Finite Group Databasebeta

← C17×#18372 #16876116

Subgroups — C17×#18372 #16876116

Order 4352.

904 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3493C211yesmaximal, characteristic
2C17×#3239C212yesmaximal, complemented
3C17×#2628C211yesmaximal, characteristic, complemented
4C17×#1618C214yesmaximal, complemented
5C17×#1617C212yesmaximal, complemented
6C17×#1616C212yesmaximal, complemented
7C17×#1512C212yesmaximal, complemented
8C17×#1464C211yesmaximal, characteristic, complemented
9C17×#556—22no—
10C17×#555—22no—
11C17×#553—22no—
12C17×#552—22no—
13C17×#544C2211yescharacteristic
14C17×#538—21no—
15C17×#538C2212yes—
16C17×#537C2211yescharacteristic
17C2×C17×C4⋊Q8C2211yescharacteristic
18C17×#529—21no—
19C17×#529C2212yes—
20C17×#525—21no—
21C17×#520—21no—
22C2×C17×#117C2211yescharacteristic
23C17×#501—22no—
24C17×#489—22no—
25C17×#484C2214yes—
26C17×#481C2214yes—
27C17×#475—22no—
28C17×#474—22no—
29C17×#432C2212yes—
30C17×#400C2212yes—
31C17×#399C2212yes—
32C17×#395—21no—
33C17×#393—21no—
34C17×#389—21no—
35C17×#382C2211yescharacteristic
36C17×#380—21no—
37C17×#328C2214yes—
38C17×#327C2214yes—
39C17×#324C2212yes—
40C17×#323C2212yes—
41C2×#41×C17—21no—
42C22×Q8×C17—21no—
43C23×C4×C17D411yescharacteristic
44C17×C4⋊Q8—22no—
45C17×C4⋊Q8—22no—
46C17×C4⋊Q8—22no—
47C17×#126—22no—
48C17×#126—22no—
49C17×#126—22no—
50C17×#126—22no—
51C17×#125—21no—
52C17×#125—22no—
53C17×#125D412yes—
54C17×#124—21no—
55C17×#124—22no—
56C17×#124—22no—
57C17×#124D412yes—
58C17×#123—22no—
59C17×#123—22no—
60C17×#123—22no—
61C17×C22⋊Q8—21no—
62C17×C22⋊Q8—21no—
63C17×C22⋊Q8—21no—
64C17×C22⋊Q8—22no—
65C17×C22⋊Q8C2312yes—
66C17×#121—21no—
67C17×#121—21no—
68C17×#121—22no—
69C17×#121—22no—
70C17×#121—22no—
71C17×#121—21no—
72C17×#121C2312yes—
73C17×#120—22no—
74C4×Q8×C17—22no—
75C4×Q8×C17—21no—
76C4×Q8×C17D412yes—
77C4×D4×C17—22no—
78C4×D4×C17—22no—
79C4×D4×C17—22no—
80C4×D4×C17—21no—
81C4×D4×C17—22no—
82C4×D4×C17—22no—
83C4×D4×C17D412yes—
84C17×#117—22no—
85C17×#117—22no—
86C17×#117—21no—
87C17×#117—22no—
88C17×#117—22no—
89C17×#117—22no—
90C2×C4⋊C4×C17—22no—
91C2×C4⋊C4×C17—21no—
92C2×C4⋊C4×C17—21no—
93C2×C4⋊C4×C17—22no—
94C2×C4⋊C4×C17D411yescharacteristic
95C2×C4⋊C4×C17C2312yes—
96C2×C4⋊C4×C17C2311yescharacteristic
97C2×C22⋊C4×C17—22no—
98C2×C22⋊C4×C17D411yescharacteristic
99C2×C22⋊C4×C17C2311yescharacteristic
100C2×C42×C17D411yescharacteristic
101C2×C42×C17C2311yescharacteristic
102C17×#107—24no—
103C17×#106—24no—
104C17×C4⋊C8—22no—
105C17×#103—24no—
106C17×#102—24no—
107C17×C22⋊C8C2314yes—
108C17×#97—22no—
109C17×#95—22no—
110C17×#95—21no—
111C17×#95—22no—
112C17×#95C2312yes—
113C24×C17C2×D411yescharacteristic
114#41×C17—42no—
115#41×C17—42no—
116C2×Q8×C17—21no—
117C2×Q8×C17—21no—
118C2×Q8×C17—42no—
119C2×Q8×C17—82no—
120C2×Q8×C17C2×D412yes—
121C2×D4×C17—21no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—22no—
125C2×D4×C17—21no—
126C2×D4×C17C2×D412yes—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—41no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2411yescharacteristic
142C2×C8×C17—24no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—22no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17—22no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—22no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—24no—
156C4⋊C4×C17—42no—
157C4⋊C4×C17—41no—
158C4⋊C4×C17—41no—
159C4⋊C4×C17—82no—
160C4⋊C4×C17C2×D412yes—
161C4⋊C4×C17C2×D412yes—
162C22⋊C4×C17—22no—
163C22⋊C4×C17—22no—
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—82no—
174C42×C17—22no—
175C42×C17—22no—
176C42×C17—21no—
177C42×C17—22no—
178C42×C17—21no—
179C42×C17C2×D412yes—
180C42×C17C2×D412yes—
181#18372C1711yesmaximal, characteristic, direct product factor, Sylow
182C23×C17—21no—
183C23×C17—22no—
184C23×C17—21no—
185C23×C17—21no—
186C23×C17—41no—
187C23×C17—41no—
188C23×C17C22×D411yescharacteristic
189Q8×C17—42no—
190Q8×C17—42no—
191Q8×C17—84no—
192Q8×C17—81no—
193D4×C17—42no—
194D4×C17—42no—
195D4×C17—42no—
196D4×C17—42no—
197D4×C17—82no—
198D4×C17—81no—
199D4×C17—82no—
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—44no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—82no—
223C2×C4×C17—82no—
224C2×C4×C17#12011yescharacteristic
225C2×C4×C17#12011yescharacteristic
226C2×C4×C17#12012yes—
227C2×C4×C17#13712yes—
228C2×C4×C17C22×D412yes—
229C8×C17—44no—
230#3493C2×C1711yescharacteristic
231#3239C2×C1712yescomplemented
232#2628C2×C1711yescharacteristic, complemented
233#1618C2×C1714yescomplemented
234#1617C2×C1712yescomplemented
235#1616C2×C1712yescomplemented
236#1512C2×C1712yescomplemented
237#1464C2×C1711yescharacteristic, complemented
238C22×C17—21no—
239C22×C17—21no—
240C22×C17—42no—
241C22×C17—41no—
242C22×C17—42no—
243C22×C17—42no—
244C22×C17—41no—
245C22×C17—42no—
246C22×C17—41no—
247C22×C17—81no—
248C22×C17C2×#12011yescharacteristic
249C22×C17C2×#13711yescharacteristic
250C22×C17#57511yescharacteristic
251C4×C17—22no—
252C4×C17—21no—
253C4×C17—21no—
254C4×C17—22no—
255C4×C17—22no—
256C4×C17—42no—
257C4×C17—41no—
258C4×C17—42no—
259C4×C17—41no—
260C4×C17—42no—
261C4×C17—82no—
262C4×C17—82no—
263#556—22no—
264#555—22no—
265#553—22no—
266#552—22no—
267#544C22×C1711yescharacteristic
268#538—21no—
269#538C22×C1712yes—
270#537C22×C1711yescharacteristic
271C2×C4⋊Q8C22×C1711yescharacteristic
272#529—21no—
273#529C22×C1712yes—
274#525—21no—
275#520—21no—
276C2×#117C22×C1711yescharacteristic
277#501—22no—
278#489—22no—
279#484C22×C1714yes—
280#481C22×C1714yes—
281#475—22no—
282#474—22no—
283#432C22×C1712yes—
284#400C22×C1712yes—
285#399C22×C1712yes—
286#395—21no—
287#393—21no—
288#389—21no—
289#382C22×C1711yescharacteristic
290#380—21no—
291#328C22×C1714yes—
292#327C22×C1714yes—
293#324C22×C1712yes—
294#323C22×C1712yes—
295C2×C17—21no—
296C2×C17—21no—
297C2×C17—82no—
298C2×C17—81no—
299C2×C17#300711yescharacteristic
300C2×C17#302111yescharacteristic
301C2×C17#302811yescharacteristic
302C2×#41—21no—
303C22×Q8—21no—
304C23×C4D4×C1711yescharacteristic
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308#126—22no—
309#126—22no—
310#126—22no—
311#126—22no—
312#125—21no—
313#125—22no—
314#125D4×C1712yes—
315#124—21no—
316#124—22no—
317#124—22no—
318#124D4×C1712yes—
319#123—22no—
320#123—22no—
321#123—22no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—22no—
326C22⋊Q8C23×C1712yes—
327#121—21no—
328#121—21no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—21no—
333#121C23×C1712yes—
334#120—22no—
335C4×Q8—22no—
336C4×Q8—21no—
337C4×Q8D4×C1712yes—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—21no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4D4×C1712yes—
345#117—22no—
346#117—22no—
347#117—21no—
348#117—22no—
349#117—22no—
350#117—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4D4×C1711yescharacteristic
356C2×C4⋊C4C23×C1712yes—
357C2×C4⋊C4C23×C1711yescharacteristic
358C2×C22⋊C4—22no—
359C2×C22⋊C4D4×C1711yescharacteristic
360C2×C22⋊C4C23×C1711yescharacteristic
361C2×C42D4×C1711yescharacteristic
362C2×C42C23×C1711yescharacteristic
363#107—24no—
364#106—24no—
365C4⋊C8—22no—
366#103—24no—
367#102—24no—
368C22⋊C8C23×C1714yes—
369#97—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95C23×C1712yes—
374C17#1837211yescharacteristic, direct product factor, Sylow
375C24C2×D4×C1711yescharacteristic
376#41—42no—
377#41—42no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—42no—
381C2×Q8—82no—
382C2×Q8C2×D4×C1712yes—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4C2×D4×C1712yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4C2×D4×C1711yescharacteristic
399C22×C4C2×D4×C1711yescharacteristic
400C22×C4C2×D4×C1711yescharacteristic
401C22×C4C2×D4×C1711yescharacteristic
402C22×C4C2×D4×C1711yescharacteristic
403C22×C4C24×C1711yescharacteristic
404C2×C8—24no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—82no—
422C4⋊C4C2×D4×C1712yes—
423C4⋊C4C2×D4×C1712yes—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—82no—
436C42—22no—
437C42—22no—
438C42—21no—
439C42—21no—
440C42—22no—
441C42C2×D4×C1712yes—
442C42C2×D4×C1712yes—
443C23—21no—
444C23—22no—
445C23—21no—
446C23—21no—
447C23—41no—
448C23—41no—
449C23C22×D4×C1711yescharacteristic
450Q8—42no—
451Q8—42no—
452Q8—84no—
453Q8—81no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—81no—
460D4—82no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4C17×#12011yescharacteristic
486C2×C4C17×#12011yescharacteristic
487C2×C4C17×#12012yes—
488C2×C4C17×#13712yes—
489C2×C4C22×D4×C1712yes—
490C8—44no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—42no—
499C22—41no—
500C22—81no—
501C22C2×C17×#12011yescharacteristic
502C22C2×C17×#13711yescharacteristic
503C22C17×#57511yescharacteristic
504C4—22no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—82no—
515C4—82no—
516C2—21no—
517C2—21no—
518C2—81no—
519C2—82no—
520C2C17×#300711yescharacteristic
521C2C17×#302111yescharacteristic
522C2C17×#302811yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC22×C4C24×C17
prime radicalC17×#3493C2
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×#3239C2semidirect
3. C17×#2628C2semidirect
4. C17×#1618C2semidirect
5. C17×#1617C2semidirect
6. C17×#1616C2semidirect
7. C17×#1512C2semidirect
8. C17×#1464C2semidirect
181. #18372C17direct (N × H)
231. #3239C2×C17semidirect
232. #2628C2×C17semidirect
233. #1618C2×C17semidirect
234. #1617C2×C17semidirect
235. #1616C2×C17semidirect
236. #1512C2×C17semidirect
237. #1464C2×C17semidirect
374. C17#18372direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34932
C17×#32392
C17×#26282
C17×#16182
C17×#16172
C17×#16162
C17×#15122
C17×#14642
#1837217