Finite Group Databasebeta

← C17×#18366 #16876110

Subgroups — C17×#18366 #16876110

Order 4352.

1048 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3493C211yesmaximal, characteristic, complemented
2C17×#3217C212yesmaximal, complemented
3C17×#2619C211yesmaximal, characteristic, complemented
4C17×#1600C214yesmaximal, complemented
5C17×#1599C212yesmaximal, complemented
6C17×#1598C212yesmaximal, complemented
7C17×#1512C212yesmaximal, complemented
8C17×#1464C211yesmaximal, characteristic, complemented
9C17×#556—22no—
10C17×#555—22no—
11C17×#553—22no—
12C17×#552—22no—
13C17×#544C2211yescharacteristic
14C17×#538—21no—
15C17×#538C2212yes—
16C17×#537C2211yescharacteristic
17C2×C17×#127C2211yescharacteristic
18C17×#529—21no—
19C17×#529C2212yes—
20C17×#525—21no—
21C2×C17×#121—21no—
22C17×#520—21no—
23C2×C17×#117C2211yescharacteristic
24C17×#496—22no—
25C17×#490—22no—
26C17×#460—22no—
27C17×#459—22no—
28C17×#450C2214yes—
29C17×#447C2214yes—
30C17×#432C2212yes—
31C17×#394—21no—
32C17×#394C2212yes—
33C17×#392—21no—
34C17×#392C2212yes—
35C17×#390—21no—
36C17×#382C2211yescharacteristic
37C17×#328C2214yes—
38C17×#327C2214yes—
39C17×#324C2212yes—
40C17×#323C2212yes—
41C2×#41×C17—21no—
42C22×D4×C17—22no—
43C22×D4×C17—21no—
44C22×D4×C17—21no—
45C22×D4×C17C2312yes—
46C23×C4×C17D411yescharacteristic
47C2×SD16×C17—24no—
48C2×D8×C17—24no—
49C17×C4⋊Q8—22no—
50C17×#127—22no—
51C17×#127—22no—
52C17×#126—22no—
53C17×#126—22no—
54C17×#126—22no—
55C17×#126—22no—
56C17×#125—21no—
57C17×#125—22no—
58C17×#125D412yescomplemented
59C17×#124—21no—
60C17×#124—22no—
61C17×#124—22no—
62C17×#124D412yescomplemented
63C17×#123—22no—
64C17×#123—22no—
65C17×#123—22no—
66C17×C22⋊Q8—21no—
67C17×C22⋊Q8—21no—
68C17×C22⋊Q8—21no—
69C17×C22⋊Q8—22no—
70C17×C22⋊Q8C2312yes—
71C17×#121—21no—
72C17×#121—21no—
73C17×#121—22no—
74C17×#121—22no—
75C17×#121—22no—
76C17×#121—21no—
77C17×#121—44no—
78C17×#121C2312yes—
79C17×#120—22no—
80C4×Q8×C17—22no—
81C4×Q8×C17—21no—
82C4×Q8×C17D412yescomplemented
83C4×D4×C17—22no—
84C4×D4×C17—22no—
85C4×D4×C17—22no—
86C4×D4×C17—21no—
87C4×D4×C17—22no—
88C4×D4×C17—22no—
89C4×D4×C17D412yescomplemented
90C17×#117—22no—
91C17×#117—22no—
92C17×#117—21no—
93C17×#117—22no—
94C17×#117—22no—
95C17×#117—22no—
96C2×C4⋊C4×C17—22no—
97C2×C4⋊C4×C17—21no—
98C2×C4⋊C4×C17D411yescharacteristic
99C2×C4⋊C4×C17C2311yescharacteristic
100C2×C22⋊C4×C17—22no—
101C2×C22⋊C4×C17—22no—
102C2×C22⋊C4×C17—22no—
103C2×C22⋊C4×C17D411yescharacteristic
104C2×C22⋊C4×C17C2311yescharacteristic
105C2×C42×C17D411yescharacteristic
106C2×C42×C17C2311yescharacteristic
107C17×C4⋊C8—22no—
108C17×#102—24no—
109C17×#102—24no—
110C17×C22⋊C8C2314yes—
111C17×#97—22no—
112C17×#95—21no—
113C17×#95C2312yes—
114C24×C17—22no—
115C24×C17C2×D411yescharacteristic
116#41×C17—42no—
117#41×C17—42no—
118C2×Q8×C17—21no—
119C2×Q8×C17—21no—
120C2×Q8×C17C2×D412yes—
121C2×D4×C17—21no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—22no—
125C2×D4×C17—21no—
126C2×D4×C17—24no—
127C2×D4×C17—42no—
128C2×D4×C17—42no—
129C2×D4×C17—44no—
130C2×D4×C17—41no—
131C2×D4×C17—41no—
132C2×D4×C17—42no—
133C2×D4×C17—82no—
134C2×D4×C17—82no—
135C2×D4×C17—82no—
136C2×D4×C17—82no—
137C2×D4×C17C2×D412yes—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—22no—
143C22×C4×C17—22no—
144C22×C4×C17—22no—
145C22×C4×C17—41no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2411yescharacteristic
152SD16×C17—84no—
153D8×C17—84no—
154C2×C8×C17—24no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—22no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17—22no—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17C2×D412yes—
169C4⋊C4×C17C2×D412yes—
170C22⋊C4×C17—22no—
171C22⋊C4×C17—22no—
172C22⋊C4×C17—22no—
173C22⋊C4×C17—22no—
174C22⋊C4×C17—22no—
175C22⋊C4×C17—22no—
176C22⋊C4×C17—22no—
177C22⋊C4×C17—22no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—22no—
180C22⋊C4×C17—22no—
181C22⋊C4×C17—44no—
182C22⋊C4×C17—82no—
183C22⋊C4×C17—82no—
184C42×C17—22no—
185C42×C17—22no—
186C42×C17—21no—
187C42×C17—22no—
188C42×C17—21no—
189C42×C17C2×D412yes—
190C42×C17C2×D412yes—
191#18366C1711yesmaximal, characteristic, direct product factor, Sylow
192C23×C17—21no—
193C23×C17—22no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—42no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17C22×D411yescharacteristic
206Q8×C17—42no—
207Q8×C17—42no—
208Q8×C17—81no—
209D4×C17—42no—
210D4×C17—42no—
211D4×C17—44no—
212D4×C17—44no—
213D4×C17—42no—
214D4×C17—42no—
215D4×C17—82no—
216D4×C17—82no—
217D4×C17—84no—
218D4×C17—82no—
219D4×C17—81no—
220D4×C17—82no—
221D4×C17—88no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—44no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—82no—
242C2×C4×C17—82no—
243C2×C4×C17#12011yescharacteristic
244C2×C4×C17#12011yescharacteristic
245C2×C4×C17#12012yes—
246C2×C4×C17AGL(1,ℤ/8)12yes—
247C2×C4×C17C22×D412yes—
248C8×C17—44no—
249#3493C2×C1711yescharacteristic, complemented
250#3217C2×C1712yescomplemented
251#2619C2×C1711yescharacteristic, complemented
252#1600C2×C1714yescomplemented
253#1599C2×C1712yescomplemented
254#1598C2×C1712yescomplemented
255#1512C2×C1712yescomplemented
256#1464C2×C1711yescharacteristic, complemented
257C22×C17—21no—
258C22×C17—21no—
259C22×C17—42no—
260C22×C17—41no—
261C22×C17—42no—
262C22×C17—42no—
263C22×C17—41no—
264C22×C17—42no—
265C22×C17—41no—
266C22×C17—81no—
267C22×C17—82no—
268C22×C17—82no—
269C22×C17—82no—
270C22×C17—82no—
271C22×C17—82no—
272C22×C17—82no—
273C22×C17—82no—
274C22×C17C2×#12011yescharacteristic
275C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C17#57511yescharacteristic
277C4×C17—22no—
278C4×C17—21no—
279C4×C17—21no—
280C4×C17—22no—
281C4×C17—22no—
282C4×C17—42no—
283C4×C17—41no—
284C4×C17—42no—
285C4×C17—41no—
286C4×C17—42no—
287C4×C17—82no—
288#556—22no—
289#555—22no—
290#553—22no—
291#552—22no—
292#544C22×C1711yescharacteristic
293#538—21no—
294#538C22×C1712yes—
295#537C22×C1711yescharacteristic
296C2×#127C22×C1711yescharacteristic
297#529—21no—
298#529C22×C1712yes—
299#525—21no—
300C2×#121—21no—
301#520—21no—
302C2×#117C22×C1711yescharacteristic
303#496—22no—
304#490—22no—
305#460—22no—
306#459—22no—
307#450C22×C1714yes—
308#447C22×C1714yes—
309#432C22×C1712yes—
310#394—21no—
311#394C22×C1712yes—
312#392—21no—
313#392C22×C1712yes—
314#390—21no—
315#382C22×C1711yescharacteristic
316#328C22×C1714yes—
317#327C22×C1714yes—
318#324C22×C1712yes—
319#323C22×C1712yes—
320C2×C17—21no—
321C2×C17—21no—
322C2×C17—82no—
323C2×C17—81no—
324C2×C17—162no—
325C2×C17#300611yescharacteristic
326C2×C17#302111yescharacteristic
327C2×C17#302711yescharacteristic
328C2×#41—21no—
329C22×D4—22no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4C23×C1712yes—
333C23×C4D4×C1711yescharacteristic
334C2×SD16—24no—
335C2×D8—24no—
336C4⋊Q8—22no—
337#127—22no—
338#127—22no—
339#126—22no—
340#126—22no—
341#126—22no—
342#126—22no—
343#125—21no—
344#125—22no—
345#125D4×C1712yescomplemented
346#124—21no—
347#124—22no—
348#124—22no—
349#124D4×C1712yescomplemented
350#123—22no—
351#123—22no—
352#123—22no—
353C22⋊Q8—21no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8C23×C1712yes—
358#121—21no—
359#121—21no—
360#121—22no—
361#121—22no—
362#121—22no—
363#121—21no—
364#121—44no—
365#121C23×C1712yes—
366#120—22no—
367C4×Q8—22no—
368C4×Q8—21no—
369C4×Q8D4×C1712yescomplemented
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4D4×C1712yescomplemented
377#117—22no—
378#117—22no—
379#117—21no—
380#117—22no—
381#117—22no—
382#117—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4D4×C1711yescharacteristic
386C2×C4⋊C4C23×C1711yescharacteristic
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4D4×C1711yescharacteristic
391C2×C22⋊C4C23×C1711yescharacteristic
392C2×C42D4×C1711yescharacteristic
393C2×C42C23×C1711yescharacteristic
394C4⋊C8—22no—
395#102—24no—
396#102—24no—
397C22⋊C8C23×C1714yes—
398#97—22no—
399#95—21no—
400#95C23×C1712yes—
401C17#1836611yescharacteristic, direct product factor, Sylow
402C24—22no—
403C24C2×D4×C1711yescharacteristic
404#41—42no—
405#41—42no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×D4×C1712yes—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—24no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—42no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×D4×C1712yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4C2×D4×C1711yescharacteristic
435C22×C4C2×D4×C1711yescharacteristic
436C22×C4C2×D4×C1711yescharacteristic
437C22×C4C2×D4×C1711yescharacteristic
438C22×C4C2×D4×C1711yescharacteristic
439C22×C4C24×C1711yescharacteristic
440SD16—84no—
441D8—84no—
442C2×C8—24no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4C2×D4×C1712yes—
457C4⋊C4C2×D4×C1712yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—44no—
470C22⋊C4—82no—
471C22⋊C4—82no—
472C42—22no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—22no—
477C42C2×D4×C1712yes—
478C42C2×D4×C1712yes—
479C23—21no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C22×D4×C1711yescharacteristic
493Q8—42no—
494Q8—42no—
495Q8—81no—
496D4—42no—
497D4—42no—
498D4—44no—
499D4—44no—
500D4—42no—
501D4—42no—
502D4—82no—
503D4—82no—
504D4—84no—
505D4—82no—
506D4—81no—
507D4—82no—
508D4—88no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4C17×#12011yescharacteristic
531C2×C4C17×#12011yescharacteristic
532C2×C4C17×#12012yes—
533C2×C4C17×AGL(1,ℤ/8)12yes—
534C2×C4C22×D4×C1712yes—
535C8—44no—
536C22—21no—
537C22—21no—
538C22—42no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22C2×C17×#12011yescharacteristic
554C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
555C22C17×#57511yescharacteristic
556C4—22no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—22no—
561C4—42no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—82no—
567C2—21no—
568C2—21no—
569C2—81no—
570C2—82no—
571C2—162no—
572C2C17×#300611yescharacteristic
573C2C17×#302111yescharacteristic
574C2C17×#302711yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
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
1. C17×#3493C2semidirect
2. C17×#3217C2semidirect
3. C17×#2619C2semidirect
4. C17×#1600C2semidirect
5. C17×#1599C2semidirect
6. C17×#1598C2semidirect
7. C17×#1512C2semidirect
8. C17×#1464C2semidirect
58. C17×#125D4semidirect
62. C17×#124D4semidirect
82. C4×Q8×C17D4semidirect
89. C4×D4×C17D4semidirect
191. #18366C17direct (N × H)
249. #3493C2×C17semidirect
250. #3217C2×C17semidirect
251. #2619C2×C17semidirect
252. #1600C2×C17semidirect
253. #1599C2×C17semidirect
254. #1598C2×C17semidirect
255. #1512C2×C17semidirect
256. #1464C2×C17semidirect
345. #125D4×C17semidirect
349. #124D4×C17semidirect
369. C4×Q8D4×C17semidirect
376. C4×D4D4×C17semidirect
401. C17#18366direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34932
C17×#32172
C17×#26192
C17×#16002
C17×#15992
C17×#15982
C17×#15122
C17×#14642
#1836617