Finite Group Databasebeta

← C17×#18351 #16876095

Subgroups — C17×#18351 #16876095

Order 4352.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3496C211yesmaximal, characteristic
2C17×#3193C212yesmaximal, complemented
3C17×#2638C211yesmaximal, characteristic, complemented
4C17×#1610C212yesmaximal, complemented
5C17×#1607C212yesmaximal, complemented
6C17×#1604C212yesmaximal, complemented
7C17×#1601C212yesmaximal, complemented
8C17×#1510C212yesmaximal, complemented
9C17×#1465C211yesmaximal, characteristic, complemented
10C2×C17×H(5,𝔽2)—21no—
11C17×#547—22no—
12C17×#546—24no—
13D42×C17—22no—
14C17×#537C2212yes—
15C17×#534—21no—
16C17×#534C2212yes—
17C17×#532—21no—
18C2×C17×C22⋊Q8C2211yescharacteristic, complemented
19C2×C17×#121—21no—
20C2×C17×#121C2212yes—
21C2×C4×D4×C17C2211yescharacteristic
22C17×#484—22no—
23C17×#480C2214yescomplemented
24C17×#468—24no—
25C17×#450—22no—
26C17×#448C2214yescomplemented
27C17×#407C2212yescomplemented
28C17×#398C2211yescharacteristic, complemented
29C17×#397—21no—
30C17×#397C2211yescharacteristic, complemented
31C17×#395C2211yescharacteristic, complemented
32C17×#394—21no—
33C17×#393C2211yescharacteristic, complemented
34C17×#387—21no—
35C17×#385—21no—
36C17×#381C2211yescharacteristic, complemented
37C17×#327C2214yescomplemented
38C17×#327C2214yescomplemented
39C17×#324C2212yescomplemented
40C17×#323C2212yescomplemented
41C17×H(5,𝔽2)—44no—
42C17×H(5,𝔽2)—44no—
43C2×#41×C17—24no—
44C2×#41×C17—22no—
45C22×Q8×C17C2311yescharacteristic
46C22×D4×C17—24no—
47C22×D4×C17—22no—
48C22×D4×C17—22no—
49C22×D4×C17—21no—
50C22×D4×C17D411yescharacteristic
51C22×D4×C17D412yes—
52C23×C4×C17D411yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C2×SD16×C17—24no—
55C2×M16×C17—22no—
56C17×C4⋊Q8—21no—
57C17×#127—21no—
58C17×#124—22no—
59C17×#123—24no—
60C17×#123—24no—
61C17×C22⋊Q8—24no—
62C17×C22⋊Q8—22no—
63C17×C22⋊Q8—22no—
64C17×#121—22no—
65C17×#121—22no—
66C17×#121—24no—
67C17×#121—22no—
68C17×#121—22no—
69C17×#121—22no—
70C17×#121—22no—
71C17×#121D414yes—
72C17×#121C2314yes—
73C17×#120—24no—
74C17×#120—24no—
75C4×D4×C17—22no—
76C4×D4×C17—22no—
77C4×D4×C17—22no—
78C4×D4×C17—22no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17D412yes—
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C4⋊C4×C17C2311yescharacteristic
86C2×C22⋊C4×C17—22no—
87C2×C22⋊C4×C17—22no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17—21no—
90C2×C22⋊C4×C17—21no—
91C2×C22⋊C4×C17—21no—
92C2×C22⋊C4×C17D411yescharacteristic
93C2×C22⋊C4×C17C2311yescharacteristic
94C2×C42×C17D411yescharacteristic
95C17×#107—24no—
96C17×#103—24no—
97C17×#102—24no—
98C17×C22⋊C8—22no—
99C17×C22⋊C8C2314yes—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95C2311yescharacteristic
106C17×#95C2311yescharacteristic
107C24×C17—22no—
108C24×C17—22no—
109C24×C17C2×D411yescharacteristic
110C24×C17C2×D411yescharacteristic
111#41×C17—44no—
112#41×C17—416no—
113#41×C17—44no—
114C2×Q8×C17—22no—
115C2×Q8×C17—22no—
116C2×Q8×C17—81no—
117C2×D4×C17—22no—
118C2×D4×C17—22no—
119C2×D4×C17—24no—
120C2×D4×C17—22no—
121C2×D4×C17—24no—
122C2×D4×C17—24no—
123C2×D4×C17—22no—
124C2×D4×C17—22no—
125C2×D4×C17—22no—
126C2×D4×C17—22no—
127C2×D4×C17—416no—
128C2×D4×C17—44no—
129C2×D4×C17—44no—
130C2×D4×C17—44no—
131C2×D4×C17—44no—
132C2×D4×C17—82no—
133C2×D4×C17—82no—
134C2×D4×C17C2×D414yes—
135C22×C4×C17—24no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—81no—
145C22×C4×C17—81no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D412yes—
151C22×C4×C17C2×D412yes—
152C22×C4×C17C2411yescharacteristic
153SD16×C17—84no—
154M16×C17—82no—
155C2×C8×C17—24no—
156C4⋊C4×C17—22no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—22no—
159C4⋊C4×C17—22no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—41no—
162C4⋊C4×C17—41no—
163C4⋊C4×C17—42no—
164C4⋊C4×C17—81no—
165C4⋊C4×C17C2×D414yes—
166C22⋊C4×C17—24no—
167C22⋊C4×C17—24no—
168C22⋊C4×C17—22no—
169C22⋊C4×C17—24no—
170C22⋊C4×C17—24no—
171C22⋊C4×C17—22no—
172C22⋊C4×C17—42no—
173C22⋊C4×C17—42no—
174C22⋊C4×C17—81no—
175C22⋊C4×C17—81no—
176C42×C17—21no—
177C42×C17—21no—
178#18351C1711yesmaximal, characteristic, direct product factor, Sylow
179C23×C17—21no—
180C23×C17—21no—
181C23×C17—21no—
182C23×C17—21no—
183C23×C17—24no—
184C23×C17—41no—
185C23×C17—41no—
186C23×C17—41no—
187C23×C17—41no—
188C23×C17—44no—
189C23×C17—44no—
190C23×C17—82no—
191C23×C17—82no—
192C23×C17—82no—
193C23×C17—82no—
194C23×C17#12012yes—
195C23×C17AGL(1,ℤ/8)12yes—
196C23×C17C22×D411yescharacteristic
197Q8×C17—44no—
198Q8×C17—44no—
199D4×C17—28no—
200D4×C17—44no—
201D4×C17—44no—
202D4×C17—44no—
203D4×C17—48no—
204D4×C17—48no—
205D4×C17—44no—
206D4×C17—44no—
207D4×C17—82no—
208D4×C17—82no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—24no—
214C2×C4×C17—42no—
215C2×C4×C17—41no—
216C2×C4×C17—42no—
217C2×C4×C17—41no—
218C2×C4×C17—42no—
219C2×C4×C17—48no—
220C2×C4×C17—44no—
221C2×C4×C17—44no—
222C2×C4×C17—41no—
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—81no—
237C2×C4×C17#12011yescharacteristic
238C2×C4×C17#12011yescharacteristic
239C2×C4×C17AGL(1,ℤ/8)12yes—
240C2×C4×C17C22×D412yes—
241C8×C17—44no—
242#3496C2×C1711yescharacteristic
243#3193C2×C1712yescomplemented
244#2638C2×C1711yescharacteristic, complemented
245#1610C2×C1712yescomplemented
246#1607C2×C1712yescomplemented
247#1604C2×C1712yescomplemented
248#1601C2×C1712yescomplemented
249#1510C2×C1712yescomplemented
250#1465C2×C1711yescharacteristic, complemented
251C22×C17—21no—
252C22×C17—21no—
253C22×C17—22no—
254C22×C17—22no—
255C22×C17—22no—
256C22×C17—22no—
257C22×C17—42no—
258C22×C17—41no—
259C22×C17—41no—
260C22×C17—44no—
261C22×C17—44no—
262C22×C17—44no—
263C22×C17—81no—
264C22×C17—81no—
265C22×C17—81no—
266C22×C17—164no—
267C22×C17—164no—
268C22×C17C2×#12011yescharacteristic
269C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
271C4×C17—22no—
272C4×C17—22no—
273C4×C17—21no—
274C4×C17—21no—
275C4×C17—44no—
276C4×C17—41no—
277C4×C17—41no—
278C4×C17—81no—
279C4×C17—81no—
280C4×C17—81no—
281C4×C17—81no—
282C2×H(5,𝔽2)—21no—
283#547—22no—
284#546—24no—
285D42—22no—
286#537C22×C1712yes—
287#534—21no—
288#534C22×C1712yes—
289#532—21no—
290C2×C22⋊Q8C22×C1711yescharacteristic, complemented
291C2×#121—21no—
292C2×#121C22×C1712yes—
293C2×C4×D4C22×C1711yescharacteristic
294#484—22no—
295#480C22×C1714yescomplemented
296#468—24no—
297#450—22no—
298#448C22×C1714yescomplemented
299#407C22×C1712yescomplemented
300#398C22×C1711yescharacteristic, complemented
301#397—21no—
302#397C22×C1711yescharacteristic, complemented
303#395C22×C1711yescharacteristic, complemented
304#394—21no—
305#393C22×C1711yescharacteristic, complemented
306#387—21no—
307#385—21no—
308#381C22×C1711yescharacteristic, complemented
309#327C22×C1714yescomplemented
310#327C22×C1714yescomplemented
311#324C22×C1712yescomplemented
312#323C22×C1712yescomplemented
313C2×C17—21no—
314C2×C17—21no—
315C2×C17—42no—
316C2×C17—42no—
317C2×C17—84no—
318C2×C17#300311yescharacteristic
319C2×C17#302211yescharacteristic
320C2×C17#302611yescharacteristic
321H(5,𝔽2)—44no—
322H(5,𝔽2)—44no—
323C2×#41—24no—
324C2×#41—22no—
325C22×Q8C23×C1711yescharacteristic
326C22×D4—24no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—21no—
330C22×D4D4×C1711yescharacteristic
331C22×D4D4×C1712yes—
332C23×C4D4×C1711yescharacteristic
333C23×C4C23×C1711yescharacteristic
334C2×SD16—24no—
335C2×M16—22no—
336C4⋊Q8—21no—
337#127—21no—
338#124—22no—
339#123—24no—
340#123—24no—
341C22⋊Q8—24no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344#121—22no—
345#121—22no—
346#121—24no—
347#121—22no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121D4×C1714yes—
352#121C23×C1714yes—
353#120—24no—
354#120—24no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4D4×C1712yes—
364C2×C4⋊C4C23×C1711yescharacteristic
365C2×C4⋊C4C23×C1711yescharacteristic
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4D4×C1711yescharacteristic
373C2×C22⋊C4C23×C1711yescharacteristic
374C2×C42D4×C1711yescharacteristic
375#107—24no—
376#103—24no—
377#102—24no—
378C22⋊C8—22no—
379C22⋊C8C23×C1714yes—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95C23×C1711yescharacteristic
386#95C23×C1711yescharacteristic
387C17#1835111yescharacteristic, direct product factor, Sylow
388C24—22no—
389C24—22no—
390C24C2×D4×C1711yescharacteristic
391C24C2×D4×C1711yescharacteristic
392#41—44no—
393#41—416no—
394#41—44no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—81no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—416no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4C2×D4×C1714yes—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4C2×D4×C1711yescharacteristic
428C22×C4C2×D4×C1711yescharacteristic
429C22×C4C2×D4×C1711yescharacteristic
430C22×C4C2×D4×C1712yes—
431C22×C4C2×D4×C1712yes—
432C22×C4C2×D4×C1711yescharacteristic
433C22×C4C24×C1711yescharacteristic
434SD16—84no—
435M16—82no—
436C2×C8—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—81no—
446C4⋊C4C2×D4×C1714yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C42—21no—
458C42—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—44no—
469C23—44no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23C17×#12012yes—
475C23C17×AGL(1,ℤ/8)12yes—
476C23C22×D4×C1711yescharacteristic
477Q8—44no—
478Q8—44no—
479D4—28no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—48no—
484D4—48no—
485D4—44no—
486D4—44no—
487D4—82no—
488D4—82no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4C17×#12011yescharacteristic
518C2×C4C17×#12011yescharacteristic
519C2×C4C17×AGL(1,ℤ/8)12yes—
520C2×C4C22×D4×C1712yes—
521C8—44no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—164no—
538C22—164no—
539C22C2×C17×#12011yescharacteristic
540C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
541C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C2—21no—
554C2—21no—
555C2—42no—
556C2—42no—
557C2—84no—
558C2C17×#300311yescharacteristic
559C2C17×#302211yescharacteristic
560C2C17×#302611yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#3496C2
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
2. C17×#3193C2semidirect
3. C17×#2638C2semidirect
4. C17×#1610C2semidirect
5. C17×#1607C2semidirect
6. C17×#1604C2semidirect
7. C17×#1601C2semidirect
8. C17×#1510C2semidirect
9. C17×#1465C2semidirect
18. C2×C17×C22⋊Q8C22semidirect
23. C17×#480C22semidirect
26. C17×#448C22semidirect
27. C17×#407C22semidirect
28. C17×#398C22semidirect
30. C17×#397C22semidirect
31. C17×#395C22semidirect
33. C17×#393C22semidirect
36. C17×#381C22semidirect
37. C17×#327C22semidirect
38. C17×#327C22semidirect
39. C17×#324C22semidirect
40. C17×#323C22semidirect
178. #18351C17direct (N × H)
243. #3193C2×C17semidirect
244. #2638C2×C17semidirect
245. #1610C2×C17semidirect
246. #1607C2×C17semidirect
247. #1604C2×C17semidirect
248. #1601C2×C17semidirect
249. #1510C2×C17semidirect
250. #1465C2×C17semidirect
290. C2×C22⋊Q8C22×C17semidirect
295. #480C22×C17semidirect
298. #448C22×C17semidirect
299. #407C22×C17semidirect
300. #398C22×C17semidirect
302. #397C22×C17semidirect
303. #395C22×C17semidirect
305. #393C22×C17semidirect
308. #381C22×C17semidirect
309. #327C22×C17semidirect
310. #327C22×C17semidirect
311. #324C22×C17semidirect
312. #323C22×C17semidirect
387. C17#18351direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34962
C17×#31932
C17×#26382
C17×#16102
C17×#16072
C17×#16042
C17×#16012
C17×#15102
C17×#14652
#1835117