608C2×C4C17×#12011yescharacteristic
609C2×C4C22×D4×C1711yescharacteristic
610C2×C4C22×D4×C1711yescharacteristic
611C2×C4C22×D4×C1711yescharacteristic
612C8—24no—
613C8—42no—
614C8—42no—
615C22—21no—
616C22—21no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—84no—
627C22C17×#45612yes—
628C22C2×C17×#12011yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—82no—
641C4C2×C17×#13711yescharacteristic
642C4C17×#57511yescharacteristic
643C4C17×#57612yes—
644C2—22no—
645C2—42no—
646C2—81no—
647C2—81no—
648C2C17×#300711yescharacteristic
649C2C17×#300811yescharacteristic
650C2C2×C17×#45611yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC42C24×C17
prime radicalC17×#3520C2
LCSG ▷ C42 ▷ 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×#3218C2semidirect
3. C17×#3165C2semidirect
4. C2×C17×#474C2semidirect
5. C17×#1654C2semidirect
6. C17×#1652C2semidirect
7. C17×#1649C2semidirect
8. C17×#1647C2semidirect
9. C17×#1552C2semidirect
10. C17×#1490C2semidirect
11. C17×#1483C2semidirect
31. C17×#500C22semidirect
32. C17×#498C22semidirect
39. C17×#462C22semidirect
40. C17×#461C22semidirect
49. C17×#332C22semidirect
50. C17×#332C22semidirect
51. C17×Q8⋊C8C22semidirect
54. C17×D4⋊C8C22semidirect
233. #18587C17direct (N × H)
291. #3218C2×C17semidirect
292. #3165C2×C17semidirect
293. C2×#474C2×C17semidirect
294. #1654C2×C17semidirect
295. #1652C2×C17semidirect
296. #1649C2×C17semidirect
297. #1647C2×C17semidirect
298. #1552C2×C17semidirect
299. #1490C2×C17semidirect
300. #1483C2×C17semidirect
349. #500C22×C17semidirect
350. #498C22×C17semidirect
357. #462C22×C17semidirect
358. #461C22×C17semidirect
367. #332C22×C17semidirect
368. #332C22×C17semidirect
369. Q8⋊C8C22×C17semidirect
372. D4⋊C8C22×C17semidirect
471. C17#18587direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#35202
C17×#32182
C17×#31652
C2×C17×#4742
C17×#16542
C17×#16522
C17×#16492
C17×#16472
C17×#15522
C17×#14902
C17×#14832
#1858717