Finite Group Databasebeta

← C17×#13420 #16871164

Subgroups — C17×#13420 #16871164

Order 4352.

600 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2697C211yesmaximal, characteristic
2C2×C17×#352C211yesmaximal, characteristic, complemented
3C2×C17×AΣL(1,𝔽16)C211yesmaximal, characteristic
4C17×#1890C211yesmaximal, characteristic, complemented
5C17×#1889C211yesmaximal, characteristic
6C17×#1856C211yesmaximal, characteristic, complemented
7C17×#1393C211yesmaximal, characteristic, complemented
8C2×C17×#123C2211yescharacteristic
9C2×C17×#120C2211yescharacteristic
10C17×#518C2211yescharacteristic
11C17×#428—21no—
12C2×C17×#100C2211yescharacteristic
13C2×C17×#99C2211yescharacteristic
14C2×C17×#99C2211yescharacteristic
15C17×#397—21no—
16C17×#396—21no—
17C17×#394—21no—
18C17×#390C2211yescharacteristic
19C17×#388—21no—
20C17×#386—21no—
21C17×#381—21no—
22C17×#379—21no—
23C17×#379—21no—
24C17×#352—22no—
25C17×AΣL(1,𝔽16)—22no—
26C17×#342—21no—
27C17×#342—21no—
28C17×#339—21no—
29C25×C17D411yescharacteristic
30C22×D4×C17—21no—
31C22×D4×C17C2311yescharacteristic
32C23×C4×C17D411yescharacteristic
33C2×M16×C17—21no—
34C17×#123—21no—
35C17×#123—21no—
36C17×#123—21no—
37C17×#123—21no—
38C17×#120—21no—
39C17×#120—21no—
40C17×#120—21no—
41C17×#120—21no—
42C4×D4×C17—41no—
43C4×D4×C17—41no—
44C17×#117—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊C4×C17—21no—
52C2×C22⋊C4×C17—21no—
53C2×C22⋊C4×C17—21no—
54C2×C22⋊C4×C17—21no—
55C2×C22⋊C4×C17—21no—
56C2×C22⋊C4×C17—21no—
57C2×C22⋊C4×C17—21no—
58C2×C22⋊C4×C17—21no—
59C2×C22⋊C4×C17—21no—
60C2×C22⋊C4×C17—41no—
61C2×C22⋊C4×C17—41no—
62C2×C22⋊C4×C17D411yescharacteristic
63C2×C22⋊C4×C17D411yescharacteristic
64C2×C42×C17D411yescharacteristic
65C17×#107—41no—
66C17×#106—41no—
67C17×#100—21no—
68C17×#100D412yes—
69C17×#99—21no—
70C17×#99—21no—
71C17×#99—21no—
72C17×#99—21no—
73C17×#95—21no—
74C17×#95—21no—
75C17×#95—21no—
76C17×#95—41no—
77C17×#95—41no—
78C24×C17—21no—
79C24×C17—21no—
80C24×C17—21no—
81C24×C17—41no—
82C24×C17—41no—
83C24×C17—42no—
84C24×C17—81no—
85C24×C17C2×D411yescharacteristic
86C24×C17C2×D411yescharacteristic
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C2×D4×C17—42no—
90C2×D4×C17—41no—
91C2×D4×C17—41no—
92C2×D4×C17—81no—
93C2×D4×C17C2×D411yescharacteristic
94C2×D4×C17#4111yescharacteristic
95C2×D4×C17#4111yescharacteristic
96C2×D4×C17#4111yescharacteristic
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—21no—
104C22×C4×C17—21no—
105C22×C4×C17—21no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—41no—
109C22×C4×C17—41no—
110C22×C4×C17—41no—
111C22×C4×C17—41no—
112C22×C4×C17C2×D411yescharacteristic
113M16×C17—22no—
114M16×C17—41no—
115C2×C8×C17—41no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17—21no—
118C4⋊C4×C17—41no—
119C4⋊C4×C17—41no—
120C4⋊C4×C17—41no—
121C4⋊C4×C17—81no—
122C22⋊C4×C17—21no—
123C22⋊C4×C17—21no—
124C22⋊C4×C17—21no—
125C22⋊C4×C17—21no—
126C22⋊C4×C17—42no—
127C22⋊C4×C17—41no—
128C22⋊C4×C17—41no—
129C22⋊C4×C17—41no—
130C22⋊C4×C17—41no—
131C22⋊C4×C17—41no—
132C22⋊C4×C17—41no—
133C22⋊C4×C17—41no—
134C22⋊C4×C17—41no—
135C22⋊C4×C17—41no—
136C22⋊C4×C17—41no—
137C22⋊C4×C17—41no—
138C22⋊C4×C17—81no—
139C22⋊C4×C17—81no—
140C22⋊C4×C17—81no—
141C22⋊C4×C17—81no—
142C22⋊C4×C17—81no—
143C22⋊C4×C17—82no—
144C22⋊C4×C17—82no—
145C42×C17—41no—
146C42×C17—41no—
147#13420C1711yesmaximal, characteristic, direct product factor, Sylow
148C23×C17—21no—
149C23×C17—21no—
150C23×C17—21no—
151C23×C17—21no—
152C23×C17—22no—
153C23×C17—41no—
154C23×C17—41no—
155C23×C17—41no—
156C23×C17—41no—
157C23×C17—41no—
158C23×C17—41no—
159C23×C17—41no—
160C23×C17—42no—
161C23×C17—42no—
162C23×C17—42no—
163C23×C17—41no—
164C23×C17—41no—
165C23×C17—41no—
166C23×C17—81no—
167C23×C17—82no—
168C23×C17—81no—
169C23×C17—82no—
170C23×C17—84no—
171C23×C17—81no—
172C23×C17#12011yescharacteristic
173C23×C17#12111yescharacteristic
174C23×C17#12111yescharacteristic
175C23×C17#12311yescharacteristic
176C23×C17#12311yescharacteristic
177D4×C17—41no—
178D4×C17—41no—
179D4×C17—41no—
180D4×C17—41no—
181D4×C17—82no—
182D4×C17—82no—
183C2×C4×C17—21no—
184C2×C4×C17—21no—
185C2×C4×C17—41no—
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—42no—
193C2×C4×C17—41no—
194C2×C4×C17—41no—
195C2×C4×C17—41no—
196C2×C4×C17—41no—
197C2×C4×C17—41no—
198C2×C4×C17—41no—
199C2×C4×C17—41no—
200C2×C4×C17—41no—
201C2×C4×C17—41no—
202C2×C4×C17—41no—
203C2×C4×C17—81no—
204C2×C4×C17—81no—
205C2×C4×C17—81no—
206C2×C4×C17—81no—
207C2×C4×C17—81no—
208C2×C4×C17—81no—
209C2×C4×C17—81no—
210C2×C4×C17—81no—
211C2×C4×C17—81no—
212C2×C4×C17—81no—
213C2×C4×C17—81no—
214C2×C4×C17#12111yescharacteristic
215C2×C4×C17#12411yescharacteristic
216C8×C17—42no—
217#2697C2×C1711yescharacteristic
218C2×#352C2×C1711yescharacteristic, complemented
219C2×AΣL(1,𝔽16)C2×C1711yescharacteristic
220#1890C2×C1711yescharacteristic, complemented
221#1889C2×C1711yescharacteristic
222#1856C2×C1711yescharacteristic, complemented
223#1393C2×C1711yescharacteristic, complemented
224C22×C17—21no—
225C22×C17—21no—
226C22×C17—21no—
227C22×C17—21no—
228C22×C17—21no—
229C22×C17—21no—
230C22×C17—41no—
231C22×C17—41no—
232C22×C17—41no—
233C22×C17—41no—
234C22×C17—42no—
235C22×C17—42no—
236C22×C17—41no—
237C22×C17—41no—
238C22×C17—41no—
239C22×C17—41no—
240C22×C17—44no—
241C22×C17—42no—
242C22×C17—81no—
243C22×C17—81no—
244C22×C17—81no—
245C22×C17—81no—
246C22×C17—82no—
247C22×C17—82no—
248C22×C17—82no—
249C22×C17—81no—
250C22×C17—81no—
251C22×C17#39411yescharacteristic
252C22×C17B(4,𝔽2)11yescharacteristic
253C22×C17#45811yescharacteristic
254C4×C17—21no—
255C4×C17—21no—
256C4×C17—81no—
257C4×C17—81no—
258C4×C17—81no—
259C4×C17—81no—
260C4×C17—81no—
261C4×C17—81no—
262C4×C17—81no—
263C4×C17—161no—
264C4×C17—161no—
265C2×#123C22×C1711yescharacteristic
266C2×#120C22×C1711yescharacteristic
267#518C22×C1711yescharacteristic
268#428—21no—
269C2×#100C22×C1711yescharacteristic
270C2×#99C22×C1711yescharacteristic
271C2×#99C22×C1711yescharacteristic
272#397—21no—
273#396—21no—
274#394—21no—
275#390C22×C1711yescharacteristic
276#388—21no—
277#386—21no—
278#381—21no—
279#379—21no—
280#379—21no—
281#352—22no—
282AΣL(1,𝔽16)—22no—
283#342—21no—
284#342—21no—
285#339—21no—
286C2×C17—21no—
287C2×C17—21no—
288C2×C17—41no—
289C2×C17—41no—
290C2×C17—41no—
291C2×C17—41no—
292C2×C17—82no—
293C2×C17#204411yescharacteristic
294C2×C17#219711yescharacteristic
295C2×C17#220011yescharacteristic
296C25D4×C1711yescharacteristic
297C22×D4—21no—
298C22×D4C23×C1711yescharacteristic
299C23×C4D4×C1711yescharacteristic
300C2×M16—21no—
301#123—21no—
302#123—21no—
303#123—21no—
304#123—21no—
305#120—21no—
306#120—21no—
307#120—21no—
308#120—21no—
309C4×D4—41no—
310C4×D4—41no—
311#117—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4D4×C1711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—41no—
328C2×C22⋊C4—41no—
329C2×C22⋊C4D4×C1711yescharacteristic
330C2×C22⋊C4D4×C1711yescharacteristic
331C2×C42D4×C1711yescharacteristic
332#107—41no—
333#106—41no—
334#100—21no—
335#100D4×C1712yes—
336#99—21no—
337#99—21no—
338#99—21no—
339#99—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—41no—
344#95—41no—
345C17#1342011yescharacteristic, direct product factor, Sylow
346C24—21no—
347C24—21no—
348C24—21no—
349C24—41no—
350C24—41no—
351C24—42no—
352C24—81no—
353C24C2×D4×C1711yescharacteristic
354C24C2×D4×C1711yescharacteristic
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—81no—
361C2×D4C2×D4×C1711yescharacteristic
362C2×D4#41×C1711yescharacteristic
363C2×D4#41×C1711yescharacteristic
364C2×D4#41×C1711yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C2×D4×C1711yescharacteristic
381M16—22no—
382M16—41no—
383C2×C8—41no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—81no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—42no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—81no—
407C22⋊C4—81no—
408C22⋊C4—81no—
409C22⋊C4—81no—
410C22⋊C4—81no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C42—41no—
414C42—41no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—81no—
434C23—82no—
435C23—81no—
436C23—82no—
437C23—84no—
438C23—81no—
439C23C17×#12011yescharacteristic
440C23C17×#12111yescharacteristic
441C23C17×#12111yescharacteristic
442C23C17×#12311yescharacteristic
443C23C17×#12311yescharacteristic
444D4—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—82no—
449D4—82no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4C17×#12111yescharacteristic
482C2×C4C17×#12411yescharacteristic
483C8—42no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—81no—
510C22—81no—
511C22C17×#39411yescharacteristic
512C22C17×B(4,𝔽2)11yescharacteristic
513C22C17×#45811yescharacteristic
514C4—21no—
515C4—21no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—161no—
524C4—161no—
525C2—21no—
526C2—21no—
527C2—41no—
528C2—41no—
529C2—41no—
530C2—41no—
531C2—82no—
532C2C17×#204411yescharacteristic
533C2C17×#219711yescharacteristic
534C2C17×#220011yescharacteristic
subgroupG/H
center, socleC22×C17B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C17
prime radicalC2×C17×#120C22
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C22×D4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C17×#352C2semidirect
4. C17×#1890C2semidirect
6. C17×#1856C2semidirect
7. C17×#1393C2semidirect
147. #13420C17direct (N × H)
218. C2×#352C2×C17semidirect
220. #1890C2×C17semidirect
222. #1856C2×C17semidirect
223. #1393C2×C17semidirect
345. C17#13420direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#26972
C2×C17×#3522
C2×C17×AΣL(1,𝔽16)2
C17×#18902
C17×#18892
C17×#18562
C17×#13932
#1342017