Finite Group Databasebeta

← C17×#22126 #16879870

Subgroups — C17×#22126 #16879870

Order 4352.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C17×#478C211yesmaximal, characteristic
2C17×#2330C211yesmaximal, characteristic, complemented
3C17×#2319C211yesmaximal, characteristic
4C17×#1989C211yesmaximal, characteristic, complemented
5C17×#1947C211yesmaximal, characteristic, complemented
6C17×#1944C211yesmaximal, characteristic, complemented
7C17×#1942C211yesmaximal, characteristic, complemented
8C17×#1936C211yesmaximal, characteristic
9C17×#1934C211yesmaximal, characteristic
10C17×#1925C211yesmaximal, characteristic
11C17×#1925C211yesmaximal, characteristic
12C17×#1921C211yesmaximal, characteristic, complemented
13C17×#1919C211yesmaximal, characteristic, complemented
14C17×#1802C211yesmaximal, characteristic
15C17×#1768C211yesmaximal, characteristic, complemented
16C2×C17×#125C2211yescharacteristic
17C2×C4×D4×C17C2211yescharacteristic
18C17×#478C418yescomplemented
19C2×C17×#107C2211yescharacteristic
20C2×C17×#107C2211yescharacteristic
21C2×C17×#107C2211yescharacteristic
22C2×C17×#106C2211yescharacteristic
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C17×#102C2211yescharacteristic
26C2×C17×#102C2211yescharacteristic
27C2×C17×#102C2211yescharacteristic
28C2×C17×#102C2211yescharacteristic
29C2×C17×#97C2211yescharacteristic
30C2×C4×C8×C17C2211yescharacteristic
31C17×#390C2211yescharacteristic
32C17×#390C2211yescharacteristic
33C17×#389—21no—
34C17×#389—21no—
35C17×#389C2211yescharacteristic
36C17×#389C2211yescharacteristic
37C17×#389C2211yescharacteristic
38C17×#389C2211yescharacteristic
39C17×#388—21no—
40C17×#388—21no—
41C17×#387—21no—
42C17×#387—21no—
43C17×#384C2211yescharacteristic
44C17×#381—21no—
45C17×#380—21no—
46C4×C4⋊C4×C17C2211yescharacteristic
47C4×C4⋊C4×C17C2211yescharacteristic
48C4×C4⋊C4×C17C2211yescharacteristic
49C17×#376C2211yescharacteristic
50C17×#340C2211yescharacteristic
51C17×#340C2211yescharacteristic
52C17×#340C2211yescharacteristic
53C17×#340C2211yescharacteristic
54C17×#340C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C17×#125—22no—
67C17×#125C2×C414yes—
68C4×D4×C17—22no—
69C4×D4×C17C2×C414yes—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C42×C17—21no—
91C2×C42×C17—21no—
92C2×C42×C17—21no—
93C2×C42×C17C2311yescharacteristic
94C2×C42×C17C2311yescharacteristic
95C2×C42×C17C2311yescharacteristic
96C17×#107—41no—
97C17×#107—41no—
98C17×#107C2×C414yes—
99C17×#106C2×C414yes—
100C17×C4⋊C8—22no—
101C17×C4⋊C8C2×C414yes—
102C17×#102—41no—
103C17×#102—41no—
104C17×#102C2×C414yes—
105C17×#102C2×C414yes—
106C17×#97—41no—
107C4×C8×C17—41no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95—21no—
115C17×#95D411yescharacteristic
116C17×#95D411yescharacteristic
117C17×#95D411yescharacteristic
118C17×#95D411yescharacteristic
119C24×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17C22×C412yes—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—24no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—81no—
135C22×C4×C17—81no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2411yescharacteristic
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17C22×C412yes—
154C2×C8×C17C22×C412yes—
155C4⋊C4×C17—24no—
156C4⋊C4×C17—24no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—81no—
169C4⋊C4×C17—81no—
170C4⋊C4×C17—81no—
171C4⋊C4×C17—81no—
172C4⋊C4×C17—81no—
173C4⋊C4×C17—81no—
174C4⋊C4×C17C22×C412yes—
175C4⋊C4×C17C22×C412yes—
176C4⋊C4×C17C22×C412yes—
177C22⋊C4×C17—24no—
178C22⋊C4×C17—81no—
179C22⋊C4×C17—81no—
180C22⋊C4×C17—81no—
181C22⋊C4×C17—81no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—81no—
187C42×C17—81no—
188C42×C17—81no—
189C42×C17C22×C412yes—
190C42×C17#4112yes—
191#22126C1711yesmaximal, characteristic, direct product factor, Sylow
192C23×C17—22no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17C22×D411yescharacteristic
200D4×C17—22no—
201D4×C17—22no—
202D4×C17—41no—
203D4×C17—41no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
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—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—84no—
242C2×C4×C17C4×D412yes—
243C2×C4×C17C4×D412yes—
244C2×C4×C17C23×C411yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C172−1+411yescharacteristic
248C2×C4×C172−1+411yescharacteristic
249C2×C4×C172−1+411yescharacteristic
250C8×C17—22no—
251C8×C17—22no—
252C8×C17—41no—
253C8×C17—41no—
254C2×#478C2×C1711yescharacteristic
255#2330C2×C1711yescharacteristic, complemented
256#2319C2×C1711yescharacteristic
257#1989C2×C1711yescharacteristic, complemented
258#1947C2×C1711yescharacteristic, complemented
259#1944C2×C1711yescharacteristic, complemented
260#1942C2×C1711yescharacteristic, complemented
261#1936C2×C1711yescharacteristic
262#1934C2×C1711yescharacteristic
263#1925C2×C1711yescharacteristic
264#1925C2×C1711yescharacteristic
265#1921C2×C1711yescharacteristic, complemented
266#1919C2×C1711yescharacteristic, complemented
267#1802C2×C1711yescharacteristic
268#1768C2×C1711yescharacteristic, complemented
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—81no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17C2×C4×D411yescharacteristic
277C22×C17#53611yescharacteristic
278C22×C17#53711yescharacteristic
279C22×C17#57611yescharacteristic
280C22×C17#57611yescharacteristic
281C22×C17#57711yescharacteristic
282C22×C17#57811yescharacteristic
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17—42no—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17—41no—
291C4×C17—41no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17—81no—
296C4×C17#51911yescharacteristic
297C4×C17#52011yescharacteristic
298C4×C17#54111yescharacteristic
299C4×C17#54211yescharacteristic
300C2×#125C22×C1711yescharacteristic
301C2×C4×D4C22×C1711yescharacteristic
302#478C4×C1718yescomplemented
303C2×#107C22×C1711yescharacteristic
304C2×#107C22×C1711yescharacteristic
305C2×#107C22×C1711yescharacteristic
306C2×#106C22×C1711yescharacteristic
307C2×C4⋊C8C22×C1711yescharacteristic
308C2×C4⋊C8C22×C1711yescharacteristic
309C2×#102C22×C1711yescharacteristic
310C2×#102C22×C1711yescharacteristic
311C2×#102C22×C1711yescharacteristic
312C2×#102C22×C1711yescharacteristic
313C2×#97C22×C1711yescharacteristic
314C2×C4×C8C22×C1711yescharacteristic
315#390C22×C1711yescharacteristic
316#390C22×C1711yescharacteristic
317#389—21no—
318#389—21no—
319#389C22×C1711yescharacteristic
320#389C22×C1711yescharacteristic
321#389C22×C1711yescharacteristic
322#389C22×C1711yescharacteristic
323#388—21no—
324#388—21no—
325#387—21no—
326#387—21no—
327#384C22×C1711yescharacteristic
328#381—21no—
329#380—21no—
330C4×C4⋊C4C22×C1711yescharacteristic
331C4×C4⋊C4C22×C1711yescharacteristic
332C4×C4⋊C4C22×C1711yescharacteristic
333#376C22×C1711yescharacteristic
334#340C22×C1711yescharacteristic
335#340C22×C1711yescharacteristic
336#340C22×C1711yescharacteristic
337#340C22×C1711yescharacteristic
338#340C22×C1711yescharacteristic
339#340C22×C1711yescharacteristic
340#340C22×C1711yescharacteristic
341#340C22×C1711yescharacteristic
342#336C22×C1711yescharacteristic
343#336C22×C1711yescharacteristic
344C2×C17—82no—
345C2×C17#232311yescharacteristic
346C2×C17#294611yescharacteristic
347C2×C17#295111yescharacteristic
348C2×C17#324111yescharacteristic
349C2×C17#324311yescharacteristic
350C2×C17#325811yescharacteristic
351C2×C17#325911yescharacteristic
352C22×D4C23×C1711yescharacteristic
353C23×C4—21no—
354C22×C8C23×C1711yescharacteristic
355C22×C8C23×C1711yescharacteristic
356C22×C8C23×C1711yescharacteristic
357C22×C8C23×C1711yescharacteristic
358#125—22no—
359#125C2×C4×C1714yes—
360C4×D4—22no—
361C4×D4C2×C4×C1714yes—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4C23×C1711yescharacteristic
371C2×C4⋊C4C23×C1711yescharacteristic
372C2×C4⋊C4C23×C1711yescharacteristic
373C2×C4⋊C4C23×C1711yescharacteristic
374C2×C4⋊C4C23×C1711yescharacteristic
375C2×C4⋊C4C23×C1711yescharacteristic
376C2×C4⋊C4C23×C1711yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42C23×C1711yescharacteristic
386C2×C42C23×C1711yescharacteristic
387C2×C42C23×C1711yescharacteristic
388#107—41no—
389#107—41no—
390#107C2×C4×C1714yes—
391#106C2×C4×C1714yes—
392C4⋊C8—22no—
393C4⋊C8C2×C4×C1714yes—
394#102—41no—
395#102—41no—
396#102C2×C4×C1714yes—
397#102C2×C4×C1714yes—
398#97—41no—
399C4×C8—41no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95D4×C1711yescharacteristic
408#95D4×C1711yescharacteristic
409#95D4×C1711yescharacteristic
410#95D4×C1711yescharacteristic
411C17#2212611yescharacteristic, direct product factor, Sylow
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C22×C4×C1712yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4C2×D4×C1711yescharacteristic
430C22×C4C2×D4×C1711yescharacteristic
431C22×C4C2×D4×C1711yescharacteristic
432C22×C4C2×D4×C1711yescharacteristic
433C22×C4C2×D4×C1711yescharacteristic
434C22×C4C2×D4×C1711yescharacteristic
435C22×C4C24×C1711yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8C22×C4×C1712yes—
447C2×C8C22×C4×C1712yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4C22×C4×C1712yes—
468C4⋊C4C22×C4×C1712yes—
469C4⋊C4C22×C4×C1712yes—
470C22⋊C4—24no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—81no—
480C42—81no—
481C42—81no—
482C42C22×C4×C1712yes—
483C42#41×C1712yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C22×D4×C1711yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
534C2×C4C4×D4×C1712yes—
535C2×C4C4×D4×C1712yes—
536C2×C4C23×C4×C1711yescharacteristic
537C2×C4C2×#41×C1711yescharacteristic
538C2×C4C17×H(5,𝔽2)11yescharacteristic
539C2×C4C17×2−1+411yescharacteristic
540C2×C4C17×2−1+411yescharacteristic
541C2×C4C17×2−1+411yescharacteristic
542C8—22no—
543C8—22no—
544C8—41no—
545C8—41no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×C4×D4×C1711yescharacteristic
554C22C17×#53611yescharacteristic
555C22C17×#53711yescharacteristic
556C22C17×#57611yescharacteristic
557C22C17×#57611yescharacteristic
558C22C17×#57711yescharacteristic
559C22C17×#57811yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4C17×#51911yescharacteristic
574C4C17×#52011yescharacteristic
575C4C17×#54111yescharacteristic
576C4C17×#54211yescharacteristic
577C2—82no—
578C2C17×#232311yescharacteristic
579C2C17×#294611yescharacteristic
580C2C17×#295111yescharacteristic
581C2C17×#324111yescharacteristic
582C2C17×#324311yescharacteristic
583C2C17×#325811yescharacteristic
584C2C17×#325911yescharacteristic
subgroupG/H
center, socleC23×C17C22×D4
commutatorC2×C4C23×C4×C17
Frattini, cosocleC22×C4C24×C17
prime radicalC22×D4×C17C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C17 ◁ C17×#376 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2330C2semidirect
4. C17×#1989C2semidirect
5. C17×#1947C2semidirect
6. C17×#1944C2semidirect
7. C17×#1942C2semidirect
12. C17×#1921C2semidirect
13. C17×#1919C2semidirect
15. C17×#1768C2semidirect
18. C17×#478C4semidirect
191. #22126C17direct (N × H)
255. #2330C2×C17semidirect
257. #1989C2×C17semidirect
258. #1947C2×C17semidirect
259. #1944C2×C17semidirect
260. #1942C2×C17semidirect
265. #1921C2×C17semidirect
266. #1919C2×C17semidirect
268. #1768C2×C17semidirect
302. #478C4×C17semidirect
411. C17#22126direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C17×#4782
C17×#23302
C17×#23192
C17×#19892
C17×#19472
C17×#19442
C17×#19422
C17×#19362
C17×#19342
C17×#19252
C17×#19212
C17×#19192
C17×#18022
C17×#17682
#2212617