Finite Group Databasebeta

← C17×#18271 #16876015

Subgroups — C17×#18271 #16876015

Order 4352.

984 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3536C211yesmaximal, characteristic
2C17×#3092C211yesmaximal, characteristic, complemented
3C2×C17×#481C211yesmaximal, characteristic, complemented
4C17×#2503C211yesmaximal, characteristic, complemented
5C17×#1616C214yesmaximal, complemented
6C17×#1615C214yesmaximal, complemented
7C17×#1507C211yesmaximal, characteristic, complemented
8C17×#1501C211yesmaximal, characteristic, complemented
9C17×#1472C211yesmaximal, characteristic, complemented
10C17×#562—22no—
11C17×#554—21no—
12C17×#548—21no—
13C17×#547—21no—
14C17×#546—22no—
15C17×#546—21no—
16C17×#543—21no—
17C17×#542—21no—
18C17×#539C2212yes—
19C17×#538—21no—
20C17×#537—21no—
21C17×#537C2212yes—
22C17×#525C2211yescharacteristic
23C2×C17×C22⋊Q8C2211yescharacteristic
24C2×C17×#121C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#484C2212yescomplemented
27C17×#483—21no—
28C17×#481—22no—
29C17×#481C2214yes—
30C17×#481C2212yescomplemented
31C17×#480—21no—
32C17×#428—21no—
33C2×C17×#106—21no—
34C17×#417—21no—
35C2×C17×#102—21no—
36C17×#407C2211yescharacteristic
37C2×C17×C22⋊C8C2211yescharacteristic
38C17×#400C2214yescomplemented
39C17×#391—21no—
40C17×#382—21no—
41C17×#381—22no—
42C4×C22⋊C4×C17C2211yescharacteristic
43C17×#328C2212yescomplemented
44C17×#327C2214yes—
45C17×#327C2212yescomplemented
46C17×#324C2214yescomplemented
47C2×#41×C17—21no—
48C2×#41×C17—22no—
49C2×#41×C17D411yescharacteristic
50C22×Q8×C17D411yescharacteristic
51C22×D4×C17—21no—
52C22×D4×C17D411yescharacteristic
53C23×C4×C17—21no—
54C23×C4×C17C2311yescharacteristic
55C2×M16×C17—21no—
56C22×C8×C17—21no—
57C17×C4⋊Q8—21no—
58C17×#126—22no—
59C17×#125—22no—
60C17×#124—22no—
61C17×#124—21no—
62C17×#124—22no—
63C17×#123—22no—
64C17×#123—22no—
65C17×#123—22no—
66C17×#123—21no—
67C17×#123—22no—
68C17×C22⋊Q8—21no—
69C17×C22⋊Q8—22no—
70C17×C22⋊Q8—22no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8—21no—
74C17×C22⋊Q8—21no—
75C17×C22⋊Q8—21no—
76C17×C22⋊Q8—21no—
77C17×C22⋊Q8—22no—
78C17×C22⋊Q8—22no—
79C17×C22⋊Q8D411yescharacteristic
80C17×C22⋊Q8C2311yescharacteristic
81C17×#121—21no—
82C17×#121—22no—
83C17×#121—22no—
84C17×#121—22no—
85C17×#121—22no—
86C17×#121—21no—
87C17×#121—21no—
88C17×#121—22no—
89C17×#121D412yes—
90C17×#121D411yescharacteristic
91C17×#121C2312yes—
92C17×#121C2311yescharacteristic
93C17×#120—22no—
94C17×#120—21no—
95C4×Q8×C17—21no—
96C4×D4×C17—22no—
97C4×D4×C17—21no—
98C4×D4×C17—21no—
99C4×D4×C17—21no—
100C4×D4×C17—22no—
101C17×#117D411yescharacteristic
102C2×C4⋊C4×C17—21no—
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×C17D411yescharacteristic
109C2×C4⋊C4×C17D411yescharacteristic
110C2×C4⋊C4×C17C2312yes—
111C2×C4⋊C4×C17C2312yes—
112C2×C22⋊C4×C17—22no—
113C2×C22⋊C4×C17—21no—
114C2×C22⋊C4×C17—21no—
115C2×C22⋊C4×C17D412yes—
116C2×C42×C17—21no—
117C17×#107—22no—
118C17×#106—24no—
119C17×#106—22no—
120C17×#103—22no—
121C17×#102—24no—
122C17×#102—22no—
123C17×C22⋊C8—21no—
124C17×C22⋊C8—21no—
125C17×C22⋊C8C2312yes—
126C17×C22⋊C8C2312yes—
127C17×#95—24no—
128C17×#95—24no—
129C17×#95C2312yes—
130C24×C17—21no—
131C24×C17C2×D411yescharacteristic
132#41×C17—42no—
133#41×C17—84no—
134#41×C17—82no—
135C2×Q8×C17—22no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—42no—
139C2×Q8×C17C2×D411yescharacteristic
140C2×D4×C17—21no—
141C2×D4×C17—22no—
142C2×D4×C17—21no—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—22no—
146C2×D4×C17—21no—
147C2×D4×C17—21no—
148C2×D4×C17—42no—
149C2×D4×C17—82no—
150C2×D4×C17C2×D412yes—
151C2×D4×C17C2×D411yescharacteristic
152C22×C4×C17—22no—
153C22×C4×C17—22no—
154C22×C4×C17—22no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—22no—
158C22×C4×C17—22no—
159C22×C4×C17—42no—
160C22×C4×C17—44no—
161C22×C4×C17—41no—
162C22×C4×C17—41no—
163C22×C4×C17C2×D412yes—
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D412yes—
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2411yescharacteristic
171M16×C17—42no—
172C2×C8×C17—22no—
173C2×C8×C17—22no—
174C2×C8×C17—42no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17—22no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—22no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—22no—
184C4⋊C4×C17—22no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—22no—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—41no—
189C4⋊C4×C17—41no—
190C4⋊C4×C17C2×D411yescharacteristic
191C4⋊C4×C17C2×D412yes—
192C4⋊C4×C17C2×D411yescharacteristic
193C22⋊C4×C17—22no—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—22no—
196C22⋊C4×C17—22no—
197C22⋊C4×C17—22no—
198C22⋊C4×C17—22no—
199C22⋊C4×C17—21no—
200C22⋊C4×C17—22no—
201C22⋊C4×C17—21no—
202C22⋊C4×C17—21no—
203C22⋊C4×C17—22no—
204C22⋊C4×C17—21no—
205C22⋊C4×C17—21no—
206C22⋊C4×C17—22no—
207C22⋊C4×C17—42no—
208C42×C17—22no—
209C42×C17—21no—
210C42×C17—42no—
211#18271C1711yesmaximal, characteristic, direct product factor, Sylow
212C23×C17—22no—
213C23×C17—22no—
214C23×C17—22no—
215C23×C17—21no—
216C23×C17—41no—
217C23×C17—42no—
218C23×C17—81no—
219C23×C17—81no—
220C23×C17#12011yescharacteristic
221C23×C17#12011yescharacteristic
222C23×C17C22×D411yescharacteristic
223Q8×C17—41no—
224Q8×C17—42no—
225Q8×C17—81no—
226Q8×C17—82no—
227D4×C17—42no—
228D4×C17—41no—
229D4×C17—44no—
230D4×C17—42no—
231D4×C17—42no—
232D4×C17—42no—
233D4×C17—82no—
234D4×C17—81no—
235D4×C17—82no—
236D4×C17—82no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—24no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—44no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—81no—
258C2×C4×C17—82no—
259C2×C4×C17—82no—
260C2×C4×C17—82no—
261C2×C4×C17—82no—
262C2×C4×C17—82no—
263C2×C4×C17#12011yescharacteristic
264C2×C4×C17#12011yescharacteristic
265C2×C4×C17C22×D411yescharacteristic
266C2×C4×C17C22×D411yescharacteristic
267C8×C17—42no—
268C8×C17—42no—
269#3536C2×C1711yescharacteristic
270#3092C2×C1711yescharacteristic, complemented
271C2×#481C2×C1711yescharacteristic, complemented
272#2503C2×C1711yescharacteristic, complemented
273#1616C2×C1714yescomplemented
274#1615C2×C1714yescomplemented
275#1507C2×C1711yescharacteristic, complemented
276#1501C2×C1711yescharacteristic, complemented
277#1472C2×C1711yescharacteristic, complemented
278C22×C17—21no—
279C22×C17—21no—
280C22×C17—21no—
281C22×C17—21no—
282C22×C17—21no—
283C22×C17—21no—
284C22×C17—21no—
285C22×C17—44no—
286C22×C17—41no—
287C22×C17—42no—
288C22×C17—41no—
289C22×C17—42no—
290C22×C17—41no—
291C22×C17—42no—
292C22×C17—162no—
293C22×C17#45412yes—
294C22×C17C2×#12011yescharacteristic
295C22×C17#57712yes—
296C4×C17—21no—
297C4×C17—21no—
298C4×C17—21no—
299C4×C17—42no—
300C4×C17—42no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—42no—
305C4×C17—82no—
306C4×C17—82no—
307C4×C17#45812yes—
308#562—22no—
309#554—21no—
310#548—21no—
311#547—21no—
312#546—21no—
313#546—22no—
314#543—21no—
315#542—21no—
316#539C22×C1712yes—
317#538—21no—
318#537—21no—
319#537C22×C1712yes—
320#525C22×C1711yescharacteristic
321C2×C22⋊Q8C22×C1711yescharacteristic
322C2×#121C22×C1711yescharacteristic
323C22×C4⋊C4C22×C1711yescharacteristic
324#484C22×C1712yescomplemented
325#483—21no—
326#481—22no—
327#481C22×C1714yes—
328#481C22×C1712yescomplemented
329#480—21no—
330#428—21no—
331C2×#106—21no—
332#417—21no—
333C2×#102—21no—
334#407C22×C1711yescharacteristic
335C2×C22⋊C8C22×C1711yescharacteristic
336#400C22×C1714yescomplemented
337#391—21no—
338#382—21no—
339#381—22no—
340C4×C22⋊C4C22×C1711yescharacteristic
341#328C22×C1712yescomplemented
342#327C22×C1714yes—
343#327C22×C1712yescomplemented
344#324C22×C1714yescomplemented
345C2×C17—22no—
346C2×C17—22no—
347C2×C17—41no—
348C2×C17—81no—
349C2×C17—82no—
350C2×C17#300911yescharacteristic
351C2×C17C2×#45411yescharacteristic
352C2×C17C2×#45811yescharacteristic
353C2×#41—21no—
354C2×#41—22no—
355C2×#41D4×C1711yescharacteristic
356C22×Q8D4×C1711yescharacteristic
357C22×D4—21no—
358C22×D4D4×C1711yescharacteristic
359C23×C4—21no—
360C23×C4C23×C1711yescharacteristic
361C2×M16—21no—
362C22×C8—21no—
363C4⋊Q8—21no—
364#126—22no—
365#125—22no—
366#124—22no—
367#124—21no—
368#124—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372#123—21no—
373#123—22no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8D4×C1711yescharacteristic
386C22⋊Q8C23×C1711yescharacteristic
387#121—21no—
388#121—22no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—21no—
394#121—22no—
395#121D4×C1712yes—
396#121D4×C1711yescharacteristic
397#121C23×C1712yes—
398#121C23×C1711yescharacteristic
399#120—22no—
400#120—21no—
401C4×Q8—21no—
402C4×D4—22no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—22no—
407#117D4×C1711yescharacteristic
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4D4×C1711yescharacteristic
415C2×C4⋊C4D4×C1711yescharacteristic
416C2×C4⋊C4C23×C1712yes—
417C2×C4⋊C4C23×C1712yes—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4D4×C1712yes—
422C2×C42—21no—
423#107—22no—
424#106—24no—
425#106—22no—
426#103—22no—
427#102—24no—
428#102—22no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8C23×C1712yes—
432C22⋊C8C23×C1712yes—
433#95—24no—
434#95—24no—
435#95C23×C1712yes—
436C17#1827111yescharacteristic, direct product factor, Sylow
437C24—21no—
438C24C2×D4×C1711yescharacteristic
439#41—42no—
440#41—84no—
441#41—82no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—42no—
446C2×Q8C2×D4×C1711yescharacteristic
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—82no—
457C2×D4C2×D4×C1711yescharacteristic
458C2×D4C2×D4×C1712yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4C2×D4×C1712yes—
471C22×C4C2×D4×C1711yescharacteristic
472C22×C4C2×D4×C1711yescharacteristic
473C22×C4C2×D4×C1711yescharacteristic
474C22×C4C2×D4×C1712yes—
475C22×C4C2×D4×C1711yescharacteristic
476C22×C4C2×D4×C1711yescharacteristic
477C22×C4C24×C1711yescharacteristic
478M16—42no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—42no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4C2×D4×C1712yes—
498C4⋊C4C2×D4×C1711yescharacteristic
499C4⋊C4C2×D4×C1711yescharacteristic
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—42no—
515C42—22no—
516C42—21no—
517C42—42no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—21no—
522C23—41no—
523C23—42no—
524C23—81no—
525C23—81no—
526C23C17×#12011yescharacteristic
527C23C17×#12011yescharacteristic
528C23C22×D4×C1711yescharacteristic
529Q8—41no—
530Q8—42no—
531Q8—81no—
532Q8—82no—
533D4—42no—
534D4—41no—
535D4—44no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—82no—
540D4—81no—
541D4—82no—
542D4—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4C17×#12011yescharacteristic
570C2×C4C17×#12011yescharacteristic
571C2×C4C22×D4×C1711yescharacteristic
572C2×C4C22×D4×C1711yescharacteristic
573C8—42no—
574C8—42no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—44no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—162no—
590C22C17×#45412yes—
591C22C2×C17×#12011yescharacteristic
592C22C17×#57712yes—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—82no—
603C4—82no—
604C4C17×#45812yes—
605C2—22no—
606C2—22no—
607C2—41no—
608C2—82no—
609C2—81no—
610C2C17×#300911yescharacteristic
611C2C2×C17×#45411yescharacteristic
612C2C2×C17×#45811yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#3536C2
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. C17×#3092C2semidirect
3. C2×C17×#481C2semidirect
4. C17×#2503C2semidirect
5. C17×#1616C2semidirect
6. C17×#1615C2semidirect
7. C17×#1507C2semidirect
8. C17×#1501C2semidirect
9. C17×#1472C2semidirect
26. C17×#484C22semidirect
30. C17×#481C22semidirect
38. C17×#400C22semidirect
43. C17×#328C22semidirect
45. C17×#327C22semidirect
46. C17×#324C22semidirect
211. #18271C17direct (N × H)
270. #3092C2×C17semidirect
271. C2×#481C2×C17semidirect
272. #2503C2×C17semidirect
273. #1616C2×C17semidirect
274. #1615C2×C17semidirect
275. #1507C2×C17semidirect
276. #1501C2×C17semidirect
277. #1472C2×C17semidirect
324. #484C22×C17semidirect
328. #481C22×C17semidirect
336. #400C22×C17semidirect
341. #328C22×C17semidirect
343. #327C22×C17semidirect
344. #324C22×C17semidirect
436. C17#18271direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#35362
C17×#30922
C2×C17×#4812
C17×#25032
C17×#16162
C17×#16152
C17×#15072
C17×#15012
C17×#14722
#1827117