Finite Group Databasebeta

← C17×#18152 #16875896

Subgroups — C17×#18152 #16875896

Order 4352.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3449C211yesmaximal, characteristic, complemented
2C17×#3109C212yesmaximal, complemented
3C17×#2598C211yesmaximal, characteristic, complemented
4C17×#1616C212yesmaximal, complemented
5C17×#1613C212yesmaximal, complemented
6C17×#1598C212yesmaximal, complemented
7C17×#1595C212yesmaximal, complemented
8C17×#1514C212yesmaximal, complemented
9C17×#1464C211yesmaximal, characteristic, complemented
10C22×#41×C17—21no—
11C17×#537—21no—
12C17×#537C2212yes—
13C17×#536—22no—
14C17×#535—22no—
15C17×#534—21no—
16C17×#534C2212yes—
17C17×#532—21no—
18C17×#532C2212yes—
19C17×#525—24no—
20C2×C4×D4×C17C2211yescharacteristic, complemented
21C2×C17×#117C2211yescharacteristic
22C17×#481C2214yescomplemented
23C17×#476—22no—
24C17×#459—22no—
25C17×#447C2214yescomplemented
26C17×#442—24no—
27C17×#432C2212yescomplemented
28C17×#400C2211yescharacteristic, complemented
29C17×#399C2211yescharacteristic
30C17×#394C2211yescharacteristic, complemented
31C17×#392C2211yescharacteristic
32C17×#388—21no—
33C17×#386—21no—
34C17×#386—21no—
35C17×#385—21no—
36C17×#383C2211yescharacteristic, complemented
37C17×#327C2214yescomplemented
38C17×#327C2214yescomplemented
39C17×#324C2212yescomplemented
40C17×#323C2212yes—
41C2×#41×C17—22no—
42C2×#41×C17—22no—
43C2×#41×C17—22no—
44C2×#41×C17—22no—
45C2×#41×C17—44no—
46C2×#41×C17—44no—
47C22×Q8×C17—21no—
48C22×D4×C17—21no—
49C22×D4×C17—22no—
50C22×D4×C17—21no—
51C22×D4×C17C2311yescharacteristic
52C23×C4×C17—22no—
53C23×C4×C17—21no—
54C23×C4×C17D411yescharacteristic
55C2×D8×C17—24no—
56C17×C4⋊Q8—21no—
57C17×C4⋊Q8D412yescomplemented
58C17×#127—21no—
59C17×#127D412yescomplemented
60C17×#124—22no—
61C17×#124—22no—
62C17×#123—24no—
63C17×#123—24no—
64C17×C22⋊Q8—24no—
65C17×C22⋊Q8—24no—
66C17×#121—24no—
67C17×#121—24no—
68C17×#121—22no—
69C17×#121—22no—
70C17×#121—22no—
71C17×#121C2314yes—
72C17×#120—24no—
73C17×#120—24no—
74C4×D4×C17—22no—
75C4×D4×C17—22no—
76C4×D4×C17—22no—
77C4×D4×C17—22no—
78C4×D4×C17—22no—
79C4×D4×C17D414yescomplemented
80C17×#117—22no—
81C17×#117—22no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17D411yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C4⋊C4×C17C2311yescharacteristic
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C22⋊C4×C17—21no—
91C2×C22⋊C4×C17D411yescharacteristic
92C2×C22⋊C4×C17C2311yescharacteristic
93C2×C42×C17D411yescharacteristic
94C2×C42×C17C2311yescharacteristic
95C17×#106—24no—
96C17×C4⋊C8—22no—
97C17×#102—24no—
98C17×#102—24no—
99C17×C22⋊C8C2314yes—
100C17×#97—22no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95C2311yescharacteristic
106C17×#95C2311yescharacteristic
107C24×C17—22no—
108C24×C17—21no—
109C24×C17C2×D411yescharacteristic
110#41×C17—48no—
111#41×C17—48no—
112#41×C17—44no—
113#41×C17—44no—
114#41×C17—44no—
115#41×C17—44no—
116C2×Q8×C17—22no—
117C2×Q8×C17—22no—
118C2×Q8×C17—44no—
119C2×D4×C17—24no—
120C2×D4×C17—22no—
121C2×D4×C17—24no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—22no—
125C2×D4×C17—22no—
126C2×D4×C17—22no—
127C2×D4×C17—44no—
128C2×D4×C17—42no—
129C2×D4×C17—44no—
130C2×D4×C17—44no—
131C2×D4×C17—42no—
132C2×D4×C17—42no—
133C2×D4×C17—81no—
134C2×D4×C17C2×D414yes—
135C22×C4×C17—22no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—22no—
139C22×C4×C17—24no—
140C22×C4×C17—22no—
141C22×C4×C17—24no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—48no—
146C22×C4×C17—42no—
147C22×C4×C17—81no—
148C22×C4×C17—81no—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2411yescharacteristic
155D8×C17—48no—
156C2×C8×C17—24no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—22no—
159C4⋊C4×C17—22no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—81no—
164C4⋊C4×C17C2×D414yes—
165C22⋊C4×C17—24no—
166C22⋊C4×C17—24no—
167C22⋊C4×C17—24no—
168C22⋊C4×C17—22no—
169C22⋊C4×C17—24no—
170C22⋊C4×C17—22no—
171C22⋊C4×C17—42no—
172C22⋊C4×C17—41no—
173C22⋊C4×C17—41no—
174C22⋊C4×C17—42no—
175C22⋊C4×C17—81no—
176C22⋊C4×C17—81no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17C2×D412yes—
180C42×C17C2×D412yes—
181#18152C1711yesmaximal, characteristic, direct product factor, Sylow
182C23×C17—21no—
183C23×C17—21no—
184C23×C17—22no—
185C23×C17—24no—
186C23×C17—41no—
187C23×C17—41no—
188C23×C17—41no—
189C23×C17—44no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—82no—
196C23×C17—82no—
197C23×C17AGL(1,ℤ/8)12yes—
198C23×C17C22×D411yescharacteristic
199Q8×C17—44no—
200Q8×C17—44no—
201D4×C17—24no—
202D4×C17—44no—
203D4×C17—42no—
204D4×C17—44no—
205D4×C17—44no—
206D4×C17—48no—
207D4×C17—44no—
208D4×C17—44no—
209D4×C17—82no—
210D4×C17—82no—
211D4×C17—82no—
212D4×C17—84no—
213C2×C4×C17—24no—
214C2×C4×C17—24no—
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—24no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—48no—
225C2×C4×C17—48no—
226C2×C4×C17—44no—
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—84no—
239C2×C4×C17#12011yescharacteristic
240C2×C4×C17#12011yescharacteristic
241C2×C4×C17#12012yes—
242C2×C4×C17AGL(1,ℤ/8)12yes—
243C2×C4×C17C22×D412yes—
244C8×C17—44no—
245#3449C2×C1711yescharacteristic, complemented
246#3109C2×C1712yescomplemented
247#2598C2×C1711yescharacteristic, complemented
248#1616C2×C1712yescomplemented
249#1613C2×C1712yescomplemented
250#1598C2×C1712yescomplemented
251#1595C2×C1712yescomplemented
252#1514C2×C1712yescomplemented
253#1464C2×C1711yescharacteristic, complemented
254C22×C17—21no—
255C22×C17—21no—
256C22×C17—22no—
257C22×C17—42no—
258C22×C17—42no—
259C22×C17—42no—
260C22×C17—41no—
261C22×C17—41no—
262C22×C17—44no—
263C22×C17—44no—
264C22×C17—42no—
265C22×C17—44no—
266C22×C17—81no—
267C22×C17—81no—
268C22×C17—81no—
269C22×C17—88no—
270C22×C17—81no—
271C22×C17—81no—
272C22×C17C2×#12011yescharacteristic
273C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
275C4×C17—21no—
276C4×C17—22no—
277C4×C17—21no—
278C4×C17—22no—
279C4×C17—22no—
280C4×C17—44no—
281C4×C17—41no—
282C4×C17—41no—
283C4×C17—81no—
284C4×C17—81no—
285C4×C17—81no—
286C22×#41—21no—
287#537—21no—
288#537C22×C1712yes—
289#536—22no—
290#535—22no—
291#534—21no—
292#534C22×C1712yes—
293#532—21no—
294#532C22×C1712yes—
295#525—24no—
296C2×C4×D4C22×C1711yescharacteristic, complemented
297C2×#117C22×C1711yescharacteristic
298#481C22×C1714yescomplemented
299#476—22no—
300#459—22no—
301#447C22×C1714yescomplemented
302#442—24no—
303#432C22×C1712yescomplemented
304#400C22×C1711yescharacteristic, complemented
305#399C22×C1711yescharacteristic
306#394C22×C1711yescharacteristic, complemented
307#392C22×C1711yescharacteristic
308#388—21no—
309#386—21no—
310#386—21no—
311#385—21no—
312#383C22×C1711yescharacteristic, complemented
313#327C22×C1714yescomplemented
314#327C22×C1714yescomplemented
315#324C22×C1712yescomplemented
316#323C22×C1712yes—
317C2×C17—21no—
318C2×C17—21no—
319C2×C17—42no—
320C2×C17—82no—
321C2×C17—84no—
322C2×C17#300311yescharacteristic
323C2×C17#302111yescharacteristic
324C2×C17#302511yescharacteristic
325C2×#41—22no—
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—44no—
330C2×#41—44no—
331C22×Q8—21no—
332C22×D4—21no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4C23×C1711yescharacteristic
336C23×C4—22no—
337C23×C4—21no—
338C23×C4D4×C1711yescharacteristic
339C2×D8—24no—
340C4⋊Q8—21no—
341C4⋊Q8D4×C1712yescomplemented
342#127—21no—
343#127D4×C1712yescomplemented
344#124—22no—
345#124—22no—
346#123—24no—
347#123—24no—
348C22⋊Q8—24no—
349C22⋊Q8—24no—
350#121—24no—
351#121—24no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121C23×C1714yes—
356#120—24no—
357#120—24no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4D4×C1714yescomplemented
364#117—22no—
365#117—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4D4×C1711yescharacteristic
368C2×C4⋊C4C23×C1711yescharacteristic
369C2×C4⋊C4C23×C1711yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4D4×C1711yescharacteristic
376C2×C22⋊C4C23×C1711yescharacteristic
377C2×C42D4×C1711yescharacteristic
378C2×C42C23×C1711yescharacteristic
379#106—24no—
380C4⋊C8—22no—
381#102—24no—
382#102—24no—
383C22⋊C8C23×C1714yes—
384#97—22no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95C23×C1711yescharacteristic
390#95C23×C1711yescharacteristic
391C17#1815211yescharacteristic, direct product factor, Sylow
392C24—22no—
393C24—21no—
394C24C2×D4×C1711yescharacteristic
395#41—48no—
396#41—48no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4C2×D4×C1714yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4C2×D4×C1711yescharacteristic
435C22×C4C2×D4×C1711yescharacteristic
436C22×C4C2×D4×C1711yescharacteristic
437C22×C4C2×D4×C1711yescharacteristic
438C22×C4C2×D4×C1711yescharacteristic
439C22×C4C24×C1711yescharacteristic
440D8—48no—
441C2×C8—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—81no—
449C4⋊C4C2×D4×C1714yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C42—21no—
463C42—21no—
464C42C2×D4×C1712yes—
465C42C2×D4×C1712yes—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—24no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—44no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—82no—
480C23—82no—
481C23C17×AGL(1,ℤ/8)12yes—
482C23C22×D4×C1711yescharacteristic
483Q8—44no—
484Q8—44no—
485D4—24no—
486D4—44no—
487D4—44no—
488D4—44no—
489D4—42no—
490D4—48no—
491D4—44no—
492D4—44no—
493D4—82no—
494D4—82no—
495D4—82no—
496D4—84no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4C17×#12011yescharacteristic
524C2×C4C17×#12011yescharacteristic
525C2×C4C17×#12012yes—
526C2×C4C17×AGL(1,ℤ/8)12yes—
527C2×C4C22×D4×C1712yes—
528C8—44no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—44no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—88no—
545C22—81no—
546C22—81no—
547C22C2×C17×#12011yescharacteristic
548C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
549C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
550C4—21no—
551C4—22no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—81no—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—82no—
565C2—84no—
566C2C17×#300311yescharacteristic
567C2C17×#302111yescharacteristic
568C2C17×#302511yescharacteristic
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×#3449C2semidirect
2. C17×#3109C2semidirect
3. C17×#2598C2semidirect
4. C17×#1616C2semidirect
5. C17×#1613C2semidirect
6. C17×#1598C2semidirect
7. C17×#1595C2semidirect
8. C17×#1514C2semidirect
9. C17×#1464C2semidirect
20. C2×C4×D4×C17C22semidirect
22. C17×#481C22semidirect
25. C17×#447C22semidirect
27. C17×#432C22semidirect
28. C17×#400C22semidirect
30. C17×#394C22semidirect
36. C17×#383C22semidirect
37. C17×#327C22semidirect
38. C17×#327C22semidirect
39. C17×#324C22semidirect
57. C17×C4⋊Q8D4semidirect
59. C17×#127D4semidirect
79. C4×D4×C17D4semidirect
181. #18152C17direct (N × H)
245. #3449C2×C17semidirect
246. #3109C2×C17semidirect
247. #2598C2×C17semidirect
248. #1616C2×C17semidirect
249. #1613C2×C17semidirect
250. #1598C2×C17semidirect
251. #1595C2×C17semidirect
252. #1514C2×C17semidirect
253. #1464C2×C17semidirect
296. C2×C4×D4C22×C17semidirect
298. #481C22×C17semidirect
301. #447C22×C17semidirect
303. #432C22×C17semidirect
304. #400C22×C17semidirect
306. #394C22×C17semidirect
312. #383C22×C17semidirect
313. #327C22×C17semidirect
314. #327C22×C17semidirect
315. #324C22×C17semidirect
341. C4⋊Q8D4×C17semidirect
343. #127D4×C17semidirect
363. C4×D4D4×C17semidirect
391. C17#18152direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34492
C17×#31092
C17×#25982
C17×#16162
C17×#16132
C17×#15982
C17×#15952
C17×#15142
C17×#14642
#1815217