Finite Group Databasebeta

← #17064370

Subgroups — #17064370

Order 4352.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic17C211yesmaximal, characteristic
2#14531179C211yesmaximal, characteristic
3#14530355C211yesmaximal, characteristic, complemented
4#14530334C211yesmaximal, characteristic, complemented
5#14530269C211yesmaximal, characteristic, complemented
6#14530217C211yesmaximal, characteristic
7#14530210C211yesmaximal, characteristic
8#14530205C211yesmaximal, characteristic, complemented
9#14529984C211yesmaximal, characteristic
10#14528676C211yesmaximal, characteristic, complemented
11#106×Dic17C211yesmaximal, characteristic, complemented
12#103×Dic17C211yesmaximal, characteristic, complemented
13#103×Dic17C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic17C211yesmaximal, characteristic
15C17×#2035C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C22×C4×Dic17C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24C2×C8×Dic17C2211yescharacteristic
25C2×C8×Dic17C2211yescharacteristic
26#11762716C2211yescharacteristic
27#11762708C2211yescharacteristic
28#11762695—21no—
29#11762684—21no—
30#11762676—21no—
31#11762674C2211yescharacteristic
32#11762674C2211yescharacteristic
33#11762670—21no—
34#11762669C2211yescharacteristic
35#11762669C2211yescharacteristic
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C22⋊C4×Dic17—21no—
41C22⋊C4×Dic17—21no—
42#11762654—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×C4×Dic17—22no—
84C2×C4×Dic17—22no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C17×#106—22no—
96C17×#103—22no—
97C17×#103—22no—
98C17×C22⋊C8—22no—
99C17×#95—21no—
100#10587506—22no—
101#10587506—22no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114C22⋊#3622—22no—
115#10587488—22no—
116C8×Dic17—22no—
117C8×Dic17—22no—
118#10587478—22no—
119#10587478—22no—
120C24×C17C2×D411yescharacteristic
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17#4112yes—
127C22×C4×C17—21no—
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×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—22no—
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×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17#4112yes—
159C4⋊C4×C17—24no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17#4112yes—
166C22⋊C4×C17—24no—
167C22⋊C4×C17—42no—
168C22⋊Dic17—42no—
169C22⋊Dic17—42no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—42no—
179C4⋊Dic17—42no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—22no—
183C4×Dic17—22no—
184C4×Dic17—42no—
185C4×Dic17—42no—
186C4×Dic17—42no—
187C4×Dic17—42no—
188C4×Dic17#4112yes—
189C4×Dic17#4112yes—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622#4112yes—
196#23759—171nomaximal, Sylow
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17#13512yes—
204C23×C17C22×D411yescharacteristic
205Q8×C17—22no—
206Q8×C17—22no—
207Q8×C17—41no—
208Q8×C17—41no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
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×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—44no—
243C2×Dic17—44no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
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#12312yes—
257C2×Dic17#12312yes—
258C2×Dic17#13512yes—
259C2×Dic17#13712yes—
260C8×C17—22no—
261C8×C17—41no—
262#3622—22no—
263#3622—41no—
264C4×C22⋊Q8—171no—
265#2283—171no—
266#2101—171no—
267#2097—171no—
268#2085—171no—
269#2077—171no—
270#2075—171no—
271#2074—171no—
272#2036—171no—
273#2035D1711yescharacteristic
274C4×#107—171no—
275C4×#106—171no—
276C4×#103—171no—
277C4×#103—171no—
278C4×C22⋊C8—171no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17C2×#12311yescharacteristic
287C22×C17#52511yescharacteristic
288C22×C17#53211yescharacteristic
289C22×C17C2×#13511yescharacteristic
290C22×C17C2×#13511yescharacteristic
291C22×C17C2×#13711yescharacteristic
292C22×C17#57511yescharacteristic
293C4×C17—42no—
294C4×C17—41no—
295C4×C17—42no—
296C4×C17—81no—
297C4×C17—81no—
298C4×C17#52911yescharacteristic
299C4×C17#52911yescharacteristic
300C4×C17#52911yescharacteristic
301C4×C17#52911yescharacteristic
302Dic17—22no—
303Dic17—41no—
304Dic17—41no—
305Dic17—81no—
306Dic17—81no—
307Dic17—81no—
308Dic17—81no—
309Dic17#48514yes—
310C2×C22⋊Q8C2×D1711yescharacteristic
311C2×C4×Q8—171no—
312C22×C42—171no—
313C2×#107—171no—
314C2×#107—171no—
315C2×#106—171no—
316C2×#106C2×D1711yescharacteristic
317C2×#103—171no—
318C2×#103—171no—
319C2×#103C2×D1711yescharacteristic
320C2×#103C2×D1711yescharacteristic
321C2×C22⋊C8—171no—
322C2×C22⋊C8C2×D1711yescharacteristic
323C2×C4×C8—171no—
324C2×C4×C8—171no—
325#391—171no—
326#389—171no—
327#387—341no—
328#385—341no—
329#383—341no—
330#382—171no—
331#382—171no—
332#381—341no—
333#380—171no—
334#380—171no—
335#380C2×D1711yescharacteristic
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—341no—
340C4×C22⋊C4—341no—
341C4×C22⋊C4—341no—
342#376—341no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340C2×D1711yescharacteristic
351#336—171no—
352#336—171no—
353C2×C17—42no—
354C2×C17#239511yescharacteristic
355C2×C17C2×#48511yescharacteristic
356C2×C17#309111yescharacteristic
357C2×C17#310211yescharacteristic
358C2×C17#310811yescharacteristic
359C2×C17#316811yescharacteristic
360C2×C17#317211yescharacteristic
361C22×Q8C22×D1711yescharacteristic
362C23×C4—171no—
363C23×C4—171no—
364C23×C4C22×D1711yescharacteristic
365C22×C8—171no—
366C22×C8—171no—
367C22×C8C22×D1711yescharacteristic
368C22×C8C22×D1711yescharacteristic
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C4×Q8—342no—
372C4×Q8—342no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C42—171no—
388C2×C42—171no—
389C2×C42—171no—
390C2×C42—171no—
391C2×C42—342no—
392C2×C42—342no—
393C2×C42—341no—
394C2×C42—341no—
395C2×C42—341no—
396C2×C42—341no—
397#107—342no—
398#107—342no—
399#106—22no—
400#106—342no—
401#103—22no—
402#103—22no—
403#103—342no—
404#103—342no—
405C22⋊C8—22no—
406C22⋊C8—342no—
407C4×C8—342no—
408C4×C8—342no—
409#95—21no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2375911yescharacteristic, complemented, Sylow
423C24D4×D1711yescharacteristic
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#6325412yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—171no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—171no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—681no—
454C22×C4D4×D1711yescharacteristic
455C22×C4C23×D1711yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—172no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8#6325412yes—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—682no—
483C4⋊C4—682no—
484C4⋊C4#6325412yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—682no—
488C22⋊C4—682no—
489C42—172no—
490C42—172no—
491C42—342no—
492C42—342no—
493C42—341no—
494C42—341no—
495C42—682no—
496C42—682no—
497C42—682no—
498C42—682no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1058760212yes—
506C23C2×D4×D1711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—172no—
531C2×C4—172no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—684no—
543C2×C4—684no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4C2×#6325411yescharacteristic
557C2×C4C2×#6325411yescharacteristic
558C2×C4C2×#6325411yescharacteristic
559C2×C4#41×D1711yescharacteristic
560C2×C4#41×D1711yescharacteristic
561C2×C4#41×D1711yescharacteristic
562C8—22no—
563C8—41no—
564C8—342no—
565C8—681no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#123×D1711yescharacteristic
574C22#1176331511yescharacteristic
575C22#1176335411yescharacteristic
576C22#135×D1711yescharacteristic
577C22C2×#1058760211yescharacteristic
578C22#137×D1711yescharacteristic
579C22#1176360811yescharacteristic
580C4—42no—
581C4—41no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—174no—
586C4—342no—
587C4—681no—
588C4—681no—
589C4—1361no—
590C4—1361no—
591C4—1361no—
592C4—1361no—
593C4#1176333611yescharacteristic
594C4#1176333811yescharacteristic
595C4#1176334011yescharacteristic
596C4#1176334011yescharacteristic
597C2—42no—
598C2#1453204011yescharacteristic
599C2D17×#48511yescharacteristic
600C2#1453869811yescharacteristic
601C2#1453880311yescharacteristic
602C2#1453885911yescharacteristic
603C2#1453934711yescharacteristic
604C2#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#23759
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. #14530355C2semidirect
4. #14530334C2semidirect
5. #14530269C2semidirect
8. #14530205C2semidirect
10. #14528676C2semidirect
11. #106×Dic17C2semidirect
12. #103×Dic17C2semidirect
13. #103×Dic17C2semidirect
422. C17#23759semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic172
#145311792
#145303552
#145303342
#145302692
#145302172
#145302102
#145302052
#145299842
#145286762
#106×Dic172
#103×Dic172
C22⋊C8×Dic172
C17×#20352
#2375917