Finite Group Databasebeta

← C17×#18454 #16876198

Subgroups — C17×#18454 #16876198

Order 4352.

936 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3538C211yesmaximal, characteristic, complemented
2C17×#3194C211yesmaximal, characteristic
3C17×#3187C211yesmaximal, characteristic, complemented
4C17×#2816C211yesmaximal, characteristic, complemented
5C17×#1611C212yesmaximal, complemented
6C17×#1609C212yesmaximal, complemented
7C17×#1599C212yesmaximal, complemented
8C17×#1597C212yesmaximal, complemented
9C17×#1516C211yesmaximal, characteristic, complemented
10C17×#1511C211yesmaximal, characteristic, complemented
11C17×#1472C211yesmaximal, characteristic, complemented
12C17×#563—22no—
13C17×#552—21no—
14C17×#552—21no—
15C17×#551—22no—
16D4×Q8×C17—21no—
17D4×Q8×C17—21no—
18C17×#544—21no—
19C17×#544C2212yes—
20C17×#541—21no—
21C17×#539—21no—
22C17×#539C2212yes—
23C17×#536—21no—
24C17×#525C2211yescharacteristic
25C2×C17×C22⋊Q8C2211yescharacteristic
26C2×C17×C22⋊Q8C2211yescharacteristic
27C2×C17×#121C2211yescharacteristic
28C17×#483C2212yes—
29C17×#483C2211yescharacteristic
30C17×#482—21no—
31C17×#481—21no—
32C17×#480C2211yescharacteristic
33C17×#469—21no—
34C17×#468—21no—
35C17×#465—22no—
36C17×#450C2212yes—
37C17×#450C2211yescharacteristic
38C17×#449—21no—
39C17×#448—21no—
40C17×#447C2211yescharacteristic
41C17×#407C2211yescharacteristic
42C2×C17×C22⋊C8C2211yescharacteristic
43C17×#399—21no—
44C17×#398C2212yes—
45C17×#394—22no—
46C17×#394C2212yes—
47C17×#393—21no—
48C17×#381C2211yescharacteristic
49C17×#328C2212yes—
50C17×#328C2212yes—
51C17×#328C2212yes—
52C17×#327C2212yes—
53C17×#324C2212yes—
54C17×#324C2212yes—
55C2×#41×C17D411yescharacteristic
56C22×Q8×C17—21no—
57C22×Q8×C17D411yescharacteristic
58C22×Q8×C17D411yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C2×SD16×C17—22no—
63C2×SD16×C17—21no—
64C2×D8×C17—21no—
65C2×M16×C17—21no—
66C22×C8×C17—21no—
67C17×C4⋊Q8—21no—
68C17×C4⋊Q8—22no—
69C17×C4⋊Q8—22no—
70C17×C4⋊Q8—22no—
71C17×#126—22no—
72C17×#126—22no—
73C17×#126—22no—
74C17×#126—22no—
75C17×#125—21no—
76C17×#125—22no—
77C17×#125—22no—
78C17×#125—22no—
79C17×#125—22no—
80C17×#124—22no—
81C17×#123—22no—
82C17×#123—22no—
83C17×#123—22no—
84C17×#123—21no—
85C17×#123—22no—
86C17×C22⋊Q8—21no—
87C17×C22⋊Q8—21no—
88C17×C22⋊Q8—22no—
89C17×C22⋊Q8—22no—
90C17×C22⋊Q8—22no—
91C17×C22⋊Q8—21no—
92C17×C22⋊Q8—21no—
93C17×C22⋊Q8—21no—
94C17×C22⋊Q8—21no—
95C17×C22⋊Q8—21no—
96C17×C22⋊Q8—21no—
97C17×C22⋊Q8—22no—
98C17×C22⋊Q8D412yes—
99C17×C22⋊Q8D411yescharacteristic
100C17×C22⋊Q8C2312yes—
101C17×C22⋊Q8C2311yescharacteristic
102C17×#121—21no—
103C17×#121—21no—
104C17×#121—21no—
105C17×#121—21no—
106C17×#121D411yescharacteristic
107C17×#121C2311yescharacteristic
108C17×#120—21no—
109C4×Q8×C17—21no—
110C4×Q8×C17—21no—
111C4×D4×C17—22no—
112C4×D4×C17—21no—
113C4×D4×C17—21no—
114C4×D4×C17—22no—
115C17×#117—21no—
116C17×#117—22no—
117C17×#117—22no—
118C17×#117D411yescharacteristic
119C2×C4⋊C4×C17—21no—
120C2×C4⋊C4×C17—22no—
121C2×C4⋊C4×C17D411yescharacteristic
122C2×C4⋊C4×C17D411yescharacteristic
123C2×C4⋊C4×C17C2311yescharacteristic
124C2×C4⋊C4×C17C2312yes—
125C2×C22⋊C4×C17—21no—
126C2×C22⋊C4×C17—21no—
127C2×C22⋊C4×C17—21no—
128C2×C22⋊C4×C17—22no—
129C2×C22⋊C4×C17—22no—
130C2×C22⋊C4×C17D412yes—
131C17×#107—21no—
132C17×#106—22no—
133C17×#106—21no—
134C17×#103—22no—
135C17×#103—21no—
136C17×#102—21no—
137C17×#102—21no—
138C17×#102—22no—
139C17×#102—21no—
140C17×C22⋊C8—21no—
141C17×C22⋊C8—21no—
142C17×C22⋊C8C2312yes—
143C17×C22⋊C8C2311yescharacteristic
144C17×C22⋊C8C2311yescharacteristic
145C17×#95—21no—
146C17×#95—22no—
147C17×#95C2312yes—
148C24×C17—21no—
149C24×C17C2×D411yescharacteristic
150#41×C17—81no—
151C2×Q8×C17—22no—
152C2×Q8×C17—21no—
153C2×Q8×C17—21no—
154C2×Q8×C17—21no—
155C2×Q8×C17—81no—
156C2×Q8×C17—82no—
157C2×Q8×C17—81no—
158C2×Q8×C17C2×D412yes—
159C2×Q8×C17C2×D411yescharacteristic
160C2×D4×C17—21no—
161C2×D4×C17—21no—
162C2×D4×C17—21no—
163C2×D4×C17—21no—
164C2×D4×C17—21no—
165C2×D4×C17—41no—
166C2×D4×C17—41no—
167C2×D4×C17—81no—
168C2×D4×C17—82no—
169C2×D4×C17C2×D411yescharacteristic
170C22×C4×C17—21no—
171C22×C4×C17—22no—
172C22×C4×C17—21no—
173C22×C4×C17—21no—
174C22×C4×C17—21no—
175C22×C4×C17—22no—
176C22×C4×C17—81no—
177C22×C4×C17C2×D412yes—
178C22×C4×C17C2×D412yes—
179C22×C4×C17C2×D411yescharacteristic
180C22×C4×C17C2×D411yescharacteristic
181C22×C4×C17C2×D411yescharacteristic
182C22×C4×C17C2×D411yescharacteristic
183C22×C4×C17C2×D411yescharacteristic
184C22×C4×C17C2411yescharacteristic
185SD16×C17—82no—
186SD16×C17—81no—
187D8×C17—81no—
188M16×C17—81no—
189C2×C8×C17—22no—
190C2×C8×C17—21no—
191C2×C8×C17—21no—
192C2×C8×C17—81no—
193C4⋊C4×C17—22no—
194C4⋊C4×C17—22no—
195C4⋊C4×C17—22no—
196C4⋊C4×C17—22no—
197C4⋊C4×C17—22no—
198C4⋊C4×C17—22no—
199C4⋊C4×C17—22no—
200C4⋊C4×C17—22no—
201C4⋊C4×C17—21no—
202C4⋊C4×C17—21no—
203C4⋊C4×C17—22no—
204C4⋊C4×C17—21no—
205C4⋊C4×C17—21no—
206C4⋊C4×C17—22no—
207C4⋊C4×C17—22no—
208C4⋊C4×C17—22no—
209C4⋊C4×C17—21no—
210C4⋊C4×C17—21no—
211C4⋊C4×C17—21no—
212C4⋊C4×C17—82no—
213C4⋊C4×C17C2×D412yes—
214C4⋊C4×C17C2×D411yescharacteristic
215C4⋊C4×C17C2×D411yescharacteristic
216C22⋊C4×C17—22no—
217C22⋊C4×C17—22no—
218C22⋊C4×C17—22no—
219C22⋊C4×C17—22no—
220C22⋊C4×C17—21no—
221C22⋊C4×C17—22no—
222C22⋊C4×C17—22no—
223C22⋊C4×C17—21no—
224C22⋊C4×C17—22no—
225C22⋊C4×C17—21no—
226C22⋊C4×C17—21no—
227C22⋊C4×C17—21no—
228C22⋊C4×C17—41no—
229C22⋊C4×C17—41no—
230C22⋊C4×C17—82no—
231C22⋊C4×C17—82no—
232C22⋊C4×C17—81no—
233C42×C17—22no—
234C42×C17—22no—
235C42×C17—22no—
236C42×C17—21no—
237#18454C1711yesmaximal, characteristic, direct product factor, Sylow
238C23×C17—22no—
239C23×C17—21no—
240C23×C17—41no—
241C23×C17—41no—
242C23×C17—42no—
243C23×C17—42no—
244C23×C17—41no—
245C23×C17—42no—
246C23×C17#12011yescharacteristic
247C23×C17#12011yescharacteristic
248C23×C17C22×D411yescharacteristic
249Q8×C17—42no—
250Q8×C17—41no—
251Q8×C17—42no—
252Q8×C17—42no—
253Q8×C17—81no—
254Q8×C17—82no—
255D4×C17—41no—
256D4×C17—41no—
257D4×C17—41no—
258D4×C17—42no—
259D4×C17—42no—
260D4×C17—42no—
261D4×C17—81no—
262D4×C17—82no—
263D4×C17—82no—
264C2×C4×C17—21no—
265C2×C4×C17—21no—
266C2×C4×C17—22no—
267C2×C4×C17—22no—
268C2×C4×C17—21no—
269C2×C4×C17—22no—
270C2×C4×C17—22no—
271C2×C4×C17—22no—
272C2×C4×C17—22no—
273C2×C4×C17—41no—
274C2×C4×C17—41no—
275C2×C4×C17—41no—
276C2×C4×C17—41no—
277C2×C4×C17—41no—
278C2×C4×C17—41no—
279C2×C4×C17—42no—
280C2×C4×C17—42no—
281C2×C4×C17—42no—
282C2×C4×C17—42no—
283C2×C4×C17—42no—
284C2×C4×C17—41no—
285C2×C4×C17—42no—
286C2×C4×C17—42no—
287C2×C4×C17—81no—
288C2×C4×C17—82no—
289C2×C4×C17#12011yescharacteristic
290C2×C4×C17#12011yescharacteristic
291C2×C4×C17C22×D411yescharacteristic
292C2×C4×C17C22×D411yescharacteristic
293C8×C17—42no—
294C8×C17—41no—
295C8×C17—41no—
296#3538C2×C1711yescharacteristic, complemented
297#3194C2×C1711yescharacteristic
298#3187C2×C1711yescharacteristic, complemented
299#2816C2×C1711yescharacteristic, complemented
300#1611C2×C1712yescomplemented
301#1609C2×C1712yescomplemented
302#1599C2×C1712yescomplemented
303#1597C2×C1712yescomplemented
304#1516C2×C1711yescharacteristic, complemented
305#1511C2×C1711yescharacteristic, complemented
306#1472C2×C1711yescharacteristic, complemented
307C22×C17—21no—
308C22×C17—21no—
309C22×C17—21no—
310C22×C17—21no—
311C22×C17—21no—
312C22×C17—21no—
313C22×C17—21no—
314C22×C17—21no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17—82no—
319C22×C17—84no—
320C22×C17—81no—
321C22×C17—81no—
322C22×C17—81no—
323C22×C17C2×#12011yescharacteristic
324C22×C17#57712yes—
325C4×C17—21no—
326C4×C17—21no—
327C4×C17—21no—
328C4×C17—21no—
329C4×C17—42no—
330C4×C17—41no—
331C4×C17—42no—
332C4×C17—41no—
333C4×C17—41no—
334C4×C17—42no—
335C4×C17—42no—
336C4×C17—82no—
337C4×C17—81no—
338#563—22no—
339#552—21no—
340#552—21no—
341#551—22no—
342D4×Q8—21no—
343D4×Q8—21no—
344#544—21no—
345#544C22×C1712yes—
346#541—21no—
347#539—21no—
348#539C22×C1712yes—
349#536—21no—
350#525C22×C1711yescharacteristic
351C2×C22⋊Q8C22×C1711yescharacteristic
352C2×C22⋊Q8C22×C1711yescharacteristic
353C2×#121C22×C1711yescharacteristic
354#483C22×C1712yes—
355#483C22×C1711yescharacteristic
356#482—21no—
357#481—21no—
358#480C22×C1711yescharacteristic
359#469—21no—
360#468—21no—
361#465—22no—
362#450C22×C1712yes—
363#450C22×C1711yescharacteristic
364#449—21no—
365#448—21no—
366#447C22×C1711yescharacteristic
367#407C22×C1711yescharacteristic
368C2×C22⋊C8C22×C1711yescharacteristic
369#399—21no—
370#398C22×C1712yes—
371#394—22no—
372#394C22×C1712yes—
373#393—21no—
374#381C22×C1711yescharacteristic
375#328C22×C1712yes—
376#328C22×C1712yes—
377#328C22×C1712yes—
378#327C22×C1712yes—
379#324C22×C1712yes—
380#324C22×C1712yes—
381C2×C17—22no—
382C2×C17—41no—
383C2×C17—41no—
384C2×C17—81no—
385C2×C17—161no—
386C2×C17#301011yescharacteristic
387C2×C17#302011yescharacteristic
388C2×C17#302611yescharacteristic
389C2×#41D4×C1711yescharacteristic
390C22×Q8—21no—
391C22×Q8D4×C1711yescharacteristic
392C22×Q8D4×C1711yescharacteristic
393C22×D4—21no—
394C22×D4C23×C1711yescharacteristic
395C23×C4C23×C1711yescharacteristic
396C2×SD16—22no—
397C2×SD16—21no—
398C2×D8—21no—
399C2×M16—21no—
400C22×C8—21no—
401C4⋊Q8—21no—
402C4⋊Q8—22no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405#126—22no—
406#126—22no—
407#126—22no—
408#126—22no—
409#125—21no—
410#125—22no—
411#125—22no—
412#125—22no—
413#125—22no—
414#124—22no—
415#123—22no—
416#123—22no—
417#123—22no—
418#123—21no—
419#123—22no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—21no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—21no—
430C22⋊Q8—21no—
431C22⋊Q8—22no—
432C22⋊Q8D4×C1712yes—
433C22⋊Q8D4×C1711yescharacteristic
434C22⋊Q8C23×C1712yes—
435C22⋊Q8C23×C1711yescharacteristic
436#121—21no—
437#121—21no—
438#121—21no—
439#121—21no—
440#121D4×C1711yescharacteristic
441#121C23×C1711yescharacteristic
442#120—21no—
443C4×Q8—21no—
444C4×Q8—21no—
445C4×D4—22no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—22no—
449#117—21no—
450#117—22no—
451#117—22no—
452#117D4×C1711yescharacteristic
453C2×C4⋊C4—21no—
454C2×C4⋊C4—22no—
455C2×C4⋊C4D4×C1711yescharacteristic
456C2×C4⋊C4D4×C1711yescharacteristic
457C2×C4⋊C4C23×C1711yescharacteristic
458C2×C4⋊C4C23×C1712yes—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—21no—
461C2×C22⋊C4—21no—
462C2×C22⋊C4—22no—
463C2×C22⋊C4—22no—
464C2×C22⋊C4D4×C1712yes—
465#107—21no—
466#106—22no—
467#106—21no—
468#103—22no—
469#103—21no—
470#102—21no—
471#102—21no—
472#102—22no—
473#102—21no—
474C22⋊C8—21no—
475C22⋊C8—21no—
476C22⋊C8C23×C1712yes—
477C22⋊C8C23×C1711yescharacteristic
478C22⋊C8C23×C1711yescharacteristic
479#95—21no—
480#95—22no—
481#95C23×C1712yes—
482C17#1845411yescharacteristic, direct product factor, Sylow
483C24—21no—
484C24C2×D4×C1711yescharacteristic
485#41—81no—
486C2×Q8—22no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—81no—
491C2×Q8—82no—
492C2×Q8—81no—
493C2×Q8C2×D4×C1712yes—
494C2×Q8C2×D4×C1711yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—41no—
501C2×D4—41no—
502C2×D4—81no—
503C2×D4—82no—
504C2×D4C2×D4×C1711yescharacteristic
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—81no—
512C22×C4C2×D4×C1712yes—
513C22×C4C2×D4×C1712yes—
514C22×C4C2×D4×C1711yescharacteristic
515C22×C4C2×D4×C1711yescharacteristic
516C22×C4C2×D4×C1711yescharacteristic
517C22×C4C2×D4×C1711yescharacteristic
518C22×C4C2×D4×C1711yescharacteristic
519C22×C4C24×C1711yescharacteristic
520SD16—82no—
521SD16—81no—
522D8—81no—
523M16—81no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—21no—
527C2×C8—81no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—82no—
548C4⋊C4C2×D4×C1712yes—
549C4⋊C4C2×D4×C1711yescharacteristic
550C4⋊C4C2×D4×C1711yescharacteristic
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—21no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—81no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—21no—
572C23—22no—
573C23—21no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23C17×#12011yescharacteristic
581C23C17×#12011yescharacteristic
582C23C22×D4×C1711yescharacteristic
583Q8—42no—
584Q8—41no—
585Q8—42no—
586Q8—42no—
587Q8—81no—
588Q8—82no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—81no—
596D4—82no—
597D4—82no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—81no—
622C2×C4—82no—
623C2×C4C17×#12011yescharacteristic
624C2×C4C17×#12011yescharacteristic
625C2×C4C22×D4×C1711yescharacteristic
626C2×C4C22×D4×C1711yescharacteristic
627C8—42no—
628C8—41no—
629C8—41no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—82no—
642C22—84no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22C2×C17×#12011yescharacteristic
647C22C17×#57712yes—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—81no—
661C2—22no—
662C2—41no—
663C2—41no—
664C2—81no—
665C2—161no—
666C2C17×#301011yescharacteristic
667C2C17×#302011yescharacteristic
668C2C17×#302611yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#3194C2
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
1. C17×#3538C2semidirect
3. C17×#3187C2semidirect
4. C17×#2816C2semidirect
5. C17×#1611C2semidirect
6. C17×#1609C2semidirect
7. C17×#1599C2semidirect
8. C17×#1597C2semidirect
9. C17×#1516C2semidirect
10. C17×#1511C2semidirect
11. C17×#1472C2semidirect
237. #18454C17direct (N × H)
296. #3538C2×C17semidirect
298. #3187C2×C17semidirect
299. #2816C2×C17semidirect
300. #1611C2×C17semidirect
301. #1609C2×C17semidirect
302. #1599C2×C17semidirect
303. #1597C2×C17semidirect
304. #1516C2×C17semidirect
305. #1511C2×C17semidirect
306. #1472C2×C17semidirect
482. C17#18454direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#35382
C17×#31942
C17×#31872
C17×#28162
C17×#16112
C17×#16092
C17×#15992
C17×#15972
C17×#15162
C17×#15112
C17×#14722
#1845417