Finite Group Databasebeta

← C17×#18441 #16876185

Subgroups — C17×#18441 #16876185

Order 4352.

888 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3559C211yesmaximal, characteristic
2C17×#3239C211yesmaximal, characteristic, complemented
3C17×#3227C211yesmaximal, characteristic, complemented
4C17×#2642C211yesmaximal, characteristic, complemented
5C17×#1616C214yesmaximal, complemented
6C17×#1615C214yesmaximal, complemented
7C17×#1505C211yesmaximal, characteristic, complemented
8C17×#1500C211yesmaximal, characteristic, complemented
9C17×#1470C211yesmaximal, characteristic, complemented
10C17×#563—21no—
11C17×#562—21no—
12C17×#554—21no—
13C17×#553—21no—
14C17×#553—21no—
15C17×#547—22no—
16C17×#547—21no—
17C17×#544—21no—
18C17×#540—21no—
19C17×#539—22no—
20C17×#539C2212yes—
21C17×#537C2212yes—
22C17×#532C2211yescharacteristic
23C2×C17×C4⋊Q8C2211yescharacteristic
24C17×#529C2211yescharacteristic
25C2×C17×#125C2211yescharacteristic
26C17×#501—21no—
27C17×#498—21no—
28C17×#489—21no—
29C17×#486—21no—
30C17×#484C2212yes—
31C17×#481C2214yes—
32C17×#481C2212yes—
33C17×#476—22no—
34C17×#475—21no—
35C17×#474—21no—
36C17×#432C2211yescharacteristic
37C17×#431C2211yescharacteristic
38C17×#400—22no—
39C17×#400C2214yes—
40C17×#391—21no—
41C17×#391—21no—
42C17×#376C2211yescharacteristic
43C17×#328C2212yes—
44C17×#327C2212yes—
45C17×#327C2214yes—
46C17×#324C2214yes—
47C2×#41×C17—21no—
48C2×#41×C17—22no—
49C22×Q8×C17—21no—
50C17×C4⋊Q8—21no—
51C17×C4⋊Q8—22no—
52C17×C4⋊Q8—21no—
53C17×C4⋊Q8—21no—
54C17×C4⋊Q8—21no—
55C17×C4⋊Q8—21no—
56C17×C4⋊Q8D411yescharacteristic
57C17×#127D411yescharacteristic
58C17×#126—22no—
59C17×#126—22no—
60C17×#126—21no—
61C17×#125—21no—
62C17×#125—22no—
63C17×#125—21no—
64C17×#125—21no—
65C17×#125—21no—
66C17×#125—21no—
67C17×#125D411yescharacteristic
68C17×#124—21no—
69C17×#124D411yescharacteristic
70C17×#123—22no—
71C17×#123—22no—
72C17×#123—22no—
73C17×#123—22no—
74C17×#123—22no—
75C17×#123—21no—
76C17×#123—22no—
77C17×C22⋊Q8—22no—
78C17×C22⋊Q8—22no—
79C17×C22⋊Q8—22no—
80C17×C22⋊Q8—22no—
81C17×C22⋊Q8—22no—
82C17×C22⋊Q8—22no—
83C17×C22⋊Q8C2311yescharacteristic
84C17×#121—21no—
85C17×#121—22no—
86C17×#121—21no—
87C17×#121—22no—
88C17×#121—22no—
89C17×#121—21no—
90C17×#121—22no—
91C17×#121C2311yescharacteristic
92C17×#121C2312yes—
93C4×Q8×C17D411yescharacteristic
94C4×D4×C17—22no—
95C4×D4×C17—21no—
96C4×D4×C17—21no—
97C4×D4×C17—21no—
98C4×D4×C17—22no—
99C4×D4×C17—21no—
100C4×D4×C17D411yescharacteristic
101C4×D4×C17D412yes—
102C17×#117—21no—
103C17×#117—21no—
104C2×C4⋊C4×C17—22no—
105C2×C4⋊C4×C17—21no—
106C2×C4⋊C4×C17—21no—
107C2×C4⋊C4×C17D411yescharacteristic
108C2×C4⋊C4×C17D412yes—
109C2×C4⋊C4×C17D411yescharacteristic
110C2×C4⋊C4×C17C2312yes—
111C2×C4⋊C4×C17C2312yes—
112C2×C42×C17—21no—
113C2×C42×C17C2311yescharacteristic
114C17×#107—22no—
115C17×#106—24no—
116C17×#106—22no—
117C17×C4⋊C8—21no—
118C17×C4⋊C8—21no—
119C17×#103—22no—
120C17×#102—22no—
121C17×#102—24no—
122C17×C22⋊C8C2312yes—
123C17×C22⋊C8C2312yes—
124C17×#97—21no—
125C4×C8×C17—21no—
126C17×#95—21no—
127C17×#95—24no—
128C17×#95—24no—
129C17×#95C2312yes—
130#41×C17—82no—
131#41×C17—84no—
132C2×Q8×C17—22no—
133C2×Q8×C17—21no—
134C2×Q8×C17—44no—
135C2×Q8×C17—41no—
136C2×Q8×C17—41no—
137C2×Q8×C17C2×D411yescharacteristic
138C2×D4×C17—21no—
139C2×D4×C17—22no—
140C2×D4×C17—22no—
141C2×D4×C17—22no—
142C2×D4×C17—21no—
143C2×D4×C17—21no—
144C2×D4×C17C2×D412yes—
145C2×D4×C17C2×D411yescharacteristic
146C22×C4×C17—22no—
147C22×C4×C17—22no—
148C22×C4×C17—22no—
149C22×C4×C17—21no—
150C22×C4×C17—22no—
151C22×C4×C17—22no—
152C22×C4×C17—22no—
153C22×C4×C17C2×D412yes—
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D412yes—
157C22×C4×C17C2411yescharacteristic
158C2×C8×C17—22no—
159C2×C8×C17—22no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—22no—
179C4⋊C4×C17—41no—
180C4⋊C4×C17—41no—
181C4⋊C4×C17C2×D411yescharacteristic
182C4⋊C4×C17C2×D412yes—
183C4⋊C4×C17C2×D411yescharacteristic
184C22⋊C4×C17—22no—
185C22⋊C4×C17—22no—
186C22⋊C4×C17—22no—
187C22⋊C4×C17—22no—
188C22⋊C4×C17—22no—
189C22⋊C4×C17—21no—
190C22⋊C4×C17—21no—
191C22⋊C4×C17—22no—
192C22⋊C4×C17—21no—
193C22⋊C4×C17—21no—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—21no—
196C42×C17—22no—
197C42×C17—21no—
198C42×C17—42no—
199C42×C17C2×D411yescharacteristic
200C42×C17C2×D411yescharacteristic
201C42×C17C2×D411yescharacteristic
202C42×C17C2×D411yescharacteristic
203#18441C1711yesmaximal, characteristic, direct product factor, Sylow
204C23×C17—21no—
205C23×C17—22no—
206C23×C17C22×D411yescharacteristic
207Q8×C17—41no—
208Q8×C17—81no—
209Q8×C17—82no—
210Q8×C17—82no—
211Q8×C17—82no—
212D4×C17—42no—
213D4×C17—41no—
214D4×C17—44no—
215D4×C17—42no—
216D4×C17—42no—
217D4×C17—81no—
218D4×C17—82no—
219D4×C17—82no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—24no—
229C2×C4×C17—41no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—44no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—82no—
238C2×C4×C17—82no—
239C2×C4×C17—82no—
240C2×C4×C17—82no—
241C2×C4×C17#12011yescharacteristic
242C2×C4×C17#12011yescharacteristic
243C2×C4×C17#12011yescharacteristic
244C2×C4×C17#12011yescharacteristic
245C2×C4×C17C22×D411yescharacteristic
246C2×C4×C17C22×D411yescharacteristic
247C8×C17—42no—
248C8×C17—42no—
249#3559C2×C1711yescharacteristic
250#3239C2×C1711yescharacteristic, complemented
251#3227C2×C1711yescharacteristic, complemented
252#2642C2×C1711yescharacteristic, complemented
253#1616C2×C1714yescomplemented
254#1615C2×C1714yescomplemented
255#1505C2×C1711yescharacteristic, complemented
256#1500C2×C1711yescharacteristic, complemented
257#1470C2×C1711yescharacteristic, complemented
258C22×C17—21no—
259C22×C17—21no—
260C22×C17—42no—
261C22×C17—41no—
262C22×C17—41no—
263C22×C17—42no—
264C22×C17—42no—
265C22×C17—41no—
266C22×C17C2×#12011yescharacteristic
267C22×C17#57712yes—
268C4×C17—21no—
269C4×C17—21no—
270C4×C17—21no—
271C4×C17—21no—
272C4×C17—21no—
273C4×C17—42no—
274C4×C17—42no—
275C4×C17—41no—
276C4×C17—41no—
277C4×C17—42no—
278C4×C17—41no—
279C4×C17—82no—
280C4×C17—82no—
281C4×C17#45412yes—
282#563—21no—
283#562—21no—
284#554—21no—
285#553—21no—
286#553—21no—
287#547—22no—
288#547—21no—
289#544—21no—
290#540—21no—
291#539—22no—
292#539C22×C1712yes—
293#537C22×C1712yes—
294#532C22×C1711yescharacteristic
295C2×C4⋊Q8C22×C1711yescharacteristic
296#529C22×C1711yescharacteristic
297C2×#125C22×C1711yescharacteristic
298#501—21no—
299#498—21no—
300#489—21no—
301#486—21no—
302#484C22×C1712yes—
303#481C22×C1714yes—
304#481C22×C1712yes—
305#476—22no—
306#475—21no—
307#474—21no—
308#432C22×C1711yescharacteristic
309#431C22×C1711yescharacteristic
310#400—22no—
311#400C22×C1714yes—
312#391—21no—
313#391—21no—
314#376C22×C1711yescharacteristic
315#328C22×C1712yes—
316#327C22×C1714yes—
317#327C22×C1712yes—
318#324C22×C1714yes—
319C2×C17—22no—
320C2×C17—81no—
321C2×C17—82no—
322C2×C17#300911yescharacteristic
323C2×C17C2×#45411yescharacteristic
324C2×C17#302811yescharacteristic
325C2×#41—21no—
326C2×#41—22no—
327C22×Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—22no—
330C4⋊Q8—21no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333C4⋊Q8—21no—
334C4⋊Q8D4×C1711yescharacteristic
335#127D4×C1711yescharacteristic
336#126—22no—
337#126—22no—
338#126—21no—
339#125—21no—
340#125—22no—
341#125—21no—
342#125—21no—
343#125—21no—
344#125—21no—
345#125D4×C1711yescharacteristic
346#124—21no—
347#124D4×C1711yescharacteristic
348#123—22no—
349#123—22no—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—21no—
354#123—22no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8C23×C1711yescharacteristic
362#121—21no—
363#121—22no—
364#121—21no—
365#121—22no—
366#121—22no—
367#121—21no—
368#121—22no—
369#121C23×C1711yescharacteristic
370#121C23×C1712yes—
371C4×Q8D4×C1711yescharacteristic
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—22no—
377C4×D4—21no—
378C4×D4D4×C1711yescharacteristic
379C4×D4D4×C1712yes—
380#117—21no—
381#117—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4D4×C1711yescharacteristic
386C2×C4⋊C4D4×C1712yes—
387C2×C4⋊C4D4×C1711yescharacteristic
388C2×C4⋊C4C23×C1712yes—
389C2×C4⋊C4C23×C1712yes—
390C2×C42—21no—
391C2×C42C23×C1711yescharacteristic
392#107—22no—
393#106—22no—
394#106—24no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397#103—22no—
398#102—22no—
399#102—24no—
400C22⋊C8C23×C1712yes—
401C22⋊C8C23×C1712yes—
402#97—21no—
403C4×C8—21no—
404#95—21no—
405#95—24no—
406#95—24no—
407#95C23×C1712yes—
408C17#1844111yescharacteristic, direct product factor, Sylow
409#41—82no—
410#41—84no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—44no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8C2×D4×C1711yescharacteristic
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4C2×D4×C1711yescharacteristic
424C2×D4C2×D4×C1712yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4C2×D4×C1712yes—
433C22×C4C2×D4×C1711yescharacteristic
434C22×C4C2×D4×C1711yescharacteristic
435C22×C4C2×D4×C1712yes—
436C22×C4C24×C1711yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C2×D4×C1712yes—
461C4⋊C4C2×D4×C1711yescharacteristic
462C4⋊C4C2×D4×C1711yescharacteristic
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C42—22no—
476C42—21no—
477C42—42no—
478C42C2×D4×C1711yescharacteristic
479C42C2×D4×C1711yescharacteristic
480C42C2×D4×C1711yescharacteristic
481C42C2×D4×C1711yescharacteristic
482C23—21no—
483C23—22no—
484C23C22×D4×C1711yescharacteristic
485Q8—41no—
486Q8—81no—
487Q8—82no—
488Q8—82no—
489Q8—82no—
490D4—42no—
491D4—41no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—81no—
496D4—82no—
497D4—82no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4C17×#12011yescharacteristic
520C2×C4C17×#12011yescharacteristic
521C2×C4C17×#12011yescharacteristic
522C2×C4C17×#12011yescharacteristic
523C2×C4C22×D4×C1711yescharacteristic
524C2×C4C22×D4×C1711yescharacteristic
525C8—42no—
526C8—42no—
527C22—21no—
528C22—21no—
529C22—42no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22C2×C17×#12011yescharacteristic
536C22C17×#57712yes—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—41no—
548C4—82no—
549C4—82no—
550C4C17×#45412yes—
551C2—22no—
552C2—81no—
553C2—82no—
554C2C17×#300911yescharacteristic
555C2C2×C17×#45411yescharacteristic
556C2C17×#302811yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#3559C2
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
2. C17×#3239C2semidirect
3. C17×#3227C2semidirect
4. C17×#2642C2semidirect
5. C17×#1616C2semidirect
6. C17×#1615C2semidirect
7. C17×#1505C2semidirect
8. C17×#1500C2semidirect
9. C17×#1470C2semidirect
203. #18441C17direct (N × H)
250. #3239C2×C17semidirect
251. #3227C2×C17semidirect
252. #2642C2×C17semidirect
253. #1616C2×C17semidirect
254. #1615C2×C17semidirect
255. #1505C2×C17semidirect
256. #1500C2×C17semidirect
257. #1470C2×C17semidirect
408. C17#18441direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#35592
C17×#32392
C17×#32272
C17×#26422
C17×#16162
C17×#16152
C17×#15052
C17×#15002
C17×#14702
#1844117