Finite Group Databasebeta

← #31029022

Subgroups — #31029022

Order 17152.

2042 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22568519C211yesmaximal, characteristic, complemented
2#22568517C212yesmaximal, complemented
3#22566181C214yesmaximal, complemented
4#22565432C214yesmaximal, complemented
5#22565431C214yesmaximal, complemented
6#22563936C212yesmaximal, complemented
7#22563935C212yesmaximal, complemented
8#22563931C212yesmaximal, complemented
9#22563929C212yesmaximal, complemented
10#22562820C214yesmaximal
11#380×D67C212yesmaximal
12C2×C4×(C2×D67)⋊C4C211yesmaximal, characteristic
13C67×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D67C2211yescharacteristic
15C2×#14524035C2211yescharacteristic, complemented
16C2×#14524034C2212yescomplemented
17C2×C4⋊C4×D67C2212yes—
18C2×C4⋊C4×D67C2212yes—
19C22×(C2×D67)⋊C4C2212yes—
20C2×C22⋊C4×D67C2214yes—
21C22×C4×Dic67C2211yescharacteristic
22#16863337C2214yescomplemented
23#16863337C2214yescomplemented
24#16863336C2214yescomplemented
25#16863334C2214yescomplemented
26#16863330C2214yescomplemented
27#16863329C2214yescomplemented
28#16863327C2214yescomplemented
29#16863326C2214yescomplemented
30#16863295C2212yescomplemented
31#16863294C2212yescomplemented
32#16863292C2212yescomplemented
33#16863283C2214yescomplemented
34#16863280C2214yescomplemented
35#16863279C2214yescomplemented
36#16863271C2214yes—
37#16863270C2214yes—
38#16863270C2214yes—
39#16863269C2214yes—
40#16863268C2214yes—
41#16863267C2214yes—
42#16863267C2214yes—
43#16863262C2211yescharacteristic, complemented
44#16863261C2212yescomplemented
45#16863256C2212yes—
46#16863255C2214yes—
47#16863255C2214yes—
48#16863255C2214yes—
49#16863254C2214yes—
50#16863253C2214yes—
51#16863253C2214yes—
52#16863253C2212yes—
53#16863253C2212yes—
54C4×(C2×D67)⋊C4C2212yes—
55C4×(C2×D67)⋊C4C2212yes—
56C4×(C2×D67)⋊C4C2212yes—
57C4×(C2×D67)⋊C4C2212yes—
58C2×C67⋊#95C2212yes—
59#95×D67C2214yes—
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
68C23×Dic67C2312yes—
69C23×Dic67C2311yescharacteristic
70C22×C4×D67—42no—
71C22×C4×D67—42no—
72C22×C4×D67Q818yes—
73C22×C4×D67C2312yes—
74C22×C4×D67C2314yes—
75C22×C4×D67C2314yes—
76C22×C4×D67C2312yes—
77#14524035—42no—
78#14524034—44no—
79C2×C4⋊Dic67C2312yes—
80C2×C4⋊Dic67C2312yes—
81C2×C4⋊Dic67C2314yes—
82C2×C4⋊Dic67C2314yes—
83C2×Dic67⋊C4C2314yes—
84C2×Dic67⋊C4C2314yes—
85C2×Dic67⋊C4C2314yes—
86C2×Dic67⋊C4C2314yes—
87C2×Dic67⋊C4C2314yes—
88C2×Dic67⋊C4C2314yes—
89C4⋊C4×D67—44no—
90C4⋊C4×D67—44no—
91C2×C22⋊Dic67C2314yes—
92C2×(C2×D67)⋊C4—28no—
93C2×(C2×D67)⋊C4C2312yes—
94C2×(C2×D67)⋊C4C2312yes—
95C2×(C2×D67)⋊C4C2312yes—
96C2×(C2×D67)⋊C4C2312yes—
97C2×(C2×D67)⋊C4C2314yes—
98C2×(C2×D67)⋊C4C2314yes—
99C22⋊C4×D67—216no—
100C2×C4×Dic67—42no—
101C2×C4×Dic67C2312yes—
102C2×C4×Dic67C2311yescharacteristic
103C2×C4×Dic67C2311yescharacteristic
104C67⋊#95C2314yes—
105C67⋊#95C2314yes—
106C67⋊#95C2314yes—
107C67⋊#95C2312yes—
108C67⋊#95C2312yes—
109C67⋊#95C2312yes—
110C67⋊#95C2312yes—
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—41no—
133C23×D67—41no—
134C23×D67—42no—
135C23×D67C2×Q814yes—
136C23×D67#4118yes—
137C23×D67C2412yes—
138C22×Dic67—24no—
139C22×Dic67—42no—
140C22×Dic67—41no—
141C22×Dic67—41no—
142C22×Dic67—42no—
143C22×Dic67C2×Q814yes—
144C22×Dic67C2412yes—
145C22×Dic67C2412yes—
146C22×Dic67C2414yes—
147C22×Dic67C2414yes—
148C22×Dic67C2412yes—
149C2×C4×D67—24no—
150C2×C4×D67—24no—
151C2×C4×D67—28no—
152C2×C4×D67—28no—
153C2×C4×D67—416no—
154C2×C4×D67—42no—
155C2×C4×D67—42no—
156C2×C4×D67—44no—
157C2×C4×D67—44no—
158C2×C4×D67—44no—
159C2×C4×D67—44no—
160C2×C4×D67C2×Q818yes—
161C2×C4×D67C2×Q818yes—
162C4⋊Dic67—24no—
163C4⋊Dic67—24no—
164C4⋊Dic67—44no—
165C4⋊Dic67—44no—
166Dic67⋊C4—44no—
167Dic67⋊C4—44no—
168Dic67⋊C4—44no—
169Dic67⋊C4—44no—
170Dic67⋊C4—44no—
171Dic67⋊C4—44no—
172C22⋊Dic67—28no—
173(C2×D67)⋊C4—28no—
174(C2×D67)⋊C4—28no—
175(C2×D67)⋊C4—24no—
176(C2×D67)⋊C4—24no—
177(C2×D67)⋊C4—24no—
178(C2×D67)⋊C4—24no—
179C4×Dic67—24no—
180C4×Dic67—41no—
181C4×Dic67—41no—
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—22no—
210C22×D67—22no—
211C22×D67—22no—
212C22×D67—22no—
213C22×D67—44no—
214C22×D67—44no—
215C22×D67—48no—
216C22×D67—48no—
217C22×D67#125116yescomplemented
218C22×D67C22×Q814yes—
219C22×D67C2×#4114yes—
220C22×D67C2×#4114yes—
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—24no—
246C2×Dic67—24no—
247C2×Dic67—22no—
248C2×Dic67—22no—
249C2×Dic67—24no—
250C2×Dic67—24no—
251C2×Dic67—24no—
252C2×Dic67—22no—
253C2×Dic67—24no—
254C2×Dic67—22no—
255C2×Dic67—44no—
256C2×Dic67—48no—
257C2×Dic67C22×Q814yes—
258C4×D67—216no—
259C4×D67—48no—
260C4×D67—88no—
261C4×D67—88no—
262C22×C67—22no—
263C22×C67—22no—
264C22×C67—24no—
265C22×C67—24no—
266C22×C67C2×#12514yes—
267C22×C67C23×Q811yescharacteristic
268C22×C67C22×#4112yes—
269C22×C67C2×H(5,𝔽2)11yescharacteristic
270C22×C67C2×H(5,𝔽2)11yescharacteristic
271C22×C67#58512yes—
272C2×D67—24no—
273C2×D67—24no—
274C2×D67—24no—
275C2×D67—24no—
276C2×D67—24no—
277C2×D67—24no—
278C2×D67—416no—
279C2×D67C2×#12518yes—
280C4×C67—22no—
281C4×C67—44no—
282C4×C67—44no—
283C4×C67—42no—
284C4×C67—42no—
285C4×C67#54314yes—
286Dic67—24no—
287Dic67—44no—
288Dic67—44no—
289Dic67—42no—
290Dic67—42no—
291#35020—671nomaximal, Sylow
292C2×C67—24no—
293C2×C67C22×#12511yescharacteristic
294C2×C67C2×#54311yescharacteristic
295C2×C67#345811yescharacteristic
296C2×C67#350512yes—
297C2×C67#350512yes—
298D67—28no—
299#2851—671no—
300#2851—672no—
301#2851D6711yescharacteristic, complemented
302#2646—674no—
303#2569—674no—
304#2569—674no—
305#2442—672no—
306#2442—672no—
307#2442—672no—
308#2442—672no—
309#2315—674no—
310C2×#380—672no—
311C2×C4×C22⋊C4—671no—
312C67#3502011yescharacteristic, complemented, Sylow
313C24×C4—671no—
314C2×#125—672no—
315C2×#125—671no—
316C2×#125C2×D6711yescharacteristic, complemented
317C22×C4⋊C4—672no—
318C22×C4⋊C4—672no—
319C22×C22⋊C4—672no—
320C22×C22⋊C4—674no—
321C22×C42—671no—
322C22×C42C2×D6711yescharacteristic
323#400—674no—
324#400—674no—
325#400—674no—
326#400—674no—
327#398—674no—
328#398—674no—
329#398—674no—
330#398—674no—
331#389—672no—
332#389—672no—
333#389—672no—
334#389C2×D6712yescomplemented
335#387—674no—
336#387—674no—
337#387—674no—
338#387C2×D6714yescomplemented
339#385—674no—
340#385—674no—
341#385—674no—
342#385—674no—
343#385—674no—
344#385—674no—
345#385—674no—
346#385C2×D6714yes—
347#382—672no—
348#382—671no—
349#382C2×D6711yescharacteristic, complemented
350#380—672no—
351#380—674no—
352#380—674no—
353#380—672no—
354#380—672no—
355#380—674no—
356#380—674no—
357#380—674no—
358#380—674no—
359#380C2×D6712yes—
360C4×C22⋊C4—672no—
361C4×C22⋊C4—672no—
362C4×C22⋊C4—672no—
363C4×C22⋊C4—672no—
364C2×#95—672no—
365C2×#95—674no—
366C25—671no—
367C23×C4—2682no—
368C23×C4—2682no—
369C23×C4—671no—
370C23×C4—678no—
371C23×C4—672no—
372C23×C4—672no—
373C23×C4—674no—
374C23×C4—674no—
375C23×C4—672no—
376C23×C4C22×D6711yescharacteristic
377C23×C4C22×D6712yes—
378#125—42no—
379#125—2682no—
380#125—2684no—
381C2×C4⋊C4—2684no—
382C2×C4⋊C4—2684no—
383C2×C4⋊C4—674no—
384C2×C4⋊C4—674no—
385C2×C4⋊C4—674no—
386C2×C4⋊C4—674no—
387C2×C4⋊C4—674no—
388C2×C4⋊C4—674no—
389C2×C4⋊C4—674no—
390C2×C4⋊C4—674no—
391C2×C4⋊C4—672no—
392C2×C4⋊C4—672no—
393C2×C4⋊C4C22×D6712yes—
394C2×C4⋊C4C22×D6712yes—
395C2×C4⋊C4C22×D6714yes—
396C2×C4⋊C4C22×D6714yes—
397C2×C22⋊C4—674no—
398C2×C22⋊C4—672no—
399C2×C22⋊C4—672no—
400C2×C22⋊C4—672no—
401C2×C22⋊C4—672no—
402C2×C22⋊C4—674no—
403C2×C22⋊C4—674no—
404C2×C22⋊C4—13416no—
405C2×C22⋊C4—1348no—
406C2×C22⋊C4C22×D6714yes—
407C2×C42—42no—
408C2×C42—2682no—
409C2×C42—671no—
410C2×C42—671no—
411C2×C42—672no—
412C2×C42C22×D6711yescharacteristic
413C2×C42C22×D6711yescharacteristic
414C2×C42C22×D6712yes—
415#95—672no—
416#95—672no—
417#95—672no—
418#95—672no—
419#95—674no—
420#95—674no—
421#95—674no—
422#95—674no—
423#95—674no—
424#95—674no—
425#95—674no—
426#95—674no—
427#95—674no—
428#95C22×D6714yes—
429#95C22×D6714yes—
430#95C22×D6714yes—
431C24—2681no—
432C24—2681no—
433C24—2682no—
434C24—674no—
435C24—678no—
436C24—672no—
437C24C23×D6711yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—2682no—
444C22×C4—2682no—
445C22×C4—26816no—
446C22×C4—2684no—
447C22×C4—2684no—
448C22×C4—2684no—
449C22×C4—2684no—
450C22×C4—2682no—
451C22×C4—2681no—
452C22×C4—2681no—
453C22×C4—2682no—
454C22×C4—674no—
455C22×C4—678no—
456C22×C4—672no—
457C22×C4—678no—
458C22×C4—672no—
459C22×C4—672no—
460C22×C4—674no—
461C22×C4—674no—
462C22×C4—1348no—
463C22×C4—1348no—
464C22×C4—1344no—
465C22×C4—1344no—
466C22×C4—1344no—
467C22×C4Q8×D6714yes—
468C22×C4C23×D6712yes—
469C22×C4C23×D6712yes—
470C22×C4C23×D6714yes—
471C22×C4C23×D6714yes—
472C22×C4C23×D6711yescharacteristic
473C22×C4C23×D6711yescharacteristic
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—2684no—
479C4⋊C4—2684no—
480C4⋊C4—2684no—
481C4⋊C4—2684no—
482C4⋊C4—2684no—
483C4⋊C4—2684no—
484C4⋊C4—2684no—
485C4⋊C4—2684no—
486C4⋊C4—1344no—
487C4⋊C4—1344no—
488C22⋊C4—28no—
489C22⋊C4—1348no—
490C22⋊C4—1348no—
491C22⋊C4—1344no—
492C22⋊C4—1344no—
493C22⋊C4—1344no—
494C22⋊C4—1344no—
495C22⋊C4—1348no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—2681no—
500C42—2681no—
501C42—1344no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—2684no—
506C23—2684no—
507C23—2688no—
508C23—2688no—
509C23—6716no—
510C23—674no—
511C23—674no—
512C23—674no—
513C23—1342no—
514C23—1342no—
515C23—1342no—
516C23—1342no—
517C23C2×Q8×D6712yes—
518C23#41×D6714yes—
519C23C24×D6711yescharacteristic
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—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—2688no—
537C2×C4—2688no—
538C2×C4—2684no—
539C2×C4—5368no—
540C2×C4—5368no—
541C2×C4—674no—
542C2×C4—1344no—
543C2×C4—1342no—
544C2×C4—1342no—
545C2×C4—1344no—
546C2×C4—1344no—
547C2×C4—1342no—
548C2×C4—1342no—
549C2×C4—1344no—
550C2×C4—1344no—
551C2×C4—1342no—
552C2×C4—1342no—
553C2×C4—1344no—
554C2×C4—1342no—
555C2×C4—1342no—
556C2×C4—13416no—
557C2×C4C2×Q8×D6712yes—
558C2×C4C2×Q8×D6712yes—
559C2×C4#1452408812yes—
560C2×C4#1452408912yes—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—26816no—
566C22—678no—
567C22—1344no—
568C22—1344no—
569C22—1344no—
570C22—1344no—
571C22—1344no—
572C22—1344no—
573C22#125×D6714yes—
574C22C22×Q8×D6711yescharacteristic
575C22C2×#41×D6712yes—
576C22C2×#1452408811yescharacteristic
577C22C2×#1452408911yescharacteristic
578C22#1686423212yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—2684no—
585C4—2682no—
586C4—2682no—
587C4—2684no—
588C4—1344no—
589C4#1686402114yes—
590C2—24no—
591C2—1348no—
592C2C2×#125×D6711yescharacteristic
593C2C2×#1686402111yescharacteristic
594C2#2257361211yescharacteristic
595C2#2257403712yes—
596C2#2257404412yes—
subgroupG/H
center, FrattiniC23C24×D67
commutator, socle, cosocleC23×C67C25
FittingC67×#2851C2
prime radicalC24×D67C23
hypercenter#2851D67
hypocenterC67#35020
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #22568519C2semidirect
2. #22568517C2semidirect
3. #22566181C2semidirect
4. #22565432C2semidirect
5. #22565431C2semidirect
6. #22563936C2semidirect
7. #22563935C2semidirect
8. #22563931C2semidirect
9. #22563929C2semidirect
13. C67×#2851C2semidirect
15. C2×#14524035C22semidirect
16. C2×#14524034C22semidirect
22. #16863337C22semidirect
23. #16863337C22semidirect
24. #16863336C22semidirect
25. #16863334C22semidirect
26. #16863330C22semidirect
27. #16863329C22semidirect
28. #16863327C22semidirect
29. #16863326C22semidirect
30. #16863295C22semidirect
31. #16863294C22semidirect
32. #16863292C22semidirect
33. #16863283C22semidirect
34. #16863280C22semidirect
35. #16863279C22semidirect
43. #16863262C22semidirect
44. #16863261C22semidirect
60. C2×#125×C67C22semidirect
62. #389×C67C22semidirect
63. #387×C67C22semidirect
65. #382×C67C22semidirect
217. C22×D67#125semidirect
301. #2851D67semidirect
312. C67#35020semidirect
316. C2×#125C2×D67semidirect
334. #389C2×D67semidirect
338. #387C2×D67semidirect
349. #382C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225685192
#225685172
#225661812
#225654322
#225654312
#225639362
#225639352
#225639312
#225639292
#225628202
#380×D672
C2×C4×(C2×D67)⋊C42
C67×#28512
#3502067