Finite Group Databasebeta

← #17064333

Subgroups — #17064333

Order 4352.

796 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic17C211yesmaximal, characteristic
2#14531186C211yesmaximal, characteristic
3#14530345C211yesmaximal, characteristic, complemented
4#14530325C211yesmaximal, characteristic, complemented
5#14530263C211yesmaximal, characteristic, complemented
6#14530220C211yesmaximal, characteristic
7#14530217C211yesmaximal, characteristic
8#14530195C211yesmaximal, characteristic, complemented
9#14529977C211yesmaximal, characteristic
10#106×Dic17C211yesmaximal, characteristic, complemented
11#14528672C211yesmaximal, characteristic, complemented
12#103×Dic17C211yesmaximal, characteristic, complemented
13#103×Dic17C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic17C211yesmaximal, characteristic
15C17×#2035C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×#10587563—21no—
18C22×C4×Dic17C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C8×Dic17C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762716C2211yescharacteristic
28#11762708C2211yescharacteristic
29#11762695—21no—
30#11762684—21no—
31#11762678C2211yescharacteristic
32#11762678C2211yescharacteristic
33#11762673—21no—
34#11762669C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762667—21no—
37C4⋊C4×Dic17—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41#11762660—21no—
42C22⋊C4×Dic17—21no—
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×C22⋊Q8C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C2×C17×#103C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×C22⋊C8C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17D411yescharacteristic
62C23×Dic17D411yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75Q8×Dic17—22no—
76Q8×Dic17—22no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×C4×Dic17—22no—
88C2×C4×Dic17—22no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17D411yescharacteristic
94C2×C4×Dic17D411yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587563—44no—
100C17×#106—22no—
101C17×#103—22no—
102C17×#103—22no—
103C17×C22⋊C8—22no—
104C17×#95—21no—
105#10587506—22no—
106#10587506—22no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115C22⋊#3622—22no—
116#10587488—22no—
117C8×Dic17—22no—
118C8×Dic17—22no—
119#10587479—22no—
120#10587479—22no—
121C24×C17C2×D411yescharacteristic
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17#4112yes—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—41no—
149C22×Dic17SD1614yes—
150C22×Dic17C2×D411yescharacteristic
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17#4112yes—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17#4112yes—
167C22⋊C4×C17—24no—
168C22⋊C4×C17—42no—
169C22⋊Dic17—42no—
170C22⋊Dic17—42no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—42no—
180C4⋊Dic17—42no—
181#63225—42no—
182#63225—42no—
183#63225—42no—
184#63225—42no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—22no—
188C4×Dic17—22no—
189C4×Dic17—42no—
190C4×Dic17—42no—
191C4×Dic17—42no—
192C4×Dic17—42no—
193C4×Dic17#4112yes—
194C4×Dic17#4112yes—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622#4112yes—
201#23756—171nomaximal, Sylow
202C23×C17—21no—
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17C2×SD1612yes—
209C23×C17C22×D411yescharacteristic
210Q8×C17—22no—
211Q8×C17—22no—
212Q8×C17—41no—
213Q8×C17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
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×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—28no—
246C2×Dic17—44no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17#12312yes—
260C2×Dic17#12312yes—
261C2×Dic17C2×SD1612yes—
262C2×Dic17C2×SD1612yes—
263C2×Dic17#13512yes—
264C2×Dic17#13712yes—
265C8×C17—22no—
266C8×C17—41no—
267#3622—22no—
268#3622—41no—
269C4×C22⋊Q8—171no—
270#2285—171no—
271#2099—171no—
272#2096—171no—
273#2084—171no—
274#2077—171no—
275#2077—171no—
276#2072—171no—
277#2035—171no—
278#2035D1711yescharacteristic
279C4×#106—171no—
280C4×#106—171no—
281C4×#103—171no—
282C4×#103—171no—
283C4×C22⋊C8—171no—
284C22×C17—22no—
285C22×C17—22no—
286C22×C17—22no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17C2×#12311yescharacteristic
292C22×C17#52511yescharacteristic
293C22×C17#53211yescharacteristic
294C22×C17C22×SD1611yescharacteristic
295C22×C17C2×#13511yescharacteristic
296C22×C17C2×#13711yescharacteristic
297C22×C17#57511yescharacteristic
298C4×C17—42no—
299C4×C17—41no—
300C4×C17—42no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17#52911yescharacteristic
304C4×C17#52911yescharacteristic
305C4×C17#52911yescharacteristic
306C4×C17#52911yescharacteristic
307Dic17—22no—
308Dic17—22no—
309Dic17—22no—
310Dic17—81no—
311Dic17—81no—
312Dic17—81no—
313Dic17—81no—
314Dic17#48314yes—
315C2×C22⋊Q8C2×D1711yescharacteristic
316C2×C4×Q8—171no—
317C2×#117—341no—
318C22×C42—171no—
319C2×#106—171no—
320C2×#106—171no—
321C2×#106—171no—
322C2×#106C2×D1711yescharacteristic
323C2×#103—171no—
324C2×#103—171no—
325C2×#103C2×D1711yescharacteristic
326C2×#103C2×D1711yescharacteristic
327C2×C22⋊C8—171no—
328C2×C22⋊C8C2×D1711yescharacteristic
329C2×C4×C8—171no—
330C2×C4×C8—171no—
331#391—171no—
332#389—171no—
333#387—341no—
334#385—341no—
335#384—171no—
336#384—171no—
337#382—341no—
338#380—171no—
339#380—171no—
340#380—341no—
341#380C2×D1711yescharacteristic
342C4×C4⋊C4—171no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—341no—
346C4×C4⋊C4—341no—
347C4×C22⋊C4—341no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340C2×D1711yescharacteristic
356#336—171no—
357#336—171no—
358C2×C17—42no—
359C2×C17#239511yescharacteristic
360C2×C17C2×#48311yescharacteristic
361C2×C17#309111yescharacteristic
362C2×C17#310311yescharacteristic
363C2×C17#310711yescharacteristic
364C2×C17#316711yescharacteristic
365C2×C17#317211yescharacteristic
366C22×Q8C22×D1711yescharacteristic
367C23×C4—171no—
368C23×C4—171no—
369C23×C4C22×D1711yescharacteristic
370C22×C8—171no—
371C22×C8—171no—
372C22×C8C22×D1711yescharacteristic
373C22×C8C22×D1711yescharacteristic
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C4×Q8—342no—
377C4×Q8—342no—
378#117—684no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4—341no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—171no—
401C2×C42—342no—
402C2×C42—342no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42—341no—
407#106—22no—
408#106—342no—
409#106—342no—
410#106—342no—
411#103—22no—
412#103—22no—
413#103—342no—
414#103—342no—
415C22⋊C8—22no—
416C22⋊C8—342no—
417C4×C8—342no—
418C4×C8—342no—
419#95—21no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428C17#2375611yescharacteristic, complemented, Sylow
429C24D4×D1711yescharacteristic
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8#6325412yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—174no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—681no—
460C22×C4D4×D1711yescharacteristic
461C22×C4C23×D1711yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—172no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8#6325412yes—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—682no—
489C4⋊C4—682no—
490C4⋊C4—682no—
491C4⋊C4—682no—
492C4⋊C4—682no—
493C4⋊C4—682no—
494C4⋊C4#6325412yes—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—682no—
498C22⋊C4—682no—
499C42—172no—
500C42—172no—
501C42—342no—
502C42—342no—
503C42—341no—
504C42—341no—
505C42—682no—
506C42—682no—
507C42—682no—
508C42—682no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23SD16×D1712yes—
516C23C2×D4×D1711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—172no—
541C2×C4—172no—
542C2×C4—172no—
543C2×C4—172no—
544C2×C4—172no—
545C2×C4—172no—
546C2×C4—348no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—684no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4C2×#6325411yescharacteristic
567C2×C4C2×#6325411yescharacteristic
568C2×C4C2×#6325411yescharacteristic
569C2×C4#41×D1711yescharacteristic
570C2×C4#41×D1711yescharacteristic
571C2×C4#41×D1711yescharacteristic
572C8—22no—
573C8—41no—
574C8—342no—
575C8—681no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22#123×D1711yescharacteristic
584C22#1176331511yescharacteristic
585C22#1176335411yescharacteristic
586C22C2×SD16×D1711yescharacteristic
587C22#135×D1711yescharacteristic
588C22#137×D1711yescharacteristic
589C22#1176360811yescharacteristic
590C4—42no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—174no—
596C4—342no—
597C4—342no—
598C4—342no—
599C4—1361no—
600C4—1361no—
601C4—1361no—
602C4—1361no—
603C4#1176333611yescharacteristic
604C4#1176333811yescharacteristic
605C4#1176334011yescharacteristic
606C4#1176334011yescharacteristic
607C2—42no—
608C2#1453204011yescharacteristic
609C2D17×#48311yescharacteristic
610C2#1453870211yescharacteristic
611C2#1453881011yescharacteristic
612C2#1453884811yescharacteristic
613C2#1453933811yescharacteristic
614C2#1453938511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2035C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2035D17
hypocenterC17#23756
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #14530345C2semidirect
4. #14530325C2semidirect
5. #14530263C2semidirect
8. #14530195C2semidirect
10. #106×Dic17C2semidirect
11. #14528672C2semidirect
12. #103×Dic17C2semidirect
13. #103×Dic17C2semidirect
428. C17#23756semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic172
#145311862
#145303452
#145303252
#145302632
#145302202
#145302172
#145301952
#145299772
#106×Dic172
#145286722
#103×Dic172
C22⋊C8×Dic172
C17×#20352
#2375617