Finite Group Databasebeta

← #31029299

Subgroups — #31029299

Order 17152.

1986 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22568559C212yesmaximal, complemented
2#22568519C211yesmaximal, characteristic, complemented
3#22566020C214yesmaximal
4#22565878C214yesmaximal, complemented
5#22565873C214yesmaximal, complemented
6#22563924C212yesmaximal, complemented
7#22563918C212yesmaximal, complemented
8#22563903C212yesmaximal, complemented
9#22563898C212yesmaximal, complemented
10#22562940C214yesmaximal, complemented
11#22562591C212yesmaximal
12C2×C4×(C2×D67)⋊C4C211yesmaximal, characteristic
13C67×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D67C2211yescharacteristic
15C2×#14524042C2212yescomplemented
16C2×#14524035C2211yescharacteristic, complemented
17C2×#14524020C2214yes—
18C2×#14524001C2212yes—
19C2×#14524001C2212yes—
20C22×(C2×D67)⋊C4C2212yes—
21C22×C4×Dic67C2211yescharacteristic
22#16863331C2214yes—
23#16863331C2214yes—
24#16863320C2214yes—
25#16863320C2214yes—
26#16863318C2214yes—
27#16863318C2214yes—
28#16863309C2214yes—
29#16863309C2214yes—
30#16863301C2214yescomplemented
31#16863300C2214yescomplemented
32#16863300C2214yescomplemented
33#16863295C2212yescomplemented
34#16863294C2212yescomplemented
35#16863293C2214yescomplemented
36#16863292C2212yescomplemented
37#16863292C2214yescomplemented
38#16863291C2214yescomplemented
39#16863281C2214yescomplemented
40#16863280C2214yescomplemented
41#16863278C2214yescomplemented
42#16863278C2214yescomplemented
43#16863277C2214yescomplemented
44#16863269C2214yes—
45#16863265C2212yescomplemented
46#16863262C2211yescharacteristic, complemented
47#16863258C2214yes—
48#16863256C2212yes—
49#16863254C2214yes—
50#16863253C2212yes—
51#16863253C2212yes—
52#16863251C2214yes—
53C22⋊C4×Dic67C2214yes—
54C4×(C2×D67)⋊C4C2212yes—
55C4×(C2×D67)⋊C4C2212yes—
56C4×(C2×D67)⋊C4C2212yes—
57C4×(C2×D67)⋊C4C2212yes—
58#16863243C2214yes—
59C2×C67⋊#95C2212yes—
60C2×#125×C67C2211yescharacteristic, complemented
61C22×C42×C67C2211yescharacteristic
62#389×C67C2212yescomplemented
63#387×C67C2214yescomplemented
64#385×C67C2214yes—
65#382×C67C2211yescharacteristic, complemented
66#380×C67C2212yes—
67C24×D67C2311yescharacteristic
68C22×Dic134C2314yes—
69C23×Dic67C2311yescharacteristic
70C23×Dic67C2312yes—
71C22×C4×D67—42no—
72C22×C4×D67—42no—
73C22×C4×D67Q818yescomplemented
74C22×C4×D67C2312yes—
75C22×C4×D67C2312yes—
76#14524042—44no—
77#14524035—42no—
78#14524020—216no—
79#14524001—44no—
80#14524001—44no—
81C2×C4⋊Dic67C2312yes—
82C2×C4⋊Dic67C2312yes—
83C2×C4⋊Dic67C2314yes—
84C2×Dic67⋊C4C2314yes—
85C2×Dic67⋊C4C2314yes—
86C2×Dic67⋊C4C2314yes—
87C2×Dic67⋊C4C2314yes—
88C2×Dic67⋊C4C2314yes—
89C2×Dic67⋊C4C2314yes—
90C2×C22⋊Dic67C2314yes—
91C2×(C2×D67)⋊C4—28no—
92C2×(C2×D67)⋊C4C2312yes—
93C2×(C2×D67)⋊C4C2312yes—
94C2×(C2×D67)⋊C4C2312yes—
95C2×(C2×D67)⋊C4C2312yes—
96C2×(C2×D67)⋊C4C2314yes—
97C2×(C2×D67)⋊C4C2314yes—
98C2×(C2×D67)⋊C4C2314yes—
99C2×(C2×D67)⋊C4C2314yes—
100C2×C4×Dic67—42no—
101C2×C4×Dic67C2312yes—
102C2×C4×Dic67C2314yes—
103C2×C4×Dic67C2314yes—
104C2×C4×Dic67C2311yescharacteristic
105C2×C4×Dic67C2311yescharacteristic
106C67⋊#95C2312yes—
107C67⋊#95C2312yes—
108C67⋊#95C2312yes—
109C67⋊#95C2312yes—
110C67⋊#95C2314yes—
111C67⋊#95C2314yes—
112C67⋊#95C2314yes—
113C67⋊#95C2314yes—
114C67⋊#95C2314yes—
115C67⋊#95C2314yes—
116C67⋊#95C2314yes—
117C23×C4×C67C2311yescharacteristic
118C23×C4×C67C2312yes—
119#125×C67—42no—
120C2×C4⋊C4×C67C2312yes—
121C2×C4⋊C4×C67C2312yes—
122C2×C4⋊C4×C67C2314yes—
123C2×C4⋊C4×C67C2314yes—
124C2×C22⋊C4×C67C2314yes—
125C2×C42×C67—42no—
126C2×C42×C67C2311yescharacteristic
127C2×C42×C67C2311yescharacteristic
128C2×C42×C67C2312yes—
129#95×C67C2314yes—
130#95×C67C2314yes—
131#95×C67C2314yes—
132C23×D67—24no—
133C23×D67—41no—
134C23×D67—41no—
135C23×D67—42no—
136C23×D67C2×Q814yes—
137C23×D67C2412yes—
138C2×Dic134—28no—
139C2×Dic134—44no—
140C2×Dic134—44no—
141C22×Dic67—42no—
142C22×Dic67—41no—
143C22×Dic67—41no—
144C22×Dic67—42no—
145C22×Dic67C2×Q814yes—
146C22×Dic67#4118yes—
147C22×Dic67C2414yes—
148C22×Dic67C2414yes—
149C22×Dic67C2412yes—
150C22×Dic67C2412yes—
151C22×Dic67C2412yes—
152C2×C4×D67—24no—
153C2×C4×D67—24no—
154C2×C4×D67—416no—
155C2×C4×D67—42no—
156C2×C4×D67—42no—
157C2×C4×D67C2×Q818yes—
158C2×C4×D67C2×Q818yes—
159C4⋊Dic67—24no—
160C4⋊Dic67—24no—
161C4⋊Dic67—28no—
162Dic67⋊C4—28no—
163Dic67⋊C4—28no—
164Dic67⋊C4—44no—
165Dic67⋊C4—44no—
166Dic67⋊C4—44no—
167Dic67⋊C4—44no—
168C22⋊Dic67—28no—
169(C2×D67)⋊C4—24no—
170(C2×D67)⋊C4—24no—
171(C2×D67)⋊C4—24no—
172(C2×D67)⋊C4—24no—
173(C2×D67)⋊C4—44no—
174(C2×D67)⋊C4—44no—
175(C2×D67)⋊C4—44no—
176(C2×D67)⋊C4—44no—
177C4×Dic67—24no—
178C4×Dic67—41no—
179C4×Dic67—41no—
180C4×Dic67—44no—
181C4×Dic67—44no—
182C24×C67C2411yescharacteristic
183C22×C4×C67—22no—
184C22×C4×C67—22no—
185C22×C4×C67—24no—
186C22×C4×C67—42no—
187C22×C4×C67—42no—
188C22×C4×C67C2×Q814yes—
189C22×C4×C67C2412yes—
190C22×C4×C67C2412yes—
191C22×C4×C67C2414yes—
192C22×C4×C67C2414yes—
193C22×C4×C67C2411yescharacteristic
194C22×C4×C67C2411yescharacteristic
195C4⋊C4×C67—24no—
196C4⋊C4×C67—24no—
197C4⋊C4×C67—44no—
198C4⋊C4×C67—44no—
199C22⋊C4×C67—28no—
200C42×C67—22no—
201C42×C67—22no—
202C42×C67—42no—
203C23×C67—21no—
204C23×C67—21no—
205C23×C67—22no—
206C23×C67C22×Q812yes—
207C23×C67C2×#4114yes—
208C23×C67C2511yescharacteristic
209C22×D67—28no—
210C22×D67—22no—
211C22×D67—22no—
212C22×D67—22no—
213C22×D67—22no—
214C22×D67—22no—
215C22×D67—22no—
216C22×D67—44no—
217C22×D67—44no—
218C22×D67—48no—
219C22×D67—48no—
220C22×D67C22×Q814yes—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—28no—
224C2×C4×C67—24no—
225C2×C4×C67—24no—
226C2×C4×C67—22no—
227C2×C4×C67—22no—
228C2×C4×C67—22no—
229C2×C4×C67—22no—
230C2×C4×C67—24no—
231C2×C4×C67—24no—
232C2×C4×C67—22no—
233C2×C4×C67—24no—
234C2×C4×C67—24no—
235C2×C4×C67—22no—
236C2×C4×C67—48no—
237C2×C4×C67C22×Q812yes—
238C2×C4×C67C22×Q812yes—
239C2×C4×C67H(5,𝔽2)12yes—
240C2×C4×C67H(5,𝔽2)12yes—
241C2×Dic67—22no—
242C2×Dic67—22no—
243C2×Dic67—22no—
244C2×Dic67—22no—
245C2×Dic67—22no—
246C2×Dic67—22no—
247C2×Dic67—24no—
248C2×Dic67—24no—
249C2×Dic67—24no—
250C2×Dic67—24no—
251C2×Dic67—24no—
252C2×Dic67—24no—
253C2×Dic67—44no—
254C2×Dic67—48no—
255C2×Dic67C22×Q814yes—
256C2×Dic67C2×#4114yes—
257C2×Dic67C2×#4114yes—
258C4×D67—216no—
259C4×D67—48no—
260Dic134—88no—
261C22×C67—22no—
262C22×C67—22no—
263C22×C67—24no—
264C22×C67—24no—
265C22×C67#53314yes—
266C22×C67C23×Q811yescharacteristic
267C22×C67C22×#4112yes—
268C22×C67C2×H(5,𝔽2)11yescharacteristic
269C22×C67C2×H(5,𝔽2)11yescharacteristic
270C22×C67#58512yes—
271C2×D67—24no—
272C2×D67—24no—
273C2×D67—24no—
274C2×D67—416no—
275C2×D67—42no—
276C2×D67—42no—
277C2×D67—42no—
278C2×D67—42no—
279C4×C67—22no—
280C4×C67—44no—
281C4×C67—44no—
282C4×C67—42no—
283C4×C67—42no—
284C4×C67#54314yes—
285Dic67—24no—
286Dic67—24no—
287Dic67—24no—
288Dic67—44no—
289Dic67—44no—
290#35033—671nomaximal, Sylow
291C2×C67—24no—
292C2×C67C2×#53311yescharacteristic
293C2×C67C2×#54311yescharacteristic
294C2×C67#345811yescharacteristic
295C2×C67#350512yes—
296C2×C67#350612yes—
297D67—44no—
298#2857—672no—
299#2851—671no—
300#2851D6711yescharacteristic, complemented
301#2626—674no—
302#2608—674no—
303#2607—674no—
304#2441—672no—
305#2441—672no—
306#2439—672no—
307#2439—672no—
308#2329—674no—
309#2280—672no—
310C2×C4×C22⋊C4—671no—
311C67#3503311yescharacteristic, complemented, Sylow
312C24×C4—671no—
313C2×C4⋊Q8—672no—
314C2×#125—671no—
315C2×#125C2×D6711yescharacteristic, complemented
316C2×C22⋊Q8—674no—
317C2×#117—672no—
318C2×#117—672no—
319C22×C22⋊C4—672no—
320C22×C42—671no—
321C22×C42C2×D6711yescharacteristic
322#399—674no—
323#399—674no—
324#396—674no—
325#396—674no—
326#396—674no—
327#396—674no—
328#393—674no—
329#393—674no—
330#391—674no—
331#391—674no—
332#391—674no—
333#389—674no—
334#389—674no—
335#389—672no—
336#389—672no—
337#389—672no—
338#389—674no—
339#389C2×D6712yescomplemented
340#387—674no—
341#387—674no—
342#387—674no—
343#387—674no—
344#387—674no—
345#387C2×D6714yescomplemented
346#385—674no—
347#385C2×D6714yes—
348#384—672no—
349#382—671no—
350#382C2×D6711yescharacteristic, complemented
351#381—674no—
352#380—672no—
353#380—672no—
354#380—672no—
355#380—674no—
356#380C2×D6712yes—
357#379—674no—
358C4×C22⋊C4—674no—
359C4×C22⋊C4—672no—
360C4×C22⋊C4—672no—
361C4×C22⋊C4—672no—
362C4×C22⋊C4—672no—
363C4×C22⋊C4—674no—
364C2×#95—672no—
365C25—671no—
366C22×Q8—674no—
367C23×C4—2682no—
368C23×C4—2682no—
369C23×C4—672no—
370C23×C4—671no—
371C23×C4—678no—
372C23×C4—672no—
373C23×C4—672no—
374C23×C4C22×D6711yescharacteristic
375C23×C4C22×D6712yes—
376C4⋊Q8—2684no—
377#125—42no—
378#125—2682no—
379C22⋊Q8—13416no—
380#117—2684no—
381#117—2684no—
382C2×C4⋊C4—674no—
383C2×C4⋊C4—674no—
384C2×C4⋊C4—674no—
385C2×C4⋊C4—674no—
386C2×C4⋊C4—672no—
387C2×C4⋊C4—672no—
388C2×C4⋊C4—674no—
389C2×C4⋊C4—674no—
390C2×C4⋊C4—674no—
391C2×C4⋊C4C22×D6712yes—
392C2×C4⋊C4C22×D6712yes—
393C2×C4⋊C4C22×D6714yes—
394C2×C4⋊C4C22×D6714yes—
395C2×C22⋊C4—674no—
396C2×C22⋊C4—674no—
397C2×C22⋊C4—674no—
398C2×C22⋊C4—674no—
399C2×C22⋊C4—674no—
400C2×C22⋊C4—672no—
401C2×C22⋊C4—672no—
402C2×C22⋊C4—672no—
403C2×C22⋊C4—672no—
404C2×C22⋊C4—1348no—
405C2×C22⋊C4C22×D6714yes—
406C2×C42—42no—
407C2×C42—2682no—
408C2×C42—674no—
409C2×C42—674no—
410C2×C42—671no—
411C2×C42—671no—
412C2×C42—672no—
413C2×C42C22×D6711yescharacteristic
414C2×C42C22×D6711yescharacteristic
415C2×C42C22×D6712yes—
416#95—672no—
417#95—672no—
418#95—672no—
419#95—672no—
420#95—674no—
421#95—674no—
422#95—674no—
423#95—674no—
424#95—674no—
425#95—674no—
426#95—674no—
427#95C22×D6714yes—
428#95C22×D6714yes—
429#95C22×D6714yes—
430C24—2681no—
431C24—2681no—
432C24—2682no—
433C24—674no—
434C24—672no—
435C24—1344no—
436C24C23×D6711yescharacteristic
437C2×Q8—2684no—
438C2×Q8—2684no—
439C2×Q8—1348no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—2682no—
446C22×C4—2682no—
447C22×C4—26816no—
448C22×C4—2682no—
449C22×C4—2681no—
450C22×C4—2681no—
451C22×C4—2682no—
452C22×C4—674no—
453C22×C4—678no—
454C22×C4—678no—
455C22×C4—672no—
456C22×C4—672no—
457C22×C4—672no—
458C22×C4—678no—
459C22×C4—674no—
460C22×C4—674no—
461C22×C4—1344no—
462C22×C4—1344no—
463C22×C4Q8×D6714yes—
464C22×C4C23×D6712yes—
465C22×C4C23×D6712yes—
466C22×C4C23×D6714yes—
467C22×C4C23×D6714yes—
468C22×C4C23×D6711yescharacteristic
469C22×C4C23×D6711yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—2684no—
475C4⋊C4—2684no—
476C4⋊C4—2684no—
477C4⋊C4—2684no—
478C4⋊C4—1348no—
479C4⋊C4—1348no—
480C4⋊C4—1348no—
481C4⋊C4—1344no—
482C4⋊C4—1344no—
483C22⋊C4—28no—
484C22⋊C4—2684no—
485C22⋊C4—2684no—
486C22⋊C4—2684no—
487C22⋊C4—2684no—
488C22⋊C4—1344no—
489C22⋊C4—1344no—
490C22⋊C4—1344no—
491C22⋊C4—1344no—
492C22⋊C4—1348no—
493C42—22no—
494C42—22no—
495C42—42no—
496C42—2681no—
497C42—2681no—
498C42—2684no—
499C42—2684no—
500C42—1344no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—2688no—
505C23—2688no—
506C23—2684no—
507C23—2684no—
508C23—674no—
509C23—1348no—
510C23—1342no—
511C23—1342no—
512C23—1342no—
513C23—1342no—
514C23—1342no—
515C23—1342no—
516C23C2×Q8×D6712yes—
517C23#41×D6714yes—
518C23C24×D6711yescharacteristic
519Q8—5368no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—2688no—
537C2×C4—2684no—
538C2×C4—2688no—
539C2×C4—674no—
540C2×C4—674no—
541C2×C4—674no—
542C2×C4—1344no—
543C2×C4—1344no—
544C2×C4—1344no—
545C2×C4—1344no—
546C2×C4—1344no—
547C2×C4—1344no—
548C2×C4—1342no—
549C2×C4—1342no—
550C2×C4—1342no—
551C2×C4—1342no—
552C2×C4—1342no—
553C2×C4—1342no—
554C2×C4—13416no—
555C2×C4C2×Q8×D6712yes—
556C2×C4C2×Q8×D6712yes—
557C2×C4#1452408812yes—
558C2×C4#1452408912yes—
559C22—22no—
560C22—22no—
561C22—24no—
562C22—24no—
563C22—26816no—
564C22—2682no—
565C22—2682no—
566C22—2682no—
567C22—2682no—
568C22—1344no—
569C22—1344no—
570C22—1344no—
571C22#1686395214yes—
572C22C22×Q8×D6711yescharacteristic
573C22C2×#41×D6712yes—
574C22C2×#1452408811yescharacteristic
575C22C2×#1452408911yescharacteristic
576C22#1686423212yes—
577C4—22no—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—2684no—
583C4—2684no—
584C4—1344no—
585C4—1344no—
586C4—1344no—
587C4#1686402114yes—
588C2—24no—
589C2—2684no—
590C2C2×#1686395211yescharacteristic
591C2C2×#1686402111yescharacteristic
592C2#2257361211yescharacteristic
593C2#2257404412yes—
594C2#2257405212yes—
subgroupG/H
center, FrattiniC23C24×D67
commutator, socle, cosocleC23×C67C25
FittingC67×#2851C2
prime radicalC24×D67C23
hypercenter#2851D67
hypocenterC67#35033
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #22568559C2semidirect
2. #22568519C2semidirect
4. #22565878C2semidirect
5. #22565873C2semidirect
6. #22563924C2semidirect
7. #22563918C2semidirect
8. #22563903C2semidirect
9. #22563898C2semidirect
10. #22562940C2semidirect
13. C67×#2851C2semidirect
15. C2×#14524042C22semidirect
16. C2×#14524035C22semidirect
30. #16863301C22semidirect
31. #16863300C22semidirect
32. #16863300C22semidirect
33. #16863295C22semidirect
34. #16863294C22semidirect
35. #16863293C22semidirect
36. #16863292C22semidirect
37. #16863292C22semidirect
38. #16863291C22semidirect
39. #16863281C22semidirect
40. #16863280C22semidirect
41. #16863278C22semidirect
42. #16863278C22semidirect
43. #16863277C22semidirect
45. #16863265C22semidirect
46. #16863262C22semidirect
60. C2×#125×C67C22semidirect
62. #389×C67C22semidirect
63. #387×C67C22semidirect
65. #382×C67C22semidirect
73. C22×C4×D67Q8semidirect
300. #2851D67semidirect
311. C67#35033semidirect
315. C2×#125C2×D67semidirect
339. #389C2×D67semidirect
345. #387C2×D67semidirect
350. #382C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225685592
#225685192
#225660202
#225658782
#225658732
#225639242
#225639182
#225639032
#225638982
#225629402
#225625912
C2×C4×(C2×D67)⋊C42
C67×#28512
#3503367