Finite Group Databasebeta

← C17×#61876 #16919620

Subgroups — C17×#61876 #16919620

Order 4352.

1712 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3548C211yesmaximal, characteristic, complemented
2C17×#3504C211yesmaximal, characteristic, complemented
3C17×#3389C214yesmaximal, complemented
4C17×#3388C214yesmaximal, complemented
5C17×#3368C214yesmaximal, complemented
6C17×#3366C214yesmaximal, complemented
7C17×#3241C214yesmaximal, complemented
8C17×#3234C212yesmaximal, complemented
9C17×#3119C214yesmaximal, complemented
10C17×#3115C212yesmaximal, complemented
11C17×#2987C211yesmaximal, characteristic, complemented
12C17×#562—21no—
13C17×#562—21no—
14C17×#557C2214yes—
15C17×#554—24no—
16C17×#554C2214yes—
17C17×#554C2214yes—
18C17×#550C2214yes—
19C17×#547—22no—
20C17×#544—22no—
21C17×#544C2212yes—
22C17×#540C2214yes—
23C17×#537—24no—
24C17×#533—21no—
25C17×#533C2211yescharacteristic
26C17×#532C2212yes—
27C2×C17×#125—21no—
28C2×C17×#125C2211yescharacteristic
29C17×#520C2212yes—
30C4×#41×C17C2211yescharacteristic
31C17×#501C2214yes—
32C17×#501C2214yes—
33C17×#499C2214yes—
34C17×#490C2214yes—
35C17×#490C2214yes—
36C17×#486C2214yes—
37C17×#482C2214yes—
38C17×#481C2214yes—
39C17×#480C2214yes—
40C17×#479C2214yes—
41C17×#478C2214yes—
42C17×#478C2214yes—
43C17×#478C2214yes—
44C17×#478C2214yes—
45C17×#478C2214yes—
46C17×#476C2214yes—
47C17×#475C2212yes—
48C17×#474C2214yes—
49C17×#474C2212yes—
50C17×#460C2212yes—
51C17×#460C2214yes—
52C17×#459C2212yes—
53C17×#459C2214yes—
54C17×#449C2214yes—
55C17×#446C2214yes—
56C17×#446C2214yes—
57C17×#442C2214yes—
58C17×#442C2214yes—
59C17×#440C2214yes—
60C17×#440C2214yes—
61C4×SD16×C17C2214yes—
62C4×D8×C17C2214yes—
63C17×#433C2214yes—
64C17×#432C2212yes—
65C2×#41×C17—24no—
66C2×#41×C17—22no—
67C2×#41×C17C2311yescharacteristic
68C2×SD16×C17C2314yes—
69C2×D8×C17C2314yes—
70C17×C4⋊Q8—24no—
71C17×C4⋊Q8—22no—
72C17×C4⋊Q8D412yes—
73C17×C4⋊Q8C2312yes—
74C17×C4⋊Q8C2314yes—
75C17×#127C2312yes—
76C17×#127C2314yes—
77C17×#125—22no—
78C17×#125—24no—
79C17×#125—24no—
80C17×#125—21no—
81C17×#125—21no—
82C17×#125D411yescharacteristic
83C17×#125D411yescharacteristic
84C17×#125C2314yes—
85C17×#125C2314yes—
86C17×#125C2314yes—
87C17×#124—22no—
88C17×#123—24no—
89C17×#123—24no—
90C17×C22⋊Q8—24no—
91C17×C22⋊Q8—24no—
92C17×C22⋊Q8—24no—
93C17×C22⋊Q8—24no—
94C17×C22⋊Q8—24no—
95C17×C22⋊Q8—24no—
96C17×C22⋊Q8—24no—
97C17×C22⋊Q8D414yes—
98C17×#121—22no—
99C17×#121—24no—
100C17×#121—24no—
101C17×#121—24no—
102C17×#121—24no—
103C17×#121—24no—
104C17×#121C2314yes—
105C4×Q8×C17—24no—
106C4×Q8×C17C2312yes—
107C4×Q8×C17C2314yes—
108C4×Q8×C17C2314yes—
109C4×D4×C17—22no—
110C4×D4×C17—24no—
111C4×D4×C17—22no—
112C4×D4×C17C2312yes—
113C4×D4×C17C2314yes—
114C4×D4×C17C2314yes—
115C4×D4×C17C2314yes—
116C4×D4×C17C2314yes—
117C17×#117—22no—
118C17×#117—21no—
119C17×#117C2311yescharacteristic
120C17×#117C2312yes—
121C2×C4⋊C4×C17—22no—
122C2×C4⋊C4×C17—22no—
123C2×C4⋊C4×C17—24no—
124C2×C4⋊C4×C17—24no—
125C2×C4⋊C4×C17C2312yes—
126C2×C42×C17—21no—
127C2×C42×C17C2311yescharacteristic
128C17×#107C2314yes—
129C17×#107C2314yes—
130C17×#107C2314yes—
131C17×#106C2314yes—
132C17×#106C2314yes—
133C17×#106C2314yes—
134C17×C4⋊C8C2312yes—
135C17×C4⋊C8C2312yes—
136C17×C4⋊C8C2314yes—
137C17×C4⋊C8C2314yes—
138C17×#103C2314yes—
139C17×#102C2314yes—
140C17×#102C2314yes—
141C17×#102C2314yes—
142C17×#102C2314yes—
143C17×#102C2314yes—
144C17×#102C2314yes—
145C17×#102C2314yes—
146C17×C22⋊C8C2314yes—
147C17×#97C2314yes—
148C17×#97C2314yes—
149C4×C8×C17C2314yes—
150#41×C17—22no—
151#41×C17—22no—
152#41×C17—84no—
153#41×C17—88no—
154C2×Q8×C17—22no—
155C2×Q8×C17—24no—
156C2×Q8×C17C2×D414yes—
157C2×Q8×C17C2411yescharacteristic
158C2×D4×C17—21no—
159C2×D4×C17—24no—
160C2×D4×C17—22no—
161C2×D4×C17—24no—
162C2×D4×C17—24no—
163C2×D4×C17—22no—
164C2×D4×C17C2414yes—
165C2×D4×C17C2411yescharacteristic
166C22×C4×C17—21no—
167C22×C4×C17—22no—
168C22×C4×C17—24no—
169C22×C4×C17—22no—
170C22×C4×C17—24no—
171C22×C4×C17—24no—
172C22×C4×C17—24no—
173C22×C4×C17C2×D412yes—
174C22×C4×C17C2411yescharacteristic
175SD16×C17—28no—
176D8×C17—28no—
177C2×C8×C17C2414yes—
178C2×C8×C17C2414yes—
179C4⋊C4×C17—24no—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—24no—
182C4⋊C4×C17—24no—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—24no—
185C4⋊C4×C17—22no—
186C4⋊C4×C17—24no—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—24no—
189C4⋊C4×C17—22no—
190C4⋊C4×C17C2×D414yes—
191C4⋊C4×C17C2×D414yes—
192C4⋊C4×C17C2×D414yes—
193C4⋊C4×C17C2×D412yes—
194C4⋊C4×C17C2×D412yes—
195C4⋊C4×C17C2414yes—
196C4⋊C4×C17C2412yes—
197C4⋊C4×C17C2414yes—
198C22⋊C4×C17—21no—
199C22⋊C4×C17—22no—
200C22⋊C4×C17—24no—
201C22⋊C4×C17—24no—
202C22⋊C4×C17—24no—
203C22⋊C4×C17—24no—
204C22⋊C4×C17—24no—
205C22⋊C4×C17C2×D414yes—
206C42×C17—21no—
207C42×C17—22no—
208C42×C17—24no—
209C42×C17C2×D411yescharacteristic
210C42×C17C2×D411yescharacteristic
211C42×C17C2414yes—
212C42×C17C2412yes—
213#61876C1711yesmaximal, characteristic, direct product factor, Sylow
214C23×C17—21no—
215C23×C17—24no—
216C23×C17C22×D411yescharacteristic
217Q8×C17—21no—
218Q8×C17—44no—
219Q8×C17—84no—
220Q8×C17—82no—
221Q8×C172−1+412yes—
222D4×C17—24no—
223D4×C17—21no—
224D4×C17—24no—
225D4×C17—24no—
226D4×C17—44no—
227D4×C17—82no—
228D4×C17—84no—
229D4×C17—84no—
230D4×C17—84no—
231D4×C172−1+412yes—
232C2×C4×C17—24no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—42no—
240C2×C4×C17—48no—
241C2×C4×C17—82no—
242C2×C4×C17—84no—
243C2×C4×C17—84no—
244C2×C4×C17—84no—
245C2×C4×C17—82no—
246C2×C4×C17C22×D411yescharacteristic
247C2×C4×C17C22×D412yes—
248C2×C4×C17C22×D414yes—
249C2×C4×C17C22×D412yes—
250C2×C4×C17C22×D414yes—
251C2×C4×C17C2511yescharacteristic
252C8×C17—24no—
253C8×C17—24no—
254#3548C2×C1711yescharacteristic, complemented
255#3504C2×C1711yescharacteristic, complemented
256#3389C2×C1714yescomplemented
257#3388C2×C1714yescomplemented
258#3368C2×C1714yescomplemented
259#3366C2×C1714yescomplemented
260#3241C2×C1714yescomplemented
261#3234C2×C1712yescomplemented
262#3119C2×C1714yescomplemented
263#3115C2×C1712yescomplemented
264#2987C2×C1711yescharacteristic, complemented
265C22×C17—22no—
266C22×C17—21no—
267C22×C17—22no—
268C22×C17—42no—
269C22×C17—44no—
270C22×C17—44no—
271C22×C17—44no—
272C22×C17C23×D411yescharacteristic
273C4×C17—22no—
274C4×C17—22no—
275C4×C17—24no—
276C4×C17—24no—
277C4×C17—21no—
278C4×C17—22no—
279C4×C17—44no—
280C4×C17—44no—
281C4×C17—42no—
282C4×C17C2×2−1+411yescharacteristic
283C4×C17C2×2−1+411yescharacteristic
284#562—21no—
285#562—21no—
286#557C22×C1714yes—
287#554—24no—
288#554C22×C1714yes—
289#554C22×C1714yes—
290#550C22×C1714yes—
291#547—22no—
292#544—22no—
293#544C22×C1712yes—
294#540C22×C1714yes—
295#537—24no—
296#533—21no—
297#533C22×C1711yescharacteristic
298#532C22×C1712yes—
299C2×#125—21no—
300C2×#125C22×C1711yescharacteristic
301#520C22×C1712yes—
302C4×#41C22×C1711yescharacteristic
303#501C22×C1714yes—
304#501C22×C1714yes—
305#499C22×C1714yes—
306#490C22×C1714yes—
307#490C22×C1714yes—
308#486C22×C1714yes—
309#482C22×C1714yes—
310#481C22×C1714yes—
311#480C22×C1714yes—
312#479C22×C1714yes—
313#478C22×C1714yes—
314#478C22×C1714yes—
315#478C22×C1714yes—
316#478C22×C1714yes—
317#478C22×C1714yes—
318#476C22×C1714yes—
319#475C22×C1712yes—
320#474C22×C1714yes—
321#474C22×C1712yes—
322#460C22×C1712yes—
323#460C22×C1714yes—
324#459C22×C1712yes—
325#459C22×C1714yes—
326#449C22×C1714yes—
327#446C22×C1714yes—
328#446C22×C1714yes—
329#442C22×C1714yes—
330#442C22×C1714yes—
331#440C22×C1714yes—
332#440C22×C1714yes—
333C4×SD16C22×C1714yes—
334C4×D8C22×C1714yes—
335#433C22×C1714yes—
336#432C22×C1712yes—
337C2×C17—41no—
338C2×C17—42no—
339C2×C17—84no—
340C2×C17#348611yescharacteristic
341C2×C17#358512yes—
342C2×#41—24no—
343C2×#41—22no—
344C2×#41C23×C1711yescharacteristic
345C2×SD16C23×C1714yes—
346C2×D8C23×C1714yes—
347C4⋊Q8—24no—
348C4⋊Q8—22no—
349C4⋊Q8D4×C1712yes—
350C4⋊Q8C23×C1712yes—
351C4⋊Q8C23×C1714yes—
352#127C23×C1712yes—
353#127C23×C1714yes—
354#125—22no—
355#125—24no—
356#125—24no—
357#125—21no—
358#125—21no—
359#125D4×C1711yescharacteristic
360#125D4×C1711yescharacteristic
361#125C23×C1714yes—
362#125C23×C1714yes—
363#125C23×C1714yes—
364#124—22no—
365#123—24no—
366#123—24no—
367C22⋊Q8—24no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370C22⋊Q8—24no—
371C22⋊Q8—24no—
372C22⋊Q8—24no—
373C22⋊Q8—24no—
374C22⋊Q8D4×C1714yes—
375#121—22no—
376#121—24no—
377#121—24no—
378#121—24no—
379#121—24no—
380#121—24no—
381#121C23×C1714yes—
382C4×Q8—24no—
383C4×Q8C23×C1712yes—
384C4×Q8C23×C1714yes—
385C4×Q8C23×C1714yes—
386C4×D4—22no—
387C4×D4—24no—
388C4×D4—22no—
389C4×D4C23×C1712yes—
390C4×D4C23×C1714yes—
391C4×D4C23×C1714yes—
392C4×D4C23×C1714yes—
393C4×D4C23×C1714yes—
394#117—22no—
395#117—21no—
396#117C23×C1711yescharacteristic
397#117C23×C1712yes—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4C23×C1712yes—
403C2×C42—21no—
404C2×C42C23×C1711yescharacteristic
405#107C23×C1714yes—
406#107C23×C1714yes—
407#107C23×C1714yes—
408#106C23×C1714yes—
409#106C23×C1714yes—
410#106C23×C1714yes—
411C4⋊C8C23×C1712yes—
412C4⋊C8C23×C1712yes—
413C4⋊C8C23×C1714yes—
414C4⋊C8C23×C1714yes—
415#103C23×C1714yes—
416#102C23×C1714yes—
417#102C23×C1714yes—
418#102C23×C1714yes—
419#102C23×C1714yes—
420#102C23×C1714yes—
421#102C23×C1714yes—
422#102C23×C1714yes—
423C22⋊C8C23×C1714yes—
424#97C23×C1714yes—
425#97C23×C1714yes—
426C4×C8C23×C1714yes—
427C17#6187611yescharacteristic, direct product factor, Sylow
428#41—22no—
429#41—22no—
430#41—84no—
431#41—88no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8C2×D4×C1714yes—
435C2×Q8C24×C1711yescharacteristic
436C2×D4—21no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4C24×C1714yes—
443C2×D4C24×C1711yescharacteristic
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4C2×D4×C1712yes—
452C22×C4C24×C1711yescharacteristic
453SD16—28no—
454D8—28no—
455C2×C8C24×C1714yes—
456C2×C8C24×C1714yes—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4C2×D4×C1712yes—
469C4⋊C4C2×D4×C1714yes—
470C4⋊C4C2×D4×C1714yes—
471C4⋊C4C2×D4×C1712yes—
472C4⋊C4C2×D4×C1714yes—
473C4⋊C4C24×C1714yes—
474C4⋊C4C24×C1712yes—
475C4⋊C4C24×C1714yes—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4C2×D4×C1714yes—
484C42—21no—
485C42—22no—
486C42—24no—
487C42C2×D4×C1711yescharacteristic
488C42C2×D4×C1711yescharacteristic
489C42C24×C1714yes—
490C42C24×C1712yes—
491C23—21no—
492C23—24no—
493C23C22×D4×C1711yescharacteristic
494Q8—21no—
495Q8—44no—
496Q8—82no—
497Q8—84no—
498Q8C17×2−1+412yes—
499D4—24no—
500D4—21no—
501D4—24no—
502D4—24no—
503D4—44no—
504D4—82no—
505D4—84no—
506D4—84no—
507D4—84no—
508D4C17×2−1+412yes—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4C22×D4×C1711yescharacteristic
524C2×C4C22×D4×C1712yes—
525C2×C4C22×D4×C1714yes—
526C2×C4C22×D4×C1712yes—
527C2×C4C22×D4×C1714yes—
528C2×C4C25×C1711yescharacteristic
529C8—24no—
530C8—24no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22C23×D4×C1711yescharacteristic
539C4—21no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—24no—
544C4—24no—
545C4—44no—
546C4—44no—
547C4—42no—
548C4C2×C17×2−1+411yescharacteristic
549C4C2×C17×2−1+411yescharacteristic
550C2—41no—
551C2—42no—
552C2—84no—
553C2C17×#348611yescharacteristic
554C2C17×#358512yes—
subgroupG/H
center, socleC22×C17C23×D4
commutator, Frattini, cosocleC2×C4C25×C17
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C17×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#3548C2semidirect
2. C17×#3504C2semidirect
3. C17×#3389C2semidirect
4. C17×#3388C2semidirect
5. C17×#3368C2semidirect
6. C17×#3366C2semidirect
7. C17×#3241C2semidirect
8. C17×#3234C2semidirect
9. C17×#3119C2semidirect
10. C17×#3115C2semidirect
11. C17×#2987C2semidirect
213. #61876C17direct (N × H)
254. #3548C2×C17semidirect
255. #3504C2×C17semidirect
256. #3389C2×C17semidirect
257. #3388C2×C17semidirect
258. #3368C2×C17semidirect
259. #3366C2×C17semidirect
260. #3241C2×C17semidirect
261. #3234C2×C17semidirect
262. #3119C2×C17semidirect
263. #3115C2×C17semidirect
264. #2987C2×C17semidirect
427. C17#61876direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#35482
C17×#35042
C17×#33892
C17×#33882
C17×#33682
C17×#33662
C17×#32412
C17×#32342
C17×#31192
C17×#31152
C17×#29872
#6187617