Finite Group Databasebeta

← C17×#25515 #16883259

Subgroups — C17×#25515 #16883259

Order 4352.

1076 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3492C211yesmaximal, characteristic
2C17×#3056C212yesmaximal, complemented
3C17×#2503C211yesmaximal, characteristic, complemented
4C17×#2061C211yesmaximal, characteristic, complemented
5C17×#2040C211yesmaximal, characteristic, complemented
6C17×#2028C211yesmaximal, characteristic, complemented
7C17×#2007C211yesmaximal, characteristic, complemented
8C17×#1882C212yesmaximal, complemented
9C17×#1871C212yesmaximal, complemented
10C17×#1860C212yesmaximal, complemented
11C2×C17×#343C211yesmaximal, characteristic, complemented
12C2×C17×#137C2212yes—
13C22×M16×C17C2212yes—
14C17×#554—22no—
15C17×#553—22no—
16C17×#552—22no—
17C17×#547—22no—
18C17×#544C2211yescharacteristic
19C17×#539—22no—
20C17×#537C2211yescharacteristic
21C17×#525—22no—
22C17×#525C2212yes—
23C2×C17×C22⋊Q8C2211yescharacteristic
24C17×#520C2212yes—
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#470—22no—
27C17×#468—22no—
28C17×#467—22no—
29C17×#465—22no—
30C17×#428C2212yes—
31C2×C17×#104C2212yes—
32C2×C17×#104C2212yes—
33C17×#419C2212yes—
34C17×#417C2212yes—
35C17×#400—22no—
36C17×#391C2211yescharacteristic
37C17×#391C2211yescharacteristic
38C17×#382C2211yescharacteristic
39C17×#382C2211yescharacteristic
40C17×#381—22no—
41C4×C22⋊C4×C17C2211yescharacteristic
42C17×#343C2211yescharacteristic
43C17×#343C2211yescharacteristic
44C17×#343C2212yes—
45C17×#339C2212yes—
46C17×#339C2212yes—
47C2×#41×C17—22no—
48C2×#41×C17C2312yes—
49C22×Q8×C17C2311yescharacteristic
50C23×C4×C17D412yes—
51C23×C4×C17C2311yescharacteristic
52C17×#137—24no—
53C17×#137—42no—
54C2×Q16×C17—22no—
55C2×SD16×C17—22no—
56C2×M16×C17—22no—
57C2×M16×C17—24no—
58C2×M16×C17—42no—
59C2×M16×C17C2312yes—
60C2×M16×C17C2312yes—
61C22×C8×C17—22no—
62C17×C4⋊Q8—22no—
63C17×#126—24no—
64C17×#125—24no—
65C17×#125—22no—
66C17×#123—24no—
67C17×#123—24no—
68C17×#123—22no—
69C17×C22⋊Q8—22no—
70C17×C22⋊Q8—24no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8—22no—
74C17×C22⋊Q8—22no—
75C17×C22⋊Q8D412yes—
76C17×C22⋊Q8D412yes—
77C17×#121—22no—
78C17×#121—22no—
79C17×#121—22no—
80C17×#121D412yes—
81C17×#121D412yes—
82C17×#120—22no—
83C4×Q8×C17—22no—
84C4×D4×C17—24no—
85C4×D4×C17—22no—
86C4×D4×C17—24no—
87C4×D4×C17—22no—
88C4×D4×C17—22no—
89C17×#117—22no—
90C17×#117—22no—
91C17×#117C2312yes—
92C2×C4⋊C4×C17—22no—
93C2×C4⋊C4×C17—22no—
94C2×C4⋊C4×C17—22no—
95C2×C4⋊C4×C17—21no—
96C2×C4⋊C4×C17D412yes—
97C2×C4⋊C4×C17C2311yescharacteristic
98C2×C4⋊C4×C17C2311yescharacteristic
99C2×C4⋊C4×C17C2311yescharacteristic
100C2×C22⋊C4×C17—21no—
101C2×C22⋊C4×C17—21no—
102C2×C42×C17C2311yescharacteristic
103C2×C42×C17C2311yescharacteristic
104C17×#107—22no—
105C17×#106—22no—
106C17×#104—24no—
107C17×#104—42no—
108C17×#103—22no—
109C17×#103—22no—
110C17×#103—22no—
111C17×#102—22no—
112C17×#95—21no—
113C17×#95—22no—
114C17×#95—22no—
115C17×#95—22no—
116C17×#95—22no—
117C24×C17C2×D411yescharacteristic
118#41×C17—22no—
119#41×C17—22no—
120#41×C17—22no—
121#41×C17—22no—
122#41×C17—84no—
123C2×Q8×C17—22no—
124C2×Q8×C17—22no—
125C2×Q8×C17—22no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17C2×D412yes—
129C2×D4×C17—22no—
130C2×D4×C17—22no—
131C2×D4×C17—22no—
132C2×D4×C17—22no—
133C2×D4×C17C2×D412yes—
134C22×C4×C17—22no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—22no—
138C22×C4×C17—24no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—44no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D412yes—
148C22×C4×C17C2×D412yes—
149C22×C4×C17C2×D412yes—
150C22×C4×C17C2×D412yes—
151C22×C4×C17C2411yescharacteristic
152Q16×C17—44no—
153SD16×C17—44no—
154M16×C17—22no—
155M16×C17—22no—
156M16×C17—22no—
157M16×C17—44no—
158M16×C17#4114yescomplemented
159C2×C8×C17—22no—
160C2×C8×C17—22no—
161C2×C8×C17—44no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—24no—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17—24no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—42no—
173C4⋊C4×C17C2×D412yes—
174C4⋊C4×C17C2×D412yes—
175C22⋊C4×C17—24no—
176C22⋊C4×C17—24no—
177C22⋊C4×C17—22no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—22no—
180C22⋊C4×C17—22no—
181C22⋊C4×C17—22no—
182C22⋊C4×C17—42no—
183C22⋊C4×C17—41no—
184C22⋊C4×C17—41no—
185C42×C17—24no—
186C42×C17—22no—
187C42×C17—22no—
188C42×C17—21no—
189C42×C17—21no—
190#25515C1711yesmaximal, characteristic, direct product factor, Sylow
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—22no—
194C23×C17—22no—
195C23×C17—41no—
196C23×C17#12011yescharacteristic
197C23×C17#12011yescharacteristic
198C23×C17C22×D411yescharacteristic
199Q8×C17—22no—
200Q8×C17—22no—
201Q8×C17—42no—
202Q8×C17—42no—
203Q8×C17—82no—
204D4×C17—22no—
205D4×C17—22no—
206D4×C17—44no—
207D4×C17—42no—
208D4×C17—42no—
209D4×C17—44no—
210D4×C17—82no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—44no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—42no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—82no—
239C2×C4×C17—84no—
240C2×C4×C17#12012yes—
241C2×C4×C17C22×D412yes—
242C2×C4×C17C2×#4112yes—
243C2×C4×C17H(5,𝔽2)11yescharacteristic
244C2×C4×C172−1+411yescharacteristic
245C8×C17—24no—
246C8×C17—42no—
247#3492C2×C1711yescharacteristic
248#3056C2×C1712yescomplemented
249#2503C2×C1711yescharacteristic, complemented
250#2061C2×C1711yescharacteristic, complemented
251#2040C2×C1711yescharacteristic, complemented
252#2028C2×C1711yescharacteristic, complemented
253#2007C2×C1711yescharacteristic, complemented
254#1882C2×C1712yescomplemented
255#1871C2×C1712yescomplemented
256#1860C2×C1712yescomplemented
257C2×#343C2×C1711yescharacteristic, complemented
258C22×C17—21no—
259C22×C17—21no—
260C22×C17—21no—
261C22×C17—21no—
262C22×C17—21no—
263C22×C17—21no—
264C22×C17—21no—
265C22×C17—42no—
266C22×C17—42no—
267C22×C17—42no—
268C22×C17—42no—
269C22×C17—42no—
270C22×C17#45512yes—
271C22×C17C2×#12011yescharacteristic
272C22×C17#53211yescharacteristic
273C22×C17#53711yescharacteristic
274C4×C17—22no—
275C4×C17—42no—
276C4×C17—42no—
277C4×C17—42no—
278C4×C17—42no—
279C4×C17—42no—
280C4×C17—81no—
281C4×C17—81no—
282C4×C17—81no—
283C4×C17—81no—
284C4×C17#54612yes—
285C4×C17#54712yes—
286C2×#137C22×C1712yes—
287C22×M16C22×C1712yes—
288#554—22no—
289#553—22no—
290#552—22no—
291#547—22no—
292#544C22×C1711yescharacteristic
293#539—22no—
294#537C22×C1711yescharacteristic
295#525—22no—
296#525C22×C1712yes—
297C2×C22⋊Q8C22×C1711yescharacteristic
298#520C22×C1712yes—
299C22×C4⋊C4C22×C1711yescharacteristic
300#470—22no—
301#468—22no—
302#467—22no—
303#465—22no—
304#428C22×C1712yes—
305C2×#104C22×C1712yes—
306C2×#104C22×C1712yes—
307#419C22×C1712yes—
308#417C22×C1712yes—
309#400—22no—
310#391C22×C1711yescharacteristic
311#391C22×C1711yescharacteristic
312#382C22×C1711yescharacteristic
313#382C22×C1711yescharacteristic
314#381—22no—
315C4×C22⋊C4C22×C1711yescharacteristic
316#343C22×C1711yescharacteristic
317#343C22×C1712yes—
318#343C22×C1711yescharacteristic
319#339C22×C1712yes—
320#339C22×C1712yes—
321C2×C17—21no—
322C2×C17—21no—
323C2×C17—22no—
324C2×C17—41no—
325C2×C17—82no—
326C2×C17#243011yescharacteristic
327C2×C17C2×#45511yescharacteristic
328C2×C17#302011yescharacteristic
329C2×#41—22no—
330C2×#41C23×C1712yes—
331C22×Q8C23×C1711yescharacteristic
332C23×C4D4×C1712yes—
333C23×C4C23×C1711yescharacteristic
334#137—24no—
335#137—42no—
336C2×Q16—22no—
337C2×SD16—22no—
338C2×M16—22no—
339C2×M16—24no—
340C2×M16—42no—
341C2×M16C23×C1712yes—
342C2×M16C23×C1712yes—
343C22×C8—22no—
344C4⋊Q8—22no—
345#126—24no—
346#125—24no—
347#125—22no—
348#123—24no—
349#123—24no—
350#123—22no—
351C22⋊Q8—22no—
352C22⋊Q8—24no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8D4×C1712yes—
358C22⋊Q8D4×C1712yes—
359#121—22no—
360#121—22no—
361#121—22no—
362#121D4×C1712yes—
363#121D4×C1712yes—
364#120—22no—
365C4×Q8—22no—
366C4×D4—24no—
367C4×D4—22no—
368C4×D4—24no—
369C4×D4—22no—
370C4×D4—22no—
371#117—22no—
372#117—22no—
373#117C23×C1712yes—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C1712yes—
379C2×C4⋊C4C23×C1711yescharacteristic
380C2×C4⋊C4C23×C1711yescharacteristic
381C2×C4⋊C4C23×C1711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C42C23×C1711yescharacteristic
385C2×C42C23×C1711yescharacteristic
386#107—22no—
387#106—22no—
388#104—24no—
389#104—42no—
390#103—22no—
391#103—22no—
392#103—22no—
393#102—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399C17#2551511yescharacteristic, direct product factor, Sylow
400C24C2×D4×C1711yescharacteristic
401#41—22no—
402#41—22no—
403#41—22no—
404#41—22no—
405#41—84no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×D4×C1712yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4C2×D4×C1712yes—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—44no—
429C22×C4C2×D4×C1712yes—
430C22×C4C2×D4×C1712yes—
431C22×C4C2×D4×C1711yescharacteristic
432C22×C4C2×D4×C1712yes—
433C22×C4C2×D4×C1712yes—
434C22×C4C24×C1711yescharacteristic
435Q16—44no—
436SD16—44no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—44no—
441M16#41×C1714yescomplemented
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—44no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4C2×D4×C1712yes—
457C4⋊C4C2×D4×C1712yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—42no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—21no—
472C42—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23C17×#12011yescharacteristic
479C23C17×#12011yescharacteristic
480C23C22×D4×C1711yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—42no—
484Q8—42no—
485Q8—82no—
486D4—22no—
487D4—22no—
488D4—44no—
489D4—42no—
490D4—42no—
491D4—44no—
492D4—82no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4C17×#12012yes—
523C2×C4C22×D4×C1712yes—
524C2×C4C2×#41×C1712yes—
525C2×C4C17×H(5,𝔽2)11yescharacteristic
526C2×C4C17×2−1+411yescharacteristic
527C8—24no—
528C8—42no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22C17×#45512yes—
542C22C2×C17×#12011yescharacteristic
543C22C17×#53211yescharacteristic
544C22C17×#53711yescharacteristic
545C4—22no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4C17×#54612yes—
556C4C17×#54712yes—
557C2—21no—
558C2—21no—
559C2—22no—
560C2—41no—
561C2—82no—
562C2C17×#243011yescharacteristic
563C2C2×C17×#45511yescharacteristic
564C2C17×#302011yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#3492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#3056C2semidirect
3. C17×#2503C2semidirect
4. C17×#2061C2semidirect
5. C17×#2040C2semidirect
6. C17×#2028C2semidirect
7. C17×#2007C2semidirect
8. C17×#1882C2semidirect
9. C17×#1871C2semidirect
10. C17×#1860C2semidirect
11. C2×C17×#343C2semidirect
158. M16×C17#41semidirect
190. #25515C17direct (N × H)
248. #3056C2×C17semidirect
249. #2503C2×C17semidirect
250. #2061C2×C17semidirect
251. #2040C2×C17semidirect
252. #2028C2×C17semidirect
253. #2007C2×C17semidirect
254. #1882C2×C17semidirect
255. #1871C2×C17semidirect
256. #1860C2×C17semidirect
257. C2×#343C2×C17semidirect
399. C17#25515direct (N × H)
441. M16#41×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#34922
C17×#30562
C17×#25032
C17×#20612
C17×#20402
C17×#20282
C17×#20072
C17×#18822
C17×#18712
C17×#18602
C2×C17×#3432
#2551517