Finite Group Databasebeta

← #2383×D67 #31018488

Subgroups — #2383×D67 #31018488

Order 17152.

2502 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#126×D67C211yesmaximal, characteristic
2#22563402C211yesmaximal, characteristic, complemented
3#22563401C211yesmaximal, characteristic, complemented
4#22563400C213yesmaximal, complemented
5#22563399C213yesmaximal, complemented
6#22563398C211yesmaximal, characteristic, complemented
7#22563397C213yesmaximal, complemented
8#22563396C213yesmaximal, complemented
9#389×D67C213yesmaximal, complemented
10#388×D67C213yesmaximal
11#387×D67C213yesmaximal, complemented
12#385×D67C213yesmaximal
13#384×D67C211yesmaximal, characteristic, complemented
14#383×D67C211yesmaximal, characteristic, complemented
15C67×#2383C211yesmaximal, characteristic, complemented
16C23×C4×D67C2211yescharacteristic
17C2×#14524038C2213yes—
18C2×#14524037C2211yescharacteristic
19C2×#14524036C2213yes—
20#126×D67C4116yescomplemented
21C2×C4⋊C4×D67C2213yes—
22C2×C4⋊C4×D67C2213yes—
23C2×C4⋊C4×D67C2213yes—
24C2×C4⋊C4×D67C2213yes—
25C2×C22⋊C4×D67C2213yes—
26C2×C22⋊C4×D67C2213yes—
27C2×C42×D67C2213yes—
28C2×C42×D67C2211yescharacteristic
29#16863295C2213yescomplemented
30#16863294C2213yescomplemented
31#16863293C2213yescomplemented
32#16863293C2213yescomplemented
33#16863292C2213yescomplemented
34#16863291C2213yescomplemented
35#16863291C2213yescomplemented
36#16863290C2213yes—
37#16863289C2213yes—
38#16863288C2213yes—
39#16863287C2213yes—
40#16863286C2213yes—
41#16863285C2213yes—
42#16863284C2213yes—
43#16863283C2213yescomplemented
44#16863282C2213yescomplemented
45#16863281C2213yescomplemented
46#16863280C2213yescomplemented
47#16863279C2213yescomplemented
48#16863278C2213yescomplemented
49#16863277C2213yescomplemented
50#16863271C2213yes—
51#16863270C2213yes—
52#16863270C2213yes—
53#16863269C2213yes—
54#16863268C2213yes—
55#16863267C2213yes—
56#16863267C2213yes—
57#16863266C2211yescharacteristic, complemented
58#16863265C2213yescomplemented
59#16863265C2213yescomplemented
60#16863264C2211yescharacteristic, complemented
61#16863263C2213yescomplemented
62#16863263C2213yescomplemented
63#95×D67C2213yes—
64#95×D67C2213yes—
65#95×D67C2213yes—
66#95×D67C2213yes—
67C2×#126×C67C2211yescharacteristic
68#389×C67C2213yescomplemented
69#388×C67C2213yes—
70#387×C67C2213yescomplemented
71#385×C67C2213yes—
72#384×C67C2211yescharacteristic, complemented
73#383×C67C2211yescharacteristic, complemented
74C24×D67C2311yescharacteristic
75C23×Dic67C2311yescharacteristic
76C22×C4×D67—46no—
77C22×C4×D67C2313yes—
78C22×C4×D67C2311yescharacteristic
79C22×C4×D67C2311yescharacteristic
80C22×C4×D67C2313yes—
81C22×C4×D67C2313yes—
82C22×C4×D67C2313yes—
83C22×C4×D67C2311yescharacteristic
84C22×C4×D67C2311yescharacteristic
85#14524038C2×C4124yescomplemented
86#14524037C2×C418yescomplemented
87#14524036C2×C4124yescomplemented
88C2×C4⋊Dic67C2313yes—
89C2×C4⋊Dic67C2313yes—
90C2×C4⋊Dic67C2313yes—
91C2×C4⋊Dic67C2313yes—
92C2×Dic67⋊C4C2313yes—
93C2×Dic67⋊C4C2313yes—
94C2×Dic67⋊C4C2313yes—
95C2×Dic67⋊C4C2313yes—
96C2×Dic67⋊C4C2313yes—
97C2×Dic67⋊C4C2313yes—
98C2×Dic67⋊C4C2313yes—
99C2×Dic67⋊C4C2313yes—
100C4⋊C4×D67—46no—
101C4⋊C4×D67—46no—
102C4⋊C4×D67—46no—
103C4⋊C4×D67C2×C4124yescomplemented
104C2×C22⋊Dic67C2313yes—
105C2×C22⋊Dic67C2313yes—
106C2×(C2×D67)⋊C4C2313yes—
107C2×(C2×D67)⋊C4C2313yes—
108C2×(C2×D67)⋊C4C2313yes—
109C2×(C2×D67)⋊C4C2313yes—
110C22⋊C4×D67—46no—
111C22⋊C4×D67C2×C4124yes—
112C2×C4×Dic67C2313yes—
113C2×C4×Dic67C2313yes—
114C2×C4×Dic67C2313yes—
115C2×C4×Dic67C2313yes—
116C42×D67—46no—
117C42×D67C2×C418yescomplemented
118C67⋊#95C2313yes—
119C67⋊#95C2313yes—
120C67⋊#95C2313yes—
121C67⋊#95C2313yes—
122C67⋊#95C2313yes—
123C67⋊#95C2313yes—
124C67⋊#95C2313yes—
125C67⋊#95C2313yes—
126C67⋊#95C2313yes—
127C67⋊#95C2313yes—
128C67⋊#95C2313yes—
129C67⋊#95C2313yes—
130C23×C4×C67C2311yescharacteristic
131#126×C67C2×C418yescomplemented
132C2×C4⋊C4×C67C2313yes—
133C2×C4⋊C4×C67C2313yes—
134C2×C4⋊C4×C67C2313yes—
135C2×C4⋊C4×C67C2313yes—
136C2×C22⋊C4×C67C2313yes—
137C2×C22⋊C4×C67C2313yes—
138C2×C42×C67C2313yes—
139C2×C42×C67C2311yescharacteristic
140#95×C67C2313yes—
141#95×C67C2313yes—
142#95×C67C2313yes—
143#95×C67C2313yes—
144C23×D67—26no—
145C23×D67—26no—
146C23×D67C22×C414yes—
147C23×D67C2411yescharacteristic
148C23×D67C2411yescharacteristic
149C22×Dic67—43no—
150C22×Dic67C2413yes—
151C22×Dic67C2411yescharacteristic
152C22×Dic67C2411yescharacteristic
153C22×Dic67C2413yes—
154C22×Dic67C2413yes—
155C22×Dic67C2413yes—
156C2×C4×D67—26no—
157C2×C4×D67—26no—
158C2×C4×D67—26no—
159C2×C4×D67—26no—
160C2×C4×D67—26no—
161C2×C4×D67—26no—
162C2×C4×D67—26no—
163C2×C4×D67—26no—
164C2×C4×D67—26no—
165C2×C4×D67—26no—
166C2×C4×D67—212no—
167C2×C4×D67—43no—
168C2×C4×D67—43no—
169C2×C4×D67—416no—
170C2×C4×D67C22×C4112yes—
171C2×C4×D67C22×C4112yes—
172C4⋊Dic67—43no—
173C4⋊Dic67—43no—
174C4⋊Dic67—43no—
175C4⋊Dic67C22×C4112yescomplemented
176Dic67⋊C4—43no—
177Dic67⋊C4—43no—
178Dic67⋊C4—43no—
179Dic67⋊C4—43no—
180Dic67⋊C4—43no—
181Dic67⋊C4—43no—
182Dic67⋊C4C22×C4112yescomplemented
183Dic67⋊C4C22×C4112yescomplemented
184C22⋊Dic67—43no—
185C22⋊Dic67C22×C4112yes—
186(C2×D67)⋊C4—43no—
187(C2×D67)⋊C4—43no—
188(C2×D67)⋊C4C22×C4112yes—
189(C2×D67)⋊C4C22×C4112yes—
190C4×Dic67—43no—
191C4×Dic67—43no—
192C4×Dic67—43no—
193C4×Dic67C22×C4112yescomplemented
194C24×C67C2411yescharacteristic
195C22×C4×C67—43no—
196C22×C4×C67C2413yes—
197C22×C4×C67C2413yes—
198C22×C4×C67C2411yescharacteristic
199C22×C4×C67C2411yescharacteristic
200C22×C4×C67C2413yes—
201C22×C4×C67C2413yes—
202C4⋊C4×C67—43no—
203C4⋊C4×C67—43no—
204C4⋊C4×C67—43no—
205C4⋊C4×C67C22×C4112yescomplemented
206C22⋊C4×C67—43no—
207C22⋊C4×C67C22×C4112yes—
208C42×C67—43no—
209C42×C67C22×C414yescomplemented
210C23×C67—23no—
211C23×C67—23no—
212C23×C67C23×C412yes—
213C23×C67C2511yescharacteristic
214C22×D67—23no—
215C22×D67—23no—
216C22×D67—224no—
217C22×D67—44no—
218C22×D67—412no—
219C22×D67C23×C412yes—
220C22×D67C23×C412yes—
221C22×D67H(5,𝔽2)16yes—
222C22×D672−1+416yes—
223C2×C4×C67—23no—
224C2×C4×C67—23no—
225C2×C4×C67—23no—
226C2×C4×C67—23no—
227C2×C4×C67—23no—
228C2×C4×C67—23no—
229C2×C4×C67—23no—
230C2×C4×C67—23no—
231C2×C4×C67—23no—
232C2×C4×C67—23no—
233C2×C4×C67—26no—
234C2×C4×C67—44no—
235C2×C4×C67C23×C416yes—
236C2×C4×C67C23×C416yes—
237C2×Dic67—23no—
238C2×Dic67—23no—
239C2×Dic67—23no—
240C2×Dic67—23no—
241C2×Dic67—23no—
242C2×Dic67—23no—
243C2×Dic67—23no—
244C2×Dic67—23no—
245C2×Dic67—23no—
246C2×Dic67—23no—
247C2×Dic67—26no—
248C2×Dic67—44no—
249C2×Dic67C23×C416yes—
250C2×Dic67C23×C416yes—
251C4×D67—224no—
252C4×D67—224no—
253C4×D67—44no—
254C4×D67—44no—
255C4×D67—412no—
256C4×D67—44no—
257C4×D67—44no—
258C4×D67—412no—
259C22×C67—26no—
260C22×C67—41no—
261C22×C67—43no—
262C22×C67C24×C411yescharacteristic
263C22×C67C2×H(5,𝔽2)13yes—
264C22×C67C2×2−1+413yes—
265C2×D67—26no—
266C2×D67—41no—
267C2×D67—43no—
268C2×D67—416no—
269C2×D67#520112yes—
270C2×D67#56014yes—
271C2×D67#563112yes—
272C4×C67—26no—
273C4×C67—26no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67—43no—
277C4×C67—43no—
278Dic67—26no—
279Dic67—26no—
280Dic67—41no—
281Dic67—41no—
282Dic67—43no—
283Dic67—43no—
284C2×#2383—671nomaximal, Sylow
285C2×C67—42no—
286C2×C67C2×#52013yes—
287C2×C67C2×#56011yescharacteristic
288C2×C67C2×#56313yes—
289D67—42no—
290D67#238318yesdirect product factor
291C22×#126—671no—
292#2383—673no—
293#2383—673no—
294#2383—673no—
295#2383—673no—
296#2383—671no—
297#2383—671no—
298#2383—671no—
299#2383D6711yescharacteristic, direct product factor
300C2×#389—673no—
301C2×#388—673no—
302C2×#387—673no—
303C2×#385—673no—
304C2×#384—671no—
305C2×#383—671no—
306C67C2×#238311yescharacteristic, complemented, Sylow
307C24×C4—671no—
308C2×#126—6716no—
309C2×#126—673no—
310C2×#126—673no—
311C2×#126—671no—
312C2×#126C2×D6711yescharacteristic
313C22×C4⋊C4—673no—
314C22×C4⋊C4—673no—
315C22×C4⋊C4—673no—
316C22×C4⋊C4—673no—
317C22×C22⋊C4—673no—
318C22×C22⋊C4—673no—
319C22×C42—673no—
320C22×C42—671no—
321#389—673no—
322#389—673no—
323#389—673no—
324#389—673no—
325#389—673no—
326#389—673no—
327#389—673no—
328#389C2×D6713yescomplemented
329#388—673no—
330#388—673no—
331#388—673no—
332#388—673no—
333#388—673no—
334#388—673no—
335#388—673no—
336#388C2×D6713yes—
337#387—673no—
338#387—673no—
339#387—673no—
340#387—673no—
341#387—673no—
342#387—673no—
343#387—673no—
344#387C2×D6713yescomplemented
345#385—673no—
346#385—673no—
347#385—673no—
348#385—673no—
349#385—673no—
350#385—673no—
351#385—673no—
352#385C2×D6713yes—
353#384—673no—
354#384—673no—
355#384—671no—
356#384C2×D6711yescharacteristic, complemented
357#383—671no—
358#383—673no—
359#383—673no—
360#383C2×D6711yescharacteristic, complemented
361C2×#95—673no—
362C2×#95—673no—
363C2×#95—673no—
364C2×#95—673no—
365C25—671no—
366C23×C4—2686no—
367C23×C4—671no—
368C23×C4—671no—
369C23×C4—671no—
370C23×C4—673no—
371C23×C4—673no—
372C23×C4—673no—
373C23×C4—673no—
374C23×C4—671no—
375C23×C4—671no—
376C23×C4C22×D6711yescharacteristic
377#126—6724no—
378#126—6724no—
379#126—678no—
380#126C4×D6718yescomplemented
381C2×C4⋊C4—2686no—
382C2×C4⋊C4—2686no—
383C2×C4⋊C4—2686no—
384C2×C4⋊C4—673no—
385C2×C4⋊C4—673no—
386C2×C4⋊C4—673no—
387C2×C4⋊C4—673no—
388C2×C4⋊C4—673no—
389C2×C4⋊C4—673no—
390C2×C4⋊C4—673no—
391C2×C4⋊C4—673no—
392C2×C4⋊C4—673no—
393C2×C4⋊C4—673no—
394C2×C4⋊C4—673no—
395C2×C4⋊C4—673no—
396C2×C4⋊C4—6724no—
397C2×C4⋊C4C22×D6713yes—
398C2×C4⋊C4C22×D6713yes—
399C2×C4⋊C4C22×D6713yes—
400C2×C4⋊C4C22×D6713yes—
401C2×C22⋊C4—2686no—
402C2×C22⋊C4—673no—
403C2×C22⋊C4—673no—
404C2×C22⋊C4—673no—
405C2×C22⋊C4—673no—
406C2×C22⋊C4—673no—
407C2×C22⋊C4—673no—
408C2×C22⋊C4—6724no—
409C2×C22⋊C4C22×D6713yes—
410C2×C22⋊C4C22×D6713yes—
411C2×C42—2686no—
412C2×C42—673no—
413C2×C42—673no—
414C2×C42—673no—
415C2×C42—673no—
416C2×C42—678no—
417C2×C42C22×D6713yes—
418C2×C42C22×D6711yescharacteristic
419#95—673no—
420#95—673no—
421#95—673no—
422#95—673no—
423#95—673no—
424#95—673no—
425#95—673no—
426#95—673no—
427#95—673no—
428#95—673no—
429#95—673no—
430#95—673no—
431#95C22×D6713yes—
432#95C22×D6713yes—
433#95C22×D6713yes—
434#95C22×D6713yes—
435C24—671no—
436C24—674no—
437C24—671no—
438C24—1346no—
439C24—1346no—
440C24C23×D6711yescharacteristic
441C22×C4—43no—
442C22×C4—2683no—
443C22×C4—2683no—
444C22×C4—26816no—
445C22×C4—2683no—
446C22×C4—673no—
447C22×C4—673no—
448C22×C4—6712no—
449C22×C4—6712no—
450C22×C4—673no—
451C22×C4—671no—
452C22×C4—671no—
453C22×C4—673no—
454C22×C4—1346no—
455C22×C4—1346no—
456C22×C4—1346no—
457C22×C4—1346no—
458C22×C4—1346no—
459C22×C4—1346no—
460C22×C4—13412no—
461C22×C4—1346no—
462C22×C4—1346no—
463C22×C4—1346no—
464C22×C4—1346no—
465C22×C4C23×D6713yes—
466C22×C4C23×D6711yescharacteristic
467C22×C4C23×D6711yescharacteristic
468C22×C4C23×D6713yes—
469C22×C4C23×D6713yes—
470C22×C4C23×D6713yes—
471C4⋊C4—43no—
472C4⋊C4—43no—
473C4⋊C4—43no—
474C4⋊C4—2683no—
475C4⋊C4—2683no—
476C4⋊C4—2683no—
477C4⋊C4—2683no—
478C4⋊C4—2683no—
479C4⋊C4—2683no—
480C4⋊C4—2683no—
481C4⋊C4—2683no—
482C4⋊C4—2683no—
483C4⋊C4—6712no—
484C4⋊C4—6712no—
485C4⋊C4—6712no—
486C4⋊C4C2×C4×D67112yescomplemented
487C22⋊C4—43no—
488C22⋊C4—2683no—
489C22⋊C4—2683no—
490C22⋊C4—2683no—
491C22⋊C4—6712no—
492C22⋊C4—6712no—
493C22⋊C4—6712no—
494C22⋊C4C2×C4×D67112yes—
495C42—43no—
496C42—2683no—
497C42—2683no—
498C42—2683no—
499C42—6712no—
500C42C2×C4×D6714yescomplemented
501C23—23no—
502C23—23no—
503C23—2684no—
504C23—26812no—
505C23—672no—
506C23—676no—
507C23—676no—
508C23—672no—
509C23—13424no—
510C23—1343no—
511C23—1343no—
512C23C22×C4×D6712yes—
513C23C24×D6711yescharacteristic
514C2×C4—23no—
515C2×C4—23no—
516C2×C4—23no—
517C2×C4—23no—
518C2×C4—23no—
519C2×C4—23no—
520C2×C4—23no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—26no—
525C2×C4—44no—
526C2×C4—2684no—
527C2×C4—2684no—
528C2×C4—2684no—
529C2×C4—2684no—
530C2×C4—26812no—
531C2×C4—26812no—
532C2×C4—2684no—
533C2×C4—676no—
534C2×C4—676no—
535C2×C4—13424no—
536C2×C4—13424no—
537C2×C4—1343no—
538C2×C4—1343no—
539C2×C4—1343no—
540C2×C4—1343no—
541C2×C4—1343no—
542C2×C4—1343no—
543C2×C4—1346no—
544C2×C4—1343no—
545C2×C4—1343no—
546C2×C4—1343no—
547C2×C4—1343no—
548C2×C4C22×C4×D6716yes—
549C2×C4C22×C4×D6716yes—
550C22—26no—
551C22—43no—
552C22—41no—
553C22—26816no—
554C22—2683no—
555C22—2681no—
556C22—6712no—
557C22—6712no—
558C22—674no—
559C22—1346no—
560C22C23×C4×D6711yescharacteristic
561C22H(5,𝔽2)×D6713yes—
562C222−1+4×D6713yes—
563C4—26no—
564C4—26no—
565C4—43no—
566C4—41no—
567C4—41no—
568C4—43no—
569C4—2683no—
570C4—2681no—
571C4—2681no—
572C4—2683no—
573C4—1346no—
574C4—1346no—
575C2—42no—
576C2—2682no—
577C2—678no—
578C2#520×D6713yes—
579C2#560×D6711yescharacteristic
580C2#563×D6713yes—
subgroupG/H
center, FrattiniC23C24×D67
commutatorC22×C67C24×C4
FittingC67×#2383C2
socle, cosocleC23×C67C25
prime radicalC24×D67C23
hypercenter#2383D67
hypocenterC67C2×#2383
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #2383

Complements

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

NH ≅ G/Ntype
2. #22563402C2semidirect
3. #22563401C2semidirect
4. #22563400C2semidirect
5. #22563399C2semidirect
6. #22563398C2semidirect
7. #22563397C2semidirect
8. #22563396C2semidirect
9. #389×D67C2semidirect
11. #387×D67C2semidirect
13. #384×D67C2semidirect
14. #383×D67C2semidirect
15. C67×#2383C2semidirect
20. #126×D67C4semidirect
29. #16863295C22semidirect
30. #16863294C22semidirect
31. #16863293C22semidirect
32. #16863293C22semidirect
33. #16863292C22semidirect
34. #16863291C22semidirect
35. #16863291C22semidirect
43. #16863283C22semidirect
44. #16863282C22semidirect
45. #16863281C22semidirect
46. #16863280C22semidirect
47. #16863279C22semidirect
48. #16863278C22semidirect
49. #16863277C22semidirect
57. #16863266C22semidirect
58. #16863265C22semidirect
59. #16863265C22semidirect
60. #16863264C22semidirect
61. #16863263C22semidirect
62. #16863263C22semidirect
68. #389×C67C22semidirect
70. #387×C67C22semidirect
72. #384×C67C22semidirect
73. #383×C67C22semidirect
85. #14524038C2×C4semidirect
86. #14524037C2×C4semidirect
87. #14524036C2×C4semidirect
103. C4⋊C4×D67C2×C4semidirect
117. C42×D67C2×C4semidirect
131. #126×C67C2×C4semidirect
175. C4⋊Dic67C22×C4semidirect
182. Dic67⋊C4C22×C4semidirect
183. Dic67⋊C4C22×C4semidirect
193. C4×Dic67C22×C4semidirect
205. C4⋊C4×C67C22×C4semidirect
209. C42×C67C22×C4semidirect
290. D67#2383direct (N × H)
299. #2383D67direct (N × H)
306. C67C2×#2383semidirect
328. #389C2×D67semidirect
344. #387C2×D67semidirect
356. #384C2×D67semidirect
360. #383C2×D67semidirect
380. #126C4×D67semidirect
486. C4⋊C4C2×C4×D67semidirect
500. C42C2×C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#126×D672
#225634022
#225634012
#225634002
#225633992
#225633982
#225633972
#225633962
#389×D672
#388×D672
#387×D672
#385×D672
#384×D672
#383×D672
C67×#23832
C2×#238367