Finite Group Databasebeta

← C17×#18361 #16876105

Subgroups — C17×#18361 #16876105

Order 4352.

930 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3491C211yesmaximal, characteristic
2C17×#3238C212yesmaximal, complemented
3C17×#2621C211yesmaximal, characteristic, complemented
4C17×#1609C214yesmaximal, complemented
5C17×#1608C212yesmaximal, complemented
6C17×#1607C212yesmaximal
7C17×#1512C212yesmaximal, complemented
8C17×#1464C211yesmaximal, characteristic, complemented
9C17×#552—22no—
10C17×#552—22no—
11C17×#551—22no—
12C17×#551—22no—
13C17×#543C2211yescharacteristic
14C17×#539—21no—
15C17×#539C2212yes—
16C17×#534C2211yescharacteristic
17C2×C17×C4⋊Q8C2211yescharacteristic, complemented
18C17×#529—21no—
19C17×#529C2212yes—
20C2×C17×#123—21no—
21C2×C17×C22⋊Q8—21no—
22C17×#518—21no—
23C2×C17×#117C2211yescharacteristic
24C17×#501—22no—
25C17×#489—22no—
26C17×#483C2214yescomplemented
27C17×#480C2214yes—
28C17×#477—22no—
29C17×#476—22no—
30C17×#432C2212yescomplemented
31C17×#398—21no—
32C17×#398C2212yescomplemented
33C17×#397—21no—
34C17×#397C2212yes—
35C17×#391—21no—
36C17×#382C2211yescharacteristic, complemented
37C17×#328C2214yescomplemented
38C17×#327C2214yes—
39C17×#324C2212yescomplemented
40C17×#323C2212yes—
41C22×Q8×C17—21no—
42C22×D4×C17—21no—
43C23×C4×C17D411yescharacteristic
44C17×C4⋊Q8—22no—
45C17×C4⋊Q8—22no—
46C17×#126—22no—
47C17×#126—22no—
48C17×#126—22no—
49C17×#126—22no—
50C17×#125—21no—
51C17×#125—22no—
52C17×#125—22no—
53C17×#125D412yes—
54C17×#124—21no—
55C17×#124D412yes—
56C17×#123—22no—
57C17×#123—22no—
58C17×#123—22no—
59C17×#123—22no—
60C17×#123—22no—
61C17×#123—22no—
62C17×#123—22no—
63C17×C22⋊Q8—22no—
64C17×C22⋊Q8—22no—
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8—21no—
67C17×C22⋊Q8—22no—
68C17×C22⋊Q8—44no—
69C17×C22⋊Q8C2312yes—
70C17×#121—21no—
71C17×#121C2312yes—
72C17×#120—22no—
73C4×Q8×C17—21no—
74C4×Q8×C17D412yes—
75C4×D4×C17—22no—
76C4×D4×C17—22no—
77C4×D4×C17—21no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C4×D4×C17D412yes—
81C17×#117—22no—
82C17×#117—22no—
83C17×#117—22no—
84C17×#117—22no—
85C17×#117—22no—
86C2×C4⋊C4×C17—21no—
87C2×C4⋊C4×C17—21no—
88C2×C4⋊C4×C17—21no—
89C2×C4⋊C4×C17—22no—
90C2×C4⋊C4×C17—21no—
91C2×C4⋊C4×C17D411yescharacteristic
92C2×C4⋊C4×C17C2312yes—
93C2×C4⋊C4×C17C2311yescharacteristic
94C2×C22⋊C4×C17—22no—
95C2×C22⋊C4×C17—21no—
96C2×C22⋊C4×C17—21no—
97C2×C22⋊C4×C17—21no—
98C2×C22⋊C4×C17—22no—
99C2×C22⋊C4×C17D411yescharacteristic
100C2×C22⋊C4×C17C2311yescharacteristic
101C2×C42×C17D411yescharacteristic
102C2×C42×C17C2311yescharacteristic
103C17×#107—24no—
104C17×#106—24no—
105C17×C4⋊C8—22no—
106C17×#103—24no—
107C17×#102—24no—
108C17×C22⋊C8C2314yes—
109C17×#97—22no—
110C17×#95—21no—
111C17×#95—22no—
112C17×#95—22no—
113C17×#95C2312yes—
114C24×C17—21no—
115C24×C17C2×D411yescharacteristic
116C2×Q8×C17—21no—
117C2×Q8×C17—42no—
118C2×Q8×C17—42no—
119C2×Q8×C17—41no—
120C2×Q8×C17—41no—
121C2×Q8×C17C2×D412yes—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—21no—
125C2×D4×C17—42no—
126C2×D4×C17—42no—
127C2×D4×C17C2×D412yes—
128C22×C4×C17—21no—
129C22×C4×C17—22no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—22no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—41no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C2×C8×C17—24no—
146C4⋊C4×C17—22no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17—22no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—22no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—22no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—24no—
159C4⋊C4×C17—42no—
160C4⋊C4×C17—44no—
161C4⋊C4×C17—42no—
162C4⋊C4×C17C2×D412yes—
163C4⋊C4×C17C2×D412yes—
164C22⋊C4×C17—22no—
165C22⋊C4×C17—22no—
166C22⋊C4×C17—22no—
167C22⋊C4×C17—22no—
168C22⋊C4×C17—22no—
169C22⋊C4×C17—22no—
170C22⋊C4×C17—22no—
171C22⋊C4×C17—22no—
172C22⋊C4×C17—22no—
173C22⋊C4×C17—22no—
174C22⋊C4×C17—22no—
175C22⋊C4×C17—22no—
176C22⋊C4×C17—44no—
177C42×C17—22no—
178C42×C17—21no—
179C42×C17—22no—
180C42×C17—21no—
181C42×C17C2×D412yes—
182C42×C17C2×D412yes—
183#18361C1711yesmaximal, characteristic, direct product factor, Sylow
184C23×C17—21no—
185C23×C17—21no—
186C23×C17—22no—
187C23×C17—41no—
188C23×C17—41no—
189C23×C17—42no—
190C23×C17—81no—
191C23×C17—81no—
192C23×C17AGL(1,ℤ/8)12yes—
193C23×C17C22×D411yescharacteristic
194Q8×C17—42no—
195Q8×C17—82no—
196Q8×C17—82no—
197Q8×C17—81no—
198D4×C17—42no—
199D4×C17—44no—
200D4×C17—42no—
201D4×C17—42no—
202D4×C17—81no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—82no—
227C2×C4×C17—81no—
228C2×C4×C17—82no—
229C2×C4×C17—81no—
230C2×C4×C17#12012yes—
231C2×C4×C17#12011yescharacteristic
232C2×C4×C17#12011yescharacteristic
233C2×C4×C17C22×D412yes—
234C8×C17—44no—
235#3491C2×C1711yescharacteristic
236#3238C2×C1712yescomplemented
237#2621C2×C1711yescharacteristic, complemented
238#1609C2×C1714yescomplemented
239#1608C2×C1712yescomplemented
240#1607C2×C1712yes—
241#1512C2×C1712yescomplemented
242#1464C2×C1711yescharacteristic, complemented
243C22×C17—21no—
244C22×C17—21no—
245C22×C17—22no—
246C22×C17—42no—
247C22×C17—41no—
248C22×C17—41no—
249C22×C17—42no—
250C22×C17—42no—
251C22×C17—42no—
252C22×C17—81no—
253C22×C17—162no—
254C22×C17C2×#12011yescharacteristic
255C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C17#57511yescharacteristic
257C4×C17—21no—
258C4×C17—21no—
259C4×C17—22no—
260C4×C17—22no—
261C4×C17—41no—
262C4×C17—42no—
263C4×C17—41no—
264C4×C17—42no—
265C4×C17—41no—
266C4×C17—42no—
267C4×C17—82no—
268C4×C17—82no—
269#552—22no—
270#552—22no—
271#551—22no—
272#551—22no—
273#543C22×C1711yescharacteristic
274#539—21no—
275#539C22×C1712yes—
276#534C22×C1711yescharacteristic
277C2×C4⋊Q8C22×C1711yescharacteristic, complemented
278#529—21no—
279#529C22×C1712yes—
280C2×#123—21no—
281C2×C22⋊Q8—21no—
282#518—21no—
283C2×#117C22×C1711yescharacteristic
284#501—22no—
285#489—22no—
286#483C22×C1714yescomplemented
287#480C22×C1714yes—
288#477—22no—
289#476—22no—
290#432C22×C1712yescomplemented
291#398—21no—
292#398C22×C1712yescomplemented
293#397—21no—
294#397C22×C1712yes—
295#391—21no—
296#382C22×C1711yescharacteristic, complemented
297#328C22×C1714yescomplemented
298#327C22×C1714yes—
299#324C22×C1712yescomplemented
300#323C22×C1712yes—
301C2×C17—21no—
302C2×C17—21no—
303C2×C17—42no—
304C2×C17—82no—
305C2×C17#300611yescharacteristic
306C2×C17#302211yescharacteristic
307C2×C17#302811yescharacteristic
308C22×Q8—21no—
309C22×D4—21no—
310C23×C4D4×C1711yescharacteristic
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313#126—22no—
314#126—22no—
315#126—22no—
316#126—22no—
317#125—21no—
318#125—22no—
319#125—22no—
320#125D4×C1712yes—
321#124—21no—
322#124D4×C1712yes—
323#123—22no—
324#123—22no—
325#123—22no—
326#123—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—21no—
334C22⋊Q8—22no—
335C22⋊Q8—44no—
336C22⋊Q8C23×C1712yes—
337#121—21no—
338#121C23×C1712yes—
339#120—22no—
340C4×Q8—21no—
341C4×Q8D4×C1712yes—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—21no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4D4×C1712yes—
348#117—22no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4D4×C1711yescharacteristic
359C2×C4⋊C4C23×C1712yes—
360C2×C4⋊C4C23×C1711yescharacteristic
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4D4×C1711yescharacteristic
367C2×C22⋊C4C23×C1711yescharacteristic
368C2×C42D4×C1711yescharacteristic
369C2×C42C23×C1711yescharacteristic
370#107—24no—
371#106—24no—
372C4⋊C8—22no—
373#103—24no—
374#102—24no—
375C22⋊C8C23×C1714yes—
376#97—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95C23×C1712yes—
381C17#1836111yescharacteristic, direct product factor, Sylow
382C24—21no—
383C24C2×D4×C1711yescharacteristic
384C2×Q8—21no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8C2×D4×C1712yes—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4C2×D4×C1712yes—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—41no—
407C22×C4C2×D4×C1711yescharacteristic
408C22×C4C2×D4×C1711yescharacteristic
409C22×C4C2×D4×C1711yescharacteristic
410C22×C4C2×D4×C1711yescharacteristic
411C22×C4C2×D4×C1711yescharacteristic
412C22×C4C24×C1711yescharacteristic
413C2×C8—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4C2×D4×C1712yes—
431C4⋊C4C2×D4×C1712yes—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—44no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42C2×D4×C1712yes—
450C42C2×D4×C1712yes—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—81no—
458C23—81no—
459C23C17×AGL(1,ℤ/8)12yes—
460C23C22×D4×C1711yescharacteristic
461Q8—42no—
462Q8—82no—
463Q8—82no—
464Q8—81no—
465D4—42no—
466D4—44no—
467D4—42no—
468D4—42no—
469D4—81no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4C17×#12012yes—
498C2×C4C17×#12011yescharacteristic
499C2×C4C17×#12011yescharacteristic
500C2×C4C22×D4×C1712yes—
501C8—44no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—81no—
512C22—162no—
513C22C2×C17×#12011yescharacteristic
514C22C2×C17×AGL(1,ℤ/8)11yescharacteristic
515C22C17×#57511yescharacteristic
516C4—21no—
517C4—21no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—82no—
527C4—82no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—82no—
532C2C17×#300611yescharacteristic
533C2C17×#302211yescharacteristic
534C2C17×#302811yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#534C22
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×#3238C2semidirect
3. C17×#2621C2semidirect
4. C17×#1609C2semidirect
5. C17×#1608C2semidirect
7. C17×#1512C2semidirect
8. C17×#1464C2semidirect
17. C2×C17×C4⋊Q8C22semidirect
26. C17×#483C22semidirect
30. C17×#432C22semidirect
32. C17×#398C22semidirect
36. C17×#382C22semidirect
37. C17×#328C22semidirect
39. C17×#324C22semidirect
183. #18361C17direct (N × H)
236. #3238C2×C17semidirect
237. #2621C2×C17semidirect
238. #1609C2×C17semidirect
239. #1608C2×C17semidirect
241. #1512C2×C17semidirect
242. #1464C2×C17semidirect
277. C2×C4⋊Q8C22×C17semidirect
286. #483C22×C17semidirect
290. #432C22×C17semidirect
292. #398C22×C17semidirect
296. #382C22×C17semidirect
297. #328C22×C17semidirect
299. #324C22×C17semidirect
381. C17#18361direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34912
C17×#32382
C17×#26212
C17×#16092
C17×#16082
C17×#16072
C17×#15122
C17×#14642
#1836117