Finite Group Databasebeta

← C83×#19864 #35247688

Subgroups — C83×#19864 #35247688

Order 21248.

578 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3360C211yesmaximal, characteristic, complemented
2C83×#3357C211yesmaximal, characteristic, complemented
3C83×#3341C211yesmaximal, characteristic
4C83×#3340C211yesmaximal, characteristic, complemented
5C83×#1723C211yesmaximal, characteristic, complemented
6C83×#1691C211yesmaximal, characteristic, complemented
7C83×#1683C211yesmaximal, characteristic, complemented
8C83×#1676C211yesmaximal, characteristic, complemented
9C83×#1675C211yesmaximal, characteristic, complemented
10C83×#1646C211yesmaximal, characteristic, complemented
11C83×#1638C211yesmaximal, characteristic, complemented
12C83×#1634C211yesmaximal, characteristic, complemented
13C83×#1632C211yesmaximal, characteristic, complemented
14C83×#1581C211yesmaximal, characteristic, complemented
15C83×SD16⋊C8C211yesmaximal, characteristic, complemented
16#556×C83C2211yescharacteristic
17#555×C83C2211yescharacteristic
18#555×C83C2211yescharacteristic
19#554×C83C2211yescharacteristic
20#500×C83C2211yescharacteristic
21#500×C83C2211yescharacteristic
22#498×C83C2211yescharacteristic
23#494×C83C2211yescharacteristic
24#488×C83—21no—
25#487×C83C2211yescharacteristic
26#487×C83C2211yescharacteristic
27#485×C83—21no—
28#485×C83—21no—
29#482×C83—21no—
30#482×C83—21no—
31#482×C83—21no—
32#480×C83—21no—
33#479×C83—21no—
34#477×C83C2211yescharacteristic
35#477×C83C2211yescharacteristic
36#476×C83C2211yescharacteristic
37#476×C83C2211yescharacteristic
38#476×C83C2211yescharacteristic
39#476×C83C2211yescharacteristic
40#474×C83C2211yescharacteristic
41#468×C83—21no—
42#466×C83—21no—
43#465×C83—21no—
44#463×C83—21no—
45#461×C83C2211yescharacteristic
46C8×Q8×C83C2211yescharacteristic
47C4×SD16×C83—21no—
48C4×SD16×C83C2211yescharacteristic
49C4×D8×C83C2211yescharacteristic
50#436×C83C2211yescharacteristic
51C8×D4×C83C2211yescharacteristic
52C8×D4×C83C2211yescharacteristic
53#428×C83—21no—
54#427×C83—21no—
55#427×C83—21no—
56#335×C83C2211yescharacteristic
57#335×C83C2211yescharacteristic
58#333×C83C2211yescharacteristic
59#333×C83C2211yescharacteristic
60#333×C83C2211yescharacteristic
61#332×C83C2211yescharacteristic
62Q8⋊C8×C83C2211yescharacteristic
63D4⋊C8×C83C2211yescharacteristic
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×Q16×C83—41no—
68C2×SD16×C83—21no—
69C2×SD16×C83D411yescharacteristic
70C2×D8×C83D411yescharacteristic
71C2×M16×C83—21no—
72C22×C8×C83—21no—
73C22×C8×C83—21no—
74C4⋊Q8×C83C2311yescharacteristic
75C4⋊Q8×C83C2311yescharacteristic
76C4⋊Q8×C83C2311yescharacteristic
77C4⋊Q8×C83C2311yescharacteristic
78#126×C83—41no—
79#125×C83—21no—
80#124×C83—21no—
81#124×C83—41no—
82#124×C83—41no—
83C22⋊Q8×C83—21no—
84C22⋊Q8×C83—21no—
85C22⋊Q8×C83—41no—
86#121×C83—21no—
87#121×C83—21no—
88#121×C83—21no—
89#121×C83—21no—
90C4×Q8×C83—21no—
91C4×Q8×C83C2311yescharacteristic
92C4×D4×C83—21no—
93C4×D4×C83C2311yescharacteristic
94C4×D4×C83C2311yescharacteristic
95C4×D4×C83C2311yescharacteristic
96#117×C83—21no—
97#117×C83—21no—
98#117×C83—21no—
99#117×C83—21no—
100#117×C83—21no—
101C2×C4⋊C4×C83—21no—
102#107×C83—21no—
103#107×C83—21no—
104#107×C83—21no—
105#107×C83—21no—
106#107×C83—21no—
107#107×C83—21no—
108#107×C83—21no—
109#107×C83D411yescharacteristic
110#106×C83—21no—
111#106×C83—21no—
112#106×C83—21no—
113#106×C83—21no—
114#106×C83—21no—
115#106×C83—21no—
116#106×C83D411yescharacteristic
117C4⋊C8×C83—21no—
118C4⋊C8×C83C2311yescharacteristic
119C4⋊C8×C83C2311yescharacteristic
120C4⋊C8×C83C2311yescharacteristic
121C4⋊C8×C83C2311yescharacteristic
122#103×C83—21no—
123#103×C83—21no—
124#103×C83—21no—
125#103×C83—21no—
126#103×C83—21no—
127#103×C83—41no—
128#103×C83D411yescharacteristic
129#102×C83—21no—
130#102×C83—21no—
131#102×C83—21no—
132#102×C83—21no—
133#102×C83—21no—
134#102×C83D411yescharacteristic
135#102×C83D411yescharacteristic
136#102×C83D411yescharacteristic
137C22⋊C8×C83—21no—
138C22⋊C8×C83—21no—
139C22⋊C8×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Q8×C83—21no—
145C2×Q8×C83—21no—
146C2×Q8×C83—21no—
147C2×Q8×C83—21no—
148C2×Q8×C83C2×D411yescharacteristic
149C2×D4×C83—41no—
150C2×D4×C83—41no—
151C2×D4×C83C2×D411yescharacteristic
152C2×D4×C83C2×D411yescharacteristic
153C2×D4×C83C2×D411yescharacteristic
154C22×C4×C83—21no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—41no—
158Q16×C83—82no—
159SD16×C83—22no—
160SD16×C83—42no—
161D8×C83—41no—
162M16×C83—42no—
163C2×C8×C83—21no—
164C2×C8×C83—21no—
165C2×C8×C83—21no—
166C2×C8×C83—21no—
167C2×C8×C83—21no—
168C2×C8×C83—42no—
169C2×C8×C83—42no—
170C2×C8×C83C2×D411yescharacteristic
171C2×C8×C83C2×D411yescharacteristic
172C2×C8×C83C2×D411yescharacteristic
173C2×C8×C83C2×D411yescharacteristic
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—41no—
184C4⋊C4×C83—41no—
185C4⋊C4×C83—41no—
186C4⋊C4×C83C2×D411yescharacteristic
187C4⋊C4×C83C2×D411yescharacteristic
188C4⋊C4×C83C2×D411yescharacteristic
189C4⋊C4×C83C2×D411yescharacteristic
190C22⋊C4×C83—21no—
191C22⋊C4×C83—21no—
192C22⋊C4×C83—21no—
193C22⋊C4×C83—41no—
194C22⋊C4×C83—41no—
195C22⋊C4×C83—41no—
196C22⋊C4×C83—41no—
197C22⋊C4×C83—41no—
198C22⋊C4×C83—41no—
199C22⋊C4×C83—41no—
200C42×C83—21no—
201C42×C83—41no—
202C42×C83—41no—
203C42×C83—41no—
204C42×C83C2411yescharacteristic
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208Q8×C83—21no—
209Q8×C83—21no—
210Q8×C83—41no—
211Q8×C83—41no—
212Q8×C83—81no—
213Q8×C83—81no—
214Q8×C83—81no—
215D4×C83—21no—
216D4×C83—21no—
217D4×C83—21no—
218D4×C83—21no—
219D4×C83—21no—
220D4×C83—21no—
221D4×C83—161no—
222D4×C83—161no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—161no—
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83H(5,𝔽2)11yescharacteristic
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—21no—
245C8×C83—41no—
246C8×C83—41no—
247C8×C83—41no—
248C8×C83—41no—
249C8×C83—41no—
250C8×C83#13512yes—
251C22×C83—41no—
252C22×C83—41no—
253C22×C83—41no—
254C22×C83—41no—
255C22×C83—41no—
256C22×C83—41no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83D4211yescharacteristic
261C4×C83—21no—
262C4×C83—41no—
263C4×C83—41no—
264C4×C83—41no—
265C4×C83—41no—
266C4×C83—41no—
267C4×C83—81no—
268C4×C83—81no—
269C4×C83—81no—
270C4×C83—81no—
271C4×C83C2×#13511yescharacteristic
272C4×C83#57511yescharacteristic
273C4×C83#57811yescharacteristic
274C4×C83#57811yescharacteristic
275#19864C8311yesmaximal, characteristic, direct product factor, Sylow
276C2×C83—81no—
277C2×C83—81no—
278C2×C83—81no—
279C2×C83#326911yescharacteristic
280C2×C83#328311yescharacteristic
281C2×C83#328811yescharacteristic
282#3360C2×C8311yescharacteristic, complemented
283#3357C2×C8311yescharacteristic, complemented
284#3341C2×C8311yescharacteristic
285#3340C2×C8311yescharacteristic, complemented
286#1723C2×C8311yescharacteristic, complemented
287#1691C2×C8311yescharacteristic, complemented
288#1683C2×C8311yescharacteristic, complemented
289#1676C2×C8311yescharacteristic, complemented
290#1675C2×C8311yescharacteristic, complemented
291#1646C2×C8311yescharacteristic, complemented
292#1638C2×C8311yescharacteristic, complemented
293#1634C2×C8311yescharacteristic, complemented
294#1632C2×C8311yescharacteristic, complemented
295#1581C2×C8311yescharacteristic, complemented
296SD16⋊C8C2×C8311yescharacteristic, complemented
297C83#1986411yescharacteristic, direct product factor, Sylow
298#556C22×C8311yescharacteristic
299#555C22×C8311yescharacteristic
300#555C22×C8311yescharacteristic
301#554C22×C8311yescharacteristic
302#500C22×C8311yescharacteristic
303#500C22×C8311yescharacteristic
304#498C22×C8311yescharacteristic
305#494C22×C8311yescharacteristic
306#488—21no—
307#487C22×C8311yescharacteristic
308#487C22×C8311yescharacteristic
309#485—21no—
310#485—21no—
311#482—21no—
312#482—21no—
313#482—21no—
314#480—21no—
315#479—21no—
316#477C22×C8311yescharacteristic
317#477C22×C8311yescharacteristic
318#476C22×C8311yescharacteristic
319#476C22×C8311yescharacteristic
320#476C22×C8311yescharacteristic
321#476C22×C8311yescharacteristic
322#474C22×C8311yescharacteristic
323#468—21no—
324#466—21no—
325#465—21no—
326#463—21no—
327#461C22×C8311yescharacteristic
328C8×Q8C22×C8311yescharacteristic
329C4×SD16—21no—
330C4×SD16C22×C8311yescharacteristic
331C4×D8C22×C8311yescharacteristic
332#436C22×C8311yescharacteristic
333C8×D4C22×C8311yescharacteristic
334C8×D4C22×C8311yescharacteristic
335#428—21no—
336#427—21no—
337#427—21no—
338#335C22×C8311yescharacteristic
339#335C22×C8311yescharacteristic
340#333C22×C8311yescharacteristic
341#333C22×C8311yescharacteristic
342#333C22×C8311yescharacteristic
343#332C22×C8311yescharacteristic
344Q8⋊C8C22×C8311yescharacteristic
345D4⋊C8C22×C8311yescharacteristic
346D4⋊C8C22×C8311yescharacteristic
347D4⋊C8C22×C8311yescharacteristic
348#322C22×C8311yescharacteristic
349C2×Q16—41no—
350C2×SD16—21no—
351C2×SD16D4×C8311yescharacteristic
352C2×D8D4×C8311yescharacteristic
353C2×M16—21no—
354C22×C8—21no—
355C22×C8—21no—
356C4⋊Q8C23×C8311yescharacteristic
357C4⋊Q8C23×C8311yescharacteristic
358C4⋊Q8C23×C8311yescharacteristic
359C4⋊Q8C23×C8311yescharacteristic
360#126—41no—
361#125—21no—
362#124—21no—
363#124—41no—
364#124—41no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—41no—
368#121—21no—
369#121—21no—
370#121—21no—
371#121—21no—
372C4×Q8—21no—
373C4×Q8C23×C8311yescharacteristic
374C4×D4—21no—
375C4×D4C23×C8311yescharacteristic
376C4×D4C23×C8311yescharacteristic
377C4×D4C23×C8311yescharacteristic
378#117—21no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383C2×C4⋊C4—21no—
384#107—21no—
385#107—21no—
386#107—21no—
387#107—21no—
388#107—21no—
389#107—21no—
390#107—21no—
391#107D4×C8311yescharacteristic
392#106—21no—
393#106—21no—
394#106—21no—
395#106—21no—
396#106—21no—
397#106—21no—
398#106D4×C8311yescharacteristic
399C4⋊C8—21no—
400C4⋊C8C23×C8311yescharacteristic
401C4⋊C8C23×C8311yescharacteristic
402C4⋊C8C23×C8311yescharacteristic
403C4⋊C8C23×C8311yescharacteristic
404#103—21no—
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—41no—
410#103D4×C8311yescharacteristic
411#102—21no—
412#102—21no—
413#102—21no—
414#102—21no—
415#102—21no—
416#102D4×C8311yescharacteristic
417#102D4×C8311yescharacteristic
418#102D4×C8311yescharacteristic
419C22⋊C8—21no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C4×C8—21no—
423C4×C8C23×C8311yescharacteristic
424C4×C8C23×C8311yescharacteristic
425C4×C8C23×C8311yescharacteristic
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8C2×D4×C8311yescharacteristic
431C2×D4—41no—
432C2×D4—41no—
433C2×D4C2×D4×C8311yescharacteristic
434C2×D4C2×D4×C8311yescharacteristic
435C2×D4C2×D4×C8311yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—41no—
440Q16—82no—
441SD16—22no—
442SD16—42no—
443D8—41no—
444M16—42no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8C2×D4×C8311yescharacteristic
453C2×C8C2×D4×C8311yescharacteristic
454C2×C8C2×D4×C8311yescharacteristic
455C2×C8C2×D4×C8311yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4C2×D4×C8311yescharacteristic
469C4⋊C4C2×D4×C8311yescharacteristic
470C4⋊C4C2×D4×C8311yescharacteristic
471C4⋊C4C2×D4×C8311yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C42—21no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42C24×C8311yescharacteristic
487C23—21no—
488C23—21no—
489C23—21no—
490Q8—21no—
491Q8—21no—
492Q8—41no—
493Q8—41no—
494Q8—81no—
495Q8—81no—
496Q8—81no—
497D4—21no—
498D4—21no—
499D4—21no—
500D4—21no—
501D4—21no—
502D4—21no—
503D4—161no—
504D4—161no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—161no—
521C2×C4C22×D4×C8311yescharacteristic
522C2×C4C22×D4×C8311yescharacteristic
523C2×C4H(5,𝔽2)×C8311yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8#135×C8312yes—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22D42×C8311yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4C2×#135×C8311yescharacteristic
554C4#575×C8311yescharacteristic
555C4#578×C8311yescharacteristic
556C4#578×C8311yescharacteristic
557C2—81no—
558C2—81no—
559C2—81no—
560C2C83×#326911yescharacteristic
561C2C83×#328311yescharacteristic
562C2C83×#328811yescharacteristic
subgroupG/H
center, socleC22×C83D42
commutator, Frattini, cosocleC42C24×C83
prime radicalC83×#3341C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3360C2semidirect
2. C83×#3357C2semidirect
4. C83×#3340C2semidirect
5. C83×#1723C2semidirect
6. C83×#1691C2semidirect
7. C83×#1683C2semidirect
8. C83×#1676C2semidirect
9. C83×#1675C2semidirect
10. C83×#1646C2semidirect
11. C83×#1638C2semidirect
12. C83×#1634C2semidirect
13. C83×#1632C2semidirect
14. C83×#1581C2semidirect
15. C83×SD16⋊C8C2semidirect
275. #19864C83direct (N × H)
282. #3360C2×C83semidirect
283. #3357C2×C83semidirect
285. #3340C2×C83semidirect
286. #1723C2×C83semidirect
287. #1691C2×C83semidirect
288. #1683C2×C83semidirect
289. #1676C2×C83semidirect
290. #1675C2×C83semidirect
291. #1646C2×C83semidirect
292. #1638C2×C83semidirect
293. #1634C2×C83semidirect
294. #1632C2×C83semidirect
295. #1581C2×C83semidirect
296. SD16⋊C8C2×C83semidirect
297. C83#19864direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#33602
C83×#33572
C83×#33412
C83×#33402
C83×#17232
C83×#16912
C83×#16832
C83×#16762
C83×#16752
C83×#16462
C83×#16382
C83×#16342
C83×#16322
C83×#15812
C83×SD16⋊C82
#1986483