Finite Group Databasebeta

← C83×#61909 #35289733

Subgroups — C83×#61909 #35289733

Order 21248.

1710 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3502C211yesmaximal, characteristic, complemented
2C83×#3480C211yesmaximal, characteristic
3C83×#3400C213yesmaximal, complemented
4C83×#3398C213yesmaximal, complemented
5C83×#3395C211yesmaximal, characteristic, complemented
6C83×#3392C211yesmaximal, characteristic, complemented
7C83×#3375C213yesmaximal, complemented
8C83×#3369C213yesmaximal, complemented
9C83×#3363C211yesmaximal, characteristic, complemented
10C83×#3362C211yesmaximal, characteristic, complemented
11C83×#3216C213yesmaximal, complemented
12C83×#3210C213yesmaximal, complemented
13C83×#3116C213yesmaximal, complemented
14C83×#3107C213yesmaximal, complemented
15C83×#2984C211yesmaximal, characteristic, complemented
16#559×C83C2213yes—
17Q82×C83C2211yescharacteristic
18Q82×C83C2213yes—
19#557×C83—21no—
20#557×C83—23no—
21#557×C83C2211yescharacteristic
22#557×C83C2211yescharacteristic
23#557×C83C2213yes—
24#556×C83C2213yes—
25#553×C83—23no—
26#550×C83C2211yescharacteristic
27#548×C83—21no—
28#544×C83—23no—
29#533×C83C2213yes—
30#533×C83C2213yes—
31#532×C83C2213yes—
32#529×C83—23no—
33#519×C83—21no—
34C4×#41×C83—21no—
35C4×#41×C83C2213yes—
36C2×C4×Q8×C83C2211yescharacteristic
37#497×C83C2216yes—
38#494×C83C2213yes—
39#492×C83C2213yes—
40#489×C83C2216yes—
41#487×C83C2213yes—
42#486×C83C2213yes—
43#485×C83C2211yescharacteristic
44#485×C83C2213yes—
45#477×C83C2213yes—
46#476×C83C2213yes—
47#475×C83C2213yes—
48#474×C83C2213yes—
49#464×C83C2216yes—
50#464×C83C2216yes—
51#462×C83C2213yes—
52#462×C83C2213yes—
53#462×C83C2213yes—
54#462×C83C2213yes—
55#461×C83C2213yes—
56#461×C83C2213yes—
57#460×C83C2213yes—
58#460×C83C2213yes—
59#452×C83C2213yes—
60#452×C83C2213yes—
61#452×C83C2216yes—
62#446×C83C2213yes—
63#446×C83C2213yes—
64C8×Q8×C83C2211yescharacteristic
65C8×Q8×C83C2211yescharacteristic
66#441×C83C2216yes—
67#440×C83C2216yes—
68C4×Q16×C83C2213yes—
69C4×Q16×C83C2213yes—
70C4×SD16×C83C2213yes—
71C4×SD16×C83C2213yes—
72#432×C83C2213yes—
73#431×C83C2213yes—
74C2×#41×C83—21no—
75C2×#41×C83C2313yes—
76C22×Q8×C83D411yescharacteristic
77C2×Q16×C83C2313yes—
78C2×Q16×C83C2316yes—
79C2×Q16×C83C2313yes—
80C2×SD16×C83C2313yes—
81C2×SD16×C83C2316yes—
82C2×SD16×C83C2313yes—
83C4⋊Q8×C83—23no—
84C4⋊Q8×C83—23no—
85C4⋊Q8×C83—26no—
86C4⋊Q8×C83C2313yes—
87C4⋊Q8×C83C2313yes—
88C4⋊Q8×C83C2313yes—
89C4⋊Q8×C83C2313yes—
90C4⋊Q8×C83C2313yes—
91#127×C83C2313yes—
92#126×C83—26no—
93#125×C83—23no—
94#125×C83—23no—
95#125×C83—23no—
96#125×C83—26no—
97#125×C83—23no—
98#125×C83—23no—
99#124×C83—23no—
100#124×C83C2316yes—
101#123×C83—26no—
102C22⋊Q8×C83—23no—
103C22⋊Q8×C83—23no—
104C22⋊Q8×C83—23no—
105C22⋊Q8×C83C2311yescharacteristic
106C22⋊Q8×C83C2313yes—
107#121×C83—23no—
108#121×C83—23no—
109#121×C83—23no—
110C4×Q8×C83—21no—
111C4×Q8×C83—23no—
112C4×Q8×C83—21no—
113C4×Q8×C83—23no—
114C4×Q8×C83—21no—
115C4×Q8×C83—23no—
116C4×Q8×C83—23no—
117C4×Q8×C83D411yescharacteristic
118C4×Q8×C83D413yes—
119C4×Q8×C83C2316yes—
120C4×Q8×C83C2313yes—
121C4×Q8×C83C2313yes—
122C4×Q8×C83C2313yes—
123C4×Q8×C83C2313yes—
124C4×Q8×C83C2311yescharacteristic
125C4×Q8×C83C2313yes—
126C4×D4×C83—23no—
127C4×D4×C83—23no—
128C4×D4×C83—23no—
129C4×D4×C83—23no—
130C4×D4×C83—23no—
131C4×D4×C83C2313yes—
132C4×D4×C83C2313yes—
133#117×C83—23no—
134#117×C83—23no—
135#117×C83—23no—
136#117×C83—23no—
137#117×C83C2311yescharacteristic
138C2×C4⋊C4×C83D413yes—
139C2×C42×C83—23no—
140C2×C42×C83C2313yes—
141#107×C83C2311yescharacteristic
142#107×C83C2313yes—
143#106×C83C2311yescharacteristic
144#106×C83C2313yes—
145C4⋊C8×C83C2313yes—
146C4⋊C8×C83C2313yes—
147C4⋊C8×C83C2313yes—
148C4⋊C8×C83C2313yes—
149#103×C83C2313yes—
150#103×C83C2316yes—
151#103×C83C2313yes—
152#103×C83C2311yescharacteristic
153#103×C83C2313yes—
154#103×C83C2311yescharacteristic
155#103×C83C2313yes—
156#102×C83C2313yes—
157#102×C83C2316yes—
158#102×C83C2313yes—
159C22⋊C8×C83C2311yescharacteristic
160C22⋊C8×C83C2313yes—
161#97×C83C2316yes—
162C4×C8×C83C2313yes—
163C4×C8×C83C2313yes—
164#41×C83—43no—
165#41×C83—43no—
166#41×C83—44no—
167C2×Q8×C83—24no—
168C2×Q8×C83—23no—
169C2×Q8×C83—23no—
170C2×Q8×C83C2×D413yes—
171C2×Q8×C83C2×D411yescharacteristic
172C2×Q8×C83C2413yes—
173C2×Q8×C83C2411yescharacteristic
174C2×Q8×C83C2413yes—
175C2×D4×C83—23no—
176C2×D4×C83—23no—
177C2×D4×C83—23no—
178C2×D4×C83C2413yes—
179C22×C4×C83—23no—
180C22×C4×C83—23no—
181C22×C4×C83—23no—
182C22×C4×C83C2×D413yes—
183C22×C4×C83C2×D413yes—
184C22×C4×C83C2411yescharacteristic
185Q16×C83—43no—
186Q16×C83—46no—
187Q16×C83—43no—
188SD16×C83—43no—
189SD16×C83—46no—
190SD16×C83—43no—
191C2×C8×C83C2411yescharacteristic
192C2×C8×C83C2411yescharacteristic
193C2×C8×C83C2413yes—
194C2×C8×C83C2413yes—
195C4⋊C4×C83—23no—
196C4⋊C4×C83—23no—
197C4⋊C4×C83—23no—
198C4⋊C4×C83—23no—
199C4⋊C4×C83—23no—
200C4⋊C4×C83—23no—
201C4⋊C4×C83—23no—
202C4⋊C4×C83—23no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—23no—
205C4⋊C4×C83—23no—
206C4⋊C4×C83—23no—
207C4⋊C4×C83—26no—
208C4⋊C4×C83—23no—
209C4⋊C4×C83—23no—
210C4⋊C4×C83C2×D413yes—
211C4⋊C4×C83C2×D413yes—
212C4⋊C4×C83C2×D416yes—
213C4⋊C4×C83C2413yes—
214C4⋊C4×C83C2411yescharacteristic
215C4⋊C4×C83C2411yescharacteristic
216C4⋊C4×C83C2411yescharacteristic
217C22⋊C4×C83—21no—
218C22⋊C4×C83—23no—
219C22⋊C4×C83—21no—
220C22⋊C4×C83—26no—
221C22⋊C4×C83—23no—
222C22⋊C4×C83—23no—
223C22⋊C4×C83—26no—
224C42×C83—21no—
225C42×C83—23no—
226C42×C83—23no—
227C42×C83—26no—
228C42×C83—23no—
229C42×C83—23no—
230C42×C83C2×D413yes—
231C42×C83C2×D413yes—
232C42×C83C2413yes—
233C42×C83C2413yes—
234C23×C83—23no—
235C23×C83C22×D411yescharacteristic
236Q8×C83—26no—
237Q8×C83—21no—
238Q8×C83—23no—
239Q8×C83—23no—
240Q8×C83—21no—
241Q8×C83—23no—
242Q8×C83—23no—
243Q8×C83—83no—
244Q8×C83—83no—
245Q8×C83#13714yes—
246D4×C83—23no—
247D4×C83—23no—
248D4×C83—46no—
249D4×C83—46no—
250D4×C83—83no—
251C2×C4×C83—26no—
252C2×C4×C83—21no—
253C2×C4×C83—21no—
254C2×C4×C83—21no—
255C2×C4×C83—21no—
256C2×C4×C83—21no—
257C2×C4×C83—23no—
258C2×C4×C83—23no—
259C2×C4×C83—21no—
260C2×C4×C83—23no—
261C2×C4×C83—43no—
262C2×C4×C83—46no—
263C2×C4×C83—46no—
264C2×C4×C83—83no—
265C2×C4×C83C22×D413yes—
266C2×C4×C83C22×D411yescharacteristic
267C2×C4×C83C22×D413yes—
268C2×C4×C83C22×D413yes—
269C2×C4×C83C22×D413yes—
270C2×C4×C83C2511yescharacteristic
271C8×C83—21no—
272C8×C83—23no—
273C8×C83—23no—
274C8×C83—21no—
275C22×C83—21no—
276C22×C83—21no—
277C22×C83—21no—
278C22×C83—43no—
279C22×C83—43no—
280C22×C83—43no—
281C22×C83C23×D411yescharacteristic
282C4×C83—23no—
283C4×C83—23no—
284C4×C83—21no—
285C4×C83—23no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83—43no—
289C4×C83—43no—
290C4×C83—41no—
291C4×C83—43no—
292C4×C83—41no—
293C4×C83C2×#13716yes—
294C4×C83#58511yescharacteristic
295C4×C83#58511yescharacteristic
296#61909C8311yesmaximal, characteristic, direct product factor, Sylow
297C2×C83—41no—
298C2×C83—83no—
299C2×C83#348711yescharacteristic
300C2×C83C22×#13711yescharacteristic
301C2×C83#358411yescharacteristic
302#3502C2×C8311yescharacteristic, complemented
303#3480C2×C8311yescharacteristic
304#3400C2×C8313yescomplemented
305#3398C2×C8313yescomplemented
306#3395C2×C8311yescharacteristic, complemented
307#3392C2×C8311yescharacteristic, complemented
308#3375C2×C8313yescomplemented
309#3369C2×C8313yescomplemented
310#3363C2×C8311yescharacteristic, complemented
311#3362C2×C8311yescharacteristic, complemented
312#3216C2×C8313yescomplemented
313#3210C2×C8313yescomplemented
314#3116C2×C8313yescomplemented
315#3107C2×C8313yescomplemented
316#2984C2×C8311yescharacteristic, complemented
317C83#6190911yescharacteristic, direct product factor, Sylow
318#559C22×C8313yes—
319Q82C22×C8311yescharacteristic
320Q82C22×C8313yes—
321#557—21no—
322#557—23no—
323#557C22×C8311yescharacteristic
324#557C22×C8311yescharacteristic
325#557C22×C8313yes—
326#556C22×C8313yes—
327#553—23no—
328#550C22×C8311yescharacteristic
329#548—21no—
330#544—23no—
331#533C22×C8313yes—
332#533C22×C8313yes—
333#532C22×C8313yes—
334#529—23no—
335#519—21no—
336C4×#41—21no—
337C4×#41C22×C8313yes—
338C2×C4×Q8C22×C8311yescharacteristic
339#497C22×C8316yes—
340#494C22×C8313yes—
341#492C22×C8313yes—
342#489C22×C8316yes—
343#487C22×C8313yes—
344#486C22×C8313yes—
345#485C22×C8311yescharacteristic
346#485C22×C8313yes—
347#477C22×C8313yes—
348#476C22×C8313yes—
349#475C22×C8313yes—
350#474C22×C8313yes—
351#464C22×C8316yes—
352#464C22×C8316yes—
353#462C22×C8313yes—
354#462C22×C8313yes—
355#462C22×C8313yes—
356#462C22×C8313yes—
357#461C22×C8313yes—
358#461C22×C8313yes—
359#460C22×C8313yes—
360#460C22×C8313yes—
361#452C22×C8313yes—
362#452C22×C8313yes—
363#452C22×C8316yes—
364#446C22×C8313yes—
365#446C22×C8313yes—
366C8×Q8C22×C8311yescharacteristic
367C8×Q8C22×C8311yescharacteristic
368#441C22×C8316yes—
369#440C22×C8316yes—
370C4×Q16C22×C8313yes—
371C4×Q16C22×C8313yes—
372C4×SD16C22×C8313yes—
373C4×SD16C22×C8313yes—
374#432C22×C8313yes—
375#431C22×C8313yes—
376C2×#41—21no—
377C2×#41C23×C8313yes—
378C22×Q8D4×C8311yescharacteristic
379C2×Q16C23×C8313yes—
380C2×Q16C23×C8313yes—
381C2×Q16C23×C8316yes—
382C2×SD16C23×C8313yes—
383C2×SD16C23×C8313yes—
384C2×SD16C23×C8316yes—
385C4⋊Q8—23no—
386C4⋊Q8—23no—
387C4⋊Q8—26no—
388C4⋊Q8C23×C8313yes—
389C4⋊Q8C23×C8313yes—
390C4⋊Q8C23×C8313yes—
391C4⋊Q8C23×C8313yes—
392C4⋊Q8C23×C8313yes—
393#127C23×C8313yes—
394#126—26no—
395#125—23no—
396#125—23no—
397#125—23no—
398#125—26no—
399#125—23no—
400#125—23no—
401#124—23no—
402#124C23×C8316yes—
403#123—26no—
404C22⋊Q8—23no—
405C22⋊Q8—23no—
406C22⋊Q8—23no—
407C22⋊Q8C23×C8311yescharacteristic
408C22⋊Q8C23×C8313yes—
409#121—23no—
410#121—23no—
411#121—23no—
412C4×Q8—21no—
413C4×Q8—23no—
414C4×Q8—21no—
415C4×Q8—23no—
416C4×Q8—21no—
417C4×Q8—23no—
418C4×Q8—23no—
419C4×Q8D4×C8311yescharacteristic
420C4×Q8D4×C8313yes—
421C4×Q8C23×C8313yes—
422C4×Q8C23×C8313yes—
423C4×Q8C23×C8313yes—
424C4×Q8C23×C8313yes—
425C4×Q8C23×C8316yes—
426C4×Q8C23×C8311yescharacteristic
427C4×Q8C23×C8313yes—
428C4×D4—23no—
429C4×D4—23no—
430C4×D4—23no—
431C4×D4—23no—
432C4×D4—23no—
433C4×D4C23×C8313yes—
434C4×D4C23×C8313yes—
435#117—23no—
436#117—23no—
437#117—23no—
438#117—23no—
439#117C23×C8311yescharacteristic
440C2×C4⋊C4D4×C8313yes—
441C2×C42—23no—
442C2×C42C23×C8313yes—
443#107C23×C8311yescharacteristic
444#107C23×C8313yes—
445#106C23×C8311yescharacteristic
446#106C23×C8313yes—
447C4⋊C8C23×C8313yes—
448C4⋊C8C23×C8313yes—
449C4⋊C8C23×C8313yes—
450C4⋊C8C23×C8313yes—
451#103C23×C8311yescharacteristic
452#103C23×C8313yes—
453#103C23×C8311yescharacteristic
454#103C23×C8313yes—
455#103C23×C8313yes—
456#103C23×C8313yes—
457#103C23×C8316yes—
458#102C23×C8313yes—
459#102C23×C8313yes—
460#102C23×C8316yes—
461C22⋊C8C23×C8311yescharacteristic
462C22⋊C8C23×C8313yes—
463#97C23×C8316yes—
464C4×C8C23×C8313yes—
465C4×C8C23×C8313yes—
466#41—43no—
467#41—43no—
468#41—44no—
469C2×Q8—24no—
470C2×Q8—23no—
471C2×Q8—23no—
472C2×Q8C2×D4×C8313yes—
473C2×Q8C2×D4×C8311yescharacteristic
474C2×Q8C24×C8313yes—
475C2×Q8C24×C8311yescharacteristic
476C2×Q8C24×C8313yes—
477C2×D4—23no—
478C2×D4—23no—
479C2×D4—23no—
480C2×D4C24×C8313yes—
481C22×C4—23no—
482C22×C4—23no—
483C22×C4—23no—
484C22×C4C2×D4×C8313yes—
485C22×C4C2×D4×C8313yes—
486C22×C4C24×C8311yescharacteristic
487Q16—43no—
488Q16—46no—
489Q16—43no—
490SD16—43no—
491SD16—46no—
492SD16—43no—
493C2×C8C24×C8311yescharacteristic
494C2×C8C24×C8311yescharacteristic
495C2×C8C24×C8313yes—
496C2×C8C24×C8313yes—
497C4⋊C4—23no—
498C4⋊C4—23no—
499C4⋊C4—23no—
500C4⋊C4—23no—
501C4⋊C4—23no—
502C4⋊C4—23no—
503C4⋊C4—23no—
504C4⋊C4—23no—
505C4⋊C4—21no—
506C4⋊C4—23no—
507C4⋊C4—23no—
508C4⋊C4—23no—
509C4⋊C4—26no—
510C4⋊C4—23no—
511C4⋊C4—23no—
512C4⋊C4C2×D4×C8313yes—
513C4⋊C4C2×D4×C8313yes—
514C4⋊C4C2×D4×C8316yes—
515C4⋊C4C24×C8311yescharacteristic
516C4⋊C4C24×C8313yes—
517C4⋊C4C24×C8311yescharacteristic
518C4⋊C4C24×C8311yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—23no—
521C22⋊C4—21no—
522C22⋊C4—26no—
523C22⋊C4—23no—
524C22⋊C4—23no—
525C22⋊C4—26no—
526C42—21no—
527C42—23no—
528C42—23no—
529C42—26no—
530C42—23no—
531C42—23no—
532C42C2×D4×C8313yes—
533C42C2×D4×C8313yes—
534C42C24×C8313yes—
535C42C24×C8313yes—
536C23—23no—
537C23C22×D4×C8311yescharacteristic
538Q8—26no—
539Q8—21no—
540Q8—23no—
541Q8—23no—
542Q8—21no—
543Q8—23no—
544Q8—23no—
545Q8—83no—
546Q8—83no—
547Q8#137×C8314yes—
548D4—23no—
549D4—23no—
550D4—46no—
551D4—46no—
552D4—83no—
553C2×C4—26no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—23no—
560C2×C4—23no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—43no—
564C2×C4—46no—
565C2×C4—46no—
566C2×C4—83no—
567C2×C4C22×D4×C8313yes—
568C2×C4C22×D4×C8311yescharacteristic
569C2×C4C22×D4×C8313yes—
570C2×C4C22×D4×C8313yes—
571C2×C4C22×D4×C8313yes—
572C2×C4C25×C8311yescharacteristic
573C8—21no—
574C8—23no—
575C8—23no—
576C8—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—43no—
581C22—43no—
582C22—43no—
583C22C23×D4×C8311yescharacteristic
584C4—23no—
585C4—23no—
586C4—23no—
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—43no—
592C4—43no—
593C4—41no—
594C4—43no—
595C4C2×#137×C8316yes—
596C4#585×C8311yescharacteristic
597C4#585×C8311yescharacteristic
598C2—41no—
599C2—83no—
600C2C83×#348711yescharacteristic
601C2C22×#137×C8311yescharacteristic
602C2C83×#358411yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC83×#3480C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×Q8×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3502C2semidirect
3. C83×#3400C2semidirect
4. C83×#3398C2semidirect
5. C83×#3395C2semidirect
6. C83×#3392C2semidirect
7. C83×#3375C2semidirect
8. C83×#3369C2semidirect
9. C83×#3363C2semidirect
10. C83×#3362C2semidirect
11. C83×#3216C2semidirect
12. C83×#3210C2semidirect
13. C83×#3116C2semidirect
14. C83×#3107C2semidirect
15. C83×#2984C2semidirect
296. #61909C83direct (N × H)
302. #3502C2×C83semidirect
304. #3400C2×C83semidirect
305. #3398C2×C83semidirect
306. #3395C2×C83semidirect
307. #3392C2×C83semidirect
308. #3375C2×C83semidirect
309. #3369C2×C83semidirect
310. #3363C2×C83semidirect
311. #3362C2×C83semidirect
312. #3216C2×C83semidirect
313. #3210C2×C83semidirect
314. #3116C2×C83semidirect
315. #3107C2×C83semidirect
316. #2984C2×C83semidirect
317. C83#61909direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35022
C83×#34802
C83×#34002
C83×#33982
C83×#33952
C83×#33922
C83×#33752
C83×#33692
C83×#33632
C83×#33622
C83×#32162
C83×#32102
C83×#31162
C83×#31072
C83×#29842
#6190983