Finite Group Databasebeta

← C17×#10396 #16868140

Subgroups — C17×#10396 #16868140

Order 4352.

632 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×#2618C211yesmaximal, characteristic
2C17×#1874C211yesmaximal, characteristic
3C17×#1853C211yesmaximal, characteristic, complemented
4C2×C17×#327C211yesmaximal, characteristic
5C2×C17×#324C211yesmaximal, characteristic, complemented
6C17×#1441C211yesmaximal, characteristic, complemented
7C17×#1294C211yesmaximal, characteristic, complemented
8C2×C17×#123—21no—
9C2×C17×#121C2211yescharacteristic
10C22×C4⋊C4×C17C2211yescharacteristic
11C22×C4⋊C4×C17C2211yescharacteristic
12C2×C17×#106—21no—
13C2×C17×#102—21no—
14C2×C17×C22⋊C8C2211yescharacteristic
15C2×C17×C22⋊C8C2211yescharacteristic
16C17×#398—21no—
17C17×#396—21no—
18C17×#396—21no—
19C17×#394—21no—
20C17×#386—21no—
21C17×#380C2211yescharacteristic
22C2×C17×#95—21no—
23C2×C17×#95—21no—
24C2×C17×#95C2211yescharacteristic
25C17×#340—21no—
26C17×#340—21no—
27C17×#327—22no—
28C17×#324—22no—
29C22×D4×C17—21no—
30C22×D4×C17D411yescharacteristic
31C23×C4×C17—21no—
32C23×C4×C17—21no—
33C23×C4×C17D411yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C22×C8×C17—21no—
37C22×C8×C17—21no—
38C17×#123—42no—
39C17×#123—42no—
40C17×#121—21no—
41C17×#121—21no—
42C17×#121—21no—
43C17×#121—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17—21no—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—21no—
53C2×C4⋊C4×C17—21no—
54C2×C4⋊C4×C17—42no—
55C2×C4⋊C4×C17D411yescharacteristic
56C2×C4⋊C4×C17D411yescharacteristic
57C2×C4⋊C4×C17D411yescharacteristic
58C2×C22⋊C4×C17—21no—
59C2×C22⋊C4×C17—21no—
60C2×C22⋊C4×C17—21no—
61C2×C22⋊C4×C17—21no—
62C2×C22⋊C4×C17—21no—
63C2×C22⋊C4×C17—21no—
64C17×#106—42no—
65C17×#102—42no—
66C17×C22⋊C8—21no—
67C17×C22⋊C8—21no—
68C17×C22⋊C8—21no—
69C17×C22⋊C8D412yes—
70C17×#95—21no—
71C17×#95—21no—
72C17×#95—21no—
73C17×#95—21no—
74C17×#95—21no—
75C17×#95—42no—
76C17×#95—42no—
77C24×C17—21no—
78C24×C17C2×D411yescharacteristic
79C2×D4×C17—21no—
80C2×D4×C17—21no—
81C2×D4×C17—41no—
82C2×D4×C17—41no—
83C2×D4×C17—41no—
84C2×D4×C17—41no—
85C2×D4×C17—41no—
86C2×D4×C17—41no—
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—21no—
94C22×C4×C17—21no—
95C22×C4×C17—22no—
96C22×C4×C17—22no—
97C22×C4×C17—21no—
98C22×C4×C17—21no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—41no—
104C22×C4×C17—41no—
105C22×C4×C17—41no—
106C22×C4×C17—41no—
107C22×C4×C17—41no—
108C22×C4×C17—41no—
109C22×C4×C17—41no—
110C22×C4×C17—41no—
111C22×C4×C17—41no—
112C22×C4×C17—41no—
113C22×C4×C17—41no—
114C22×C4×C17—41no—
115C22×C4×C17—41no—
116C22×C4×C17—41no—
117C22×C4×C17—41no—
118C22×C4×C17—41no—
119C22×C4×C17SD1612yes—
120C22×C4×C17C2×D411yescharacteristic
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17#4111yescharacteristic
124C22×C4×C17#4111yescharacteristic
125C22×C4×C17#4111yescharacteristic
126C2×C8×C17—22no—
127C2×C8×C17—41no—
128C2×C8×C17—41no—
129C2×C8×C17—41no—
130C2×C8×C17—41no—
131C2×C8×C17—41no—
132C4⋊C4×C17—22no—
133C4⋊C4×C17—22no—
134C4⋊C4×C17—21no—
135C4⋊C4×C17—21no—
136C4⋊C4×C17—41no—
137C4⋊C4×C17—41no—
138C4⋊C4×C17—42no—
139C4⋊C4×C17—42no—
140C4⋊C4×C17—41no—
141C4⋊C4×C17—41no—
142C4⋊C4×C17—41no—
143C4⋊C4×C17—41no—
144C4⋊C4×C17—41no—
145C4⋊C4×C17—41no—
146C22⋊C4×C17—42no—
147C22⋊C4×C17—42no—
148C22⋊C4×C17—41no—
149C22⋊C4×C17—41no—
150C22⋊C4×C17—41no—
151C22⋊C4×C17—41no—
152C22⋊C4×C17—81no—
153C22⋊C4×C17—81no—
154#10396C1711yesmaximal, characteristic, direct product factor, Sylow
155C23×C17—21no—
156C23×C17—21no—
157C23×C17—21no—
158C23×C17—21no—
159C23×C17—41no—
160C23×C17—41no—
161C23×C17—41no—
162C23×C17—41no—
163C23×C17—41no—
164C23×C17—41no—
165C23×C17—41no—
166C23×C17#12011yescharacteristic
167C23×C17#12111yescharacteristic
168C23×C17#12311yescharacteristic
169C23×C17C2×SD1611yescharacteristic
170C23×C17#13511yescharacteristic
171C23×C17AGL(1,ℤ/8)11yescharacteristic
172C23×C17AGL(1,ℤ/8)11yescharacteristic
173D4×C17—42no—
174D4×C17—81no—
175D4×C17—81no—
176D4×C17—81no—
177D4×C17—81no—
178D4×C17—81no—
179C2×C4×C17—21no—
180C2×C4×C17—21no—
181C2×C4×C17—21no—
182C2×C4×C17—21no—
183C2×C4×C17—22no—
184C2×C4×C17—22no—
185C2×C4×C17—22no—
186C2×C4×C17—41no—
187C2×C4×C17—41no—
188C2×C4×C17—41no—
189C2×C4×C17—41no—
190C2×C4×C17—41no—
191C2×C4×C17—41no—
192C2×C4×C17—41no—
193C2×C4×C17—41no—
194C2×C4×C17—42no—
195C2×C4×C17—42no—
196C2×C4×C17—42no—
197C2×C4×C17—42no—
198C2×C4×C17—42no—
199C2×C4×C17—42no—
200C2×C4×C17—41no—
201C2×C4×C17—41no—
202C2×C4×C17—41no—
203C2×C4×C17—41no—
204C2×C4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—41no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—42no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—81no—
214C2×C4×C17—81no—
215C2×C4×C17—81no—
216C2×C4×C17—81no—
217C2×C4×C17—81no—
218C2×C4×C17—81no—
219C2×C4×C17—81no—
220C2×C4×C17—81no—
221C2×C4×C17—81no—
222C2×C4×C17#12111yescharacteristic
223C2×C4×C17#12111yescharacteristic
224C2×C4×C17#12311yescharacteristic
225C2×C4×C17#12411yescharacteristic
226C8×C17—42no—
227C8×C17—81no—
228#2618C2×C1711yescharacteristic
229#1874C2×C1711yescharacteristic
230#1853C2×C1711yescharacteristic, complemented
231C2×#327C2×C1711yescharacteristic
232C2×#324C2×C1711yescharacteristic, complemented
233#1441C2×C1711yescharacteristic, complemented
234#1294C2×C1711yescharacteristic, complemented
235C22×C17—21no—
236C22×C17—21no—
237C22×C17—21no—
238C22×C17—21no—
239C22×C17—21no—
240C22×C17—21no—
241C22×C17—21no—
242C22×C17—21no—
243C22×C17—21no—
244C22×C17—21no—
245C22×C17—21no—
246C22×C17—21no—
247C22×C17—81no—
248C22×C17—81no—
249C22×C17—81no—
250C22×C17—81no—
251C22×C17—81no—
252C22×C17—81no—
253C22×C17—81no—
254C22×C17#39411yescharacteristic
255C22×C17#44911yescharacteristic
256C22×C17#45011yescharacteristic
257C22×C17#45411yescharacteristic
258C22×C17#45611yescharacteristic
259C22×C17B(4,𝔽2)11yescharacteristic
260C22×C17#45811yescharacteristic
261C22×C17#46511yescharacteristic
262C22×C17#46911yescharacteristic
263C22×C17#48111yescharacteristic
264C22×C17#48211yescharacteristic
265C4×C17—21no—
266C4×C17—21no—
267C4×C17—21no—
268C4×C17—21no—
269C4×C17—42no—
270C4×C17—82no—
271C4×C17—82no—
272C4×C17—81no—
273C4×C17—81no—
274C4×C17—81no—
275C4×C17—81no—
276C2×#123—21no—
277C2×#121C22×C1711yescharacteristic
278C22×C4⋊C4C22×C1711yescharacteristic
279C22×C4⋊C4C22×C1711yescharacteristic
280C2×#106—21no—
281C2×#102—21no—
282C2×C22⋊C8C22×C1711yescharacteristic
283C2×C22⋊C8C22×C1711yescharacteristic
284#398—21no—
285#396—21no—
286#396—21no—
287#394—21no—
288#386—21no—
289#380C22×C1711yescharacteristic
290C2×#95—21no—
291C2×#95—21no—
292C2×#95C22×C1711yescharacteristic
293#340—21no—
294#340—21no—
295#327—22no—
296#324—22no—
297C2×C17—21no—
298C2×C17—21no—
299C2×C17—21no—
300C2×C17—21no—
301C2×C17—161no—
302C2×C17#159711yescharacteristic
303C2×C17#160411yescharacteristic
304C2×C17#160911yescharacteristic
305C2×C17#161611yescharacteristic
306C2×C17#203411yescharacteristic
307C2×C17#203811yescharacteristic
308C2×C17#204411yescharacteristic
309C22×D4—21no—
310C22×D4D4×C1711yescharacteristic
311C23×C4—21no—
312C23×C4—21no—
313C23×C4D4×C1711yescharacteristic
314C23×C4D4×C1711yescharacteristic
315C23×C4C23×C1711yescharacteristic
316C22×C8—21no—
317C22×C8—21no—
318#123—42no—
319#123—42no—
320#121—21no—
321#121—21no—
322#121—21no—
323#121—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4D4×C1711yescharacteristic
336C2×C4⋊C4D4×C1711yescharacteristic
337C2×C4⋊C4D4×C1711yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344#106—42no—
345#102—42no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8D4×C1712yes—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—42no—
356#95—42no—
357C17#1039611yescharacteristic, direct product factor, Sylow
358C24—21no—
359C24C2×D4×C1711yescharacteristic
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4SD16×C1712yes—
401C22×C4C2×D4×C1711yescharacteristic
402C22×C4C2×D4×C1711yescharacteristic
403C22×C4C2×D4×C1711yescharacteristic
404C22×C4#41×C1711yescharacteristic
405C22×C4#41×C1711yescharacteristic
406C22×C4#41×C1711yescharacteristic
407C2×C8—22no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23C17×#12011yescharacteristic
447C23C17×#12111yescharacteristic
448C23C17×#12311yescharacteristic
449C23C2×SD16×C1711yescharacteristic
450C23C17×#13511yescharacteristic
451C23C17×AGL(1,ℤ/8)11yescharacteristic
452C23C17×AGL(1,ℤ/8)11yescharacteristic
453D4—42no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—81no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4C17×#12111yescharacteristic
503C2×C4C17×#12111yescharacteristic
504C2×C4C17×#12311yescharacteristic
505C2×C4C17×#12411yescharacteristic
506C8—42no—
507C8—81no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22C17×#39411yescharacteristic
528C22C17×#44911yescharacteristic
529C22C17×#45011yescharacteristic
530C22C17×#45411yescharacteristic
531C22C17×#45611yescharacteristic
532C22C17×B(4,𝔽2)11yescharacteristic
533C22C17×#45811yescharacteristic
534C22C17×#46511yescharacteristic
535C22C17×#46911yescharacteristic
536C22C17×#48111yescharacteristic
537C22C17×#48211yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—82no—
544C4—81no—
545C4—81no—
546C4—82no—
547C4—81no—
548C4—81no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—161no—
554C2C17×#159711yescharacteristic
555C2C17×#160411yescharacteristic
556C2C17×#160911yescharacteristic
557C2C17×#161611yescharacteristic
558C2C17×#203411yescharacteristic
559C2C17×#203811yescharacteristic
560C2C17×#204411yescharacteristic
subgroupG/H
center, socleC23×C17#120
commutator, Frattini, cosocleC23×C4C23×C17
prime radicalC2×C17×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C17 ◁ C23×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
3. C17×#1853C2semidirect
5. C2×C17×#324C2semidirect
6. C17×#1441C2semidirect
7. C17×#1294C2semidirect
154. #10396C17direct (N × H)
230. #1853C2×C17semidirect
232. C2×#324C2×C17semidirect
233. #1441C2×C17semidirect
234. #1294C2×C17semidirect
357. C17#10396direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#26182
C17×#18742
C17×#18532
C2×C17×#3272
C2×C17×#3242
C17×#14412
C17×#12942
#1039617