Finite Group Databasebeta

← C83×#61877 #35289701

Subgroups — C83×#61877 #35289701

Order 21248.

1600 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3556C211yesmaximal, characteristic, complemented
2C83×#3504C211yesmaximal, characteristic
3C83×#3390C214yesmaximal, complemented
4C83×#3388C214yesmaximal, complemented
5C83×#3374C214yesmaximal, complemented
6C83×#3372C214yesmaximal, complemented
7C83×#3242C214yesmaximal, complemented
8C83×#3236C212yesmaximal, complemented
9C83×#3120C214yesmaximal
10C83×#3116C212yesmaximal
11C83×#2987C211yesmaximal, characteristic, complemented
12#564×C83—21no—
13#564×C83—21no—
14#557×C83—22no—
15#557×C83C2214yes—
16#557×C83C2214yes—
17#557×C83C2214yes—
18#554×C83—24no—
19#554×C83C2214yes—
20#544×C83—22no—
21#544×C83C2212yes—
22#541×C83—24no—
23#541×C83C2214yes—
24#533×C83—21no—
25#533×C83C2212yes—
26#533×C83C2211yescharacteristic
27C2×#125×C83—21no—
28C2×#125×C83C2211yescharacteristic
29#520×C83C2212yes—
30C4×#41×C83C2211yescharacteristic
31#501×C83C2214yes—
32#501×C83C2214yes—
33#499×C83C2214yes—
34#491×C83C2214yes—
35#491×C83C2214yes—
36#487×C83C2214yes—
37#485×C83C2214yes—
38#484×C83C2214yes—
39#483×C83C2214yes—
40#479×C83C2214yes—
41#479×C83C2214yes—
42#479×C83C2214yes—
43#479×C83C2214yes—
44#479×C83C2214yes—
45#478×C83C2214yes—
46#477×C83C2212yes—
47#477×C83C2214yes—
48#476×C83C2212yes—
49#475×C83C2214yes—
50#462×C83C2212yes—
51#462×C83C2214yes—
52#461×C83C2212yes—
53#461×C83C2214yes—
54#452×C83C2214yes—
55#446×C83C2214yes—
56#446×C83C2214yes—
57#441×C83C2214yes—
58#441×C83C2214yes—
59#440×C83C2214yes—
60#440×C83C2214yes—
61C4×Q16×C83C2214yes—
62C4×SD16×C83C2214yes—
63#433×C83C2214yes—
64#432×C83C2212yes—
65C2×#41×C83C2311yescharacteristic
66C2×Q16×C83C2314yes—
67C2×SD16×C83C2314yes—
68C4⋊Q8×C83—24no—
69C4⋊Q8×C83—24no—
70C4⋊Q8×C83—22no—
71C4⋊Q8×C83D412yes—
72C4⋊Q8×C83C2314yes—
73C4⋊Q8×C83C2312yes—
74C4⋊Q8×C83C2314yes—
75C4⋊Q8×C83C2312yes—
76#126×C83—24no—
77#126×C83—24no—
78#125×C83—22no—
79#125×C83—24no—
80#125×C83—24no—
81#125×C83—24no—
82#125×C83—24no—
83#125×C83—24no—
84#125×C83—24no—
85#125×C83—24no—
86#125×C83—24no—
87#125×C83—24no—
88#125×C83—21no—
89#125×C83—21no—
90#125×C83—22no—
91#125×C83D411yescharacteristic
92#125×C83D411yescharacteristic
93#125×C83C2314yes—
94#125×C83C2314yes—
95#125×C83C2314yes—
96C22⋊Q8×C83—24no—
97C22⋊Q8×C83—22no—
98C22⋊Q8×C83—24no—
99C22⋊Q8×C83—24no—
100C22⋊Q8×C83—24no—
101C22⋊Q8×C83D414yes—
102C22⋊Q8×C83C2314yes—
103C4×Q8×C83—24no—
104C4×Q8×C83—24no—
105C4×Q8×C83—24no—
106C4×Q8×C83—22no—
107C4×Q8×C83C2314yes—
108C4×Q8×C83C2314yes—
109C4×Q8×C83C2312yes—
110C4×Q8×C83C2314yes—
111C4×Q8×C83C2314yes—
112C4×Q8×C83C2314yes—
113C4×D4×C83—22no—
114C4×D4×C83—24no—
115C4×D4×C83C2312yes—
116C4×D4×C83C2314yes—
117#117×C83—22no—
118#117×C83—22no—
119#117×C83—21no—
120#117×C83C2311yescharacteristic
121#117×C83C2312yes—
122C2×C4⋊C4×C83—22no—
123C2×C4⋊C4×C83—22no—
124C2×C4⋊C4×C83—24no—
125C2×C4⋊C4×C83C2312yes—
126C2×C42×C83—21no—
127C2×C42×C83C2311yescharacteristic
128#107×C83C2314yes—
129#107×C83C2314yes—
130#107×C83C2314yes—
131#106×C83C2314yes—
132#106×C83C2314yes—
133#106×C83C2314yes—
134C4⋊C8×C83C2314yes—
135C4⋊C8×C83C2312yes—
136C4⋊C8×C83C2312yes—
137C4⋊C8×C83C2314yes—
138#103×C83C2314yes—
139#103×C83C2314yes—
140#103×C83C2314yes—
141#103×C83C2314yes—
142#103×C83C2314yes—
143#103×C83C2314yes—
144#103×C83C2314yes—
145#102×C83C2314yes—
146C22⋊C8×C83C2314yes—
147#97×C83C2314yes—
148#97×C83C2314yes—
149C4×C8×C83C2314yes—
150#41×C83—22no—
151#41×C83—22no—
152C2×Q8×C83—22no—
153C2×Q8×C83—24no—
154C2×Q8×C83C2×D414yes—
155C2×Q8×C83C2414yes—
156C2×Q8×C83C2411yescharacteristic
157C2×D4×C83—21no—
158C2×D4×C83C2411yescharacteristic
159C22×C4×C83—21no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—24no—
163C22×C4×C83C2×D412yes—
164C22×C4×C83C2411yescharacteristic
165Q16×C83—28no—
166SD16×C83—28no—
167C2×C8×C83C2414yes—
168C2×C8×C83C2414yes—
169C4⋊C4×C83—24no—
170C4⋊C4×C83—22no—
171C4⋊C4×C83—24no—
172C4⋊C4×C83—24no—
173C4⋊C4×C83—24no—
174C4⋊C4×C83—24no—
175C4⋊C4×C83—22no—
176C4⋊C4×C83—22no—
177C4⋊C4×C83—22no—
178C4⋊C4×C83—22no—
179C4⋊C4×C83—24no—
180C4⋊C4×C83—24no—
181C4⋊C4×C83—24no—
182C4⋊C4×C83—24no—
183C4⋊C4×C83—24no—
184C4⋊C4×C83—24no—
185C4⋊C4×C83—24no—
186C4⋊C4×C83—24no—
187C4⋊C4×C83—24no—
188C4⋊C4×C83C2×D412yes—
189C4⋊C4×C83C2×D414yes—
190C4⋊C4×C83C2×D412yes—
191C4⋊C4×C83C2×D414yes—
192C4⋊C4×C83C2×D414yes—
193C4⋊C4×C83C2414yes—
194C4⋊C4×C83C2414yes—
195C4⋊C4×C83C2412yes—
196C22⋊C4×C83—24no—
197C22⋊C4×C83—21no—
198C22⋊C4×C83—22no—
199C22⋊C4×C83—24no—
200C22⋊C4×C83C2×D414yes—
201C42×C83—21no—
202C42×C83—22no—
203C42×C83—24no—
204C42×C83—24no—
205C42×C83—24no—
206C42×C83—24no—
207C42×C83C2×D411yescharacteristic
208C42×C83C2×D411yescharacteristic
209C42×C83C2414yes—
210C42×C83C2412yes—
211C23×C83—21no—
212C23×C83C22×D411yescharacteristic
213Q8×C83—21no—
214Q8×C83—24no—
215Q8×C83—24no—
216Q8×C83—44no—
217Q8×C83—44no—
218Q8×C83—84no—
219Q8×C832−1+412yes—
220D4×C83—24no—
221D4×C83—21no—
222D4×C832−1+412yes—
223C2×C4×C83—24no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—22no—
230C2×C4×C83—24no—
231C2×C4×C83—42no—
232C2×C4×C83—82no—
233C2×C4×C83—84no—
234C2×C4×C83—82no—
235C2×C4×C83C22×D412yes—
236C2×C4×C83C22×D414yes—
237C2×C4×C83C22×D411yescharacteristic
238C2×C4×C83C22×D412yes—
239C2×C4×C83C22×D414yes—
240C2×C4×C83C2511yescharacteristic
241C8×C83—24no—
242C8×C83—24no—
243C22×C83—22no—
244C22×C83—21no—
245C22×C83—22no—
246C22×C83—42no—
247C22×C83C23×D411yescharacteristic
248C4×C83—22no—
249C4×C83—22no—
250C4×C83—24no—
251C4×C83—24no—
252C4×C83—21no—
253C4×C83—22no—
254C4×C83—44no—
255C4×C83—42no—
256C4×C83—44no—
257C4×C83—44no—
258C4×C83C2×2−1+411yescharacteristic
259C4×C83C2×2−1+411yescharacteristic
260#61877C8311yesmaximal, characteristic, direct product factor, Sylow
261C2×C83—41no—
262C2×C83—42no—
263C2×C83#348611yescharacteristic
264C2×C83#358612yes—
265#3556C2×C8311yescharacteristic, complemented
266#3504C2×C8311yescharacteristic
267#3390C2×C8314yescomplemented
268#3388C2×C8314yescomplemented
269#3374C2×C8314yescomplemented
270#3372C2×C8314yescomplemented
271#3242C2×C8314yescomplemented
272#3236C2×C8312yescomplemented
273#3120C2×C8314yes—
274#3116C2×C8312yes—
275#2987C2×C8311yescharacteristic, complemented
276C83#6187711yescharacteristic, direct product factor, Sylow
277#564—21no—
278#564—21no—
279#557—22no—
280#557C22×C8314yes—
281#557C22×C8314yes—
282#557C22×C8314yes—
283#554—24no—
284#554C22×C8314yes—
285#544—22no—
286#544C22×C8312yes—
287#541—24no—
288#541C22×C8314yes—
289#533—21no—
290#533C22×C8312yes—
291#533C22×C8311yescharacteristic
292C2×#125—21no—
293C2×#125C22×C8311yescharacteristic
294#520C22×C8312yes—
295C4×#41C22×C8311yescharacteristic
296#501C22×C8314yes—
297#501C22×C8314yes—
298#499C22×C8314yes—
299#491C22×C8314yes—
300#491C22×C8314yes—
301#487C22×C8314yes—
302#485C22×C8314yes—
303#484C22×C8314yes—
304#483C22×C8314yes—
305#479C22×C8314yes—
306#479C22×C8314yes—
307#479C22×C8314yes—
308#479C22×C8314yes—
309#479C22×C8314yes—
310#478C22×C8314yes—
311#477C22×C8312yes—
312#477C22×C8314yes—
313#476C22×C8312yes—
314#475C22×C8314yes—
315#462C22×C8312yes—
316#462C22×C8314yes—
317#461C22×C8312yes—
318#461C22×C8314yes—
319#452C22×C8314yes—
320#446C22×C8314yes—
321#446C22×C8314yes—
322#441C22×C8314yes—
323#441C22×C8314yes—
324#440C22×C8314yes—
325#440C22×C8314yes—
326C4×Q16C22×C8314yes—
327C4×SD16C22×C8314yes—
328#433C22×C8314yes—
329#432C22×C8312yes—
330C2×#41C23×C8311yescharacteristic
331C2×Q16C23×C8314yes—
332C2×SD16C23×C8314yes—
333C4⋊Q8—24no—
334C4⋊Q8—24no—
335C4⋊Q8—22no—
336C4⋊Q8D4×C8312yes—
337C4⋊Q8C23×C8312yes—
338C4⋊Q8C23×C8314yes—
339C4⋊Q8C23×C8312yes—
340C4⋊Q8C23×C8314yes—
341#126—24no—
342#126—24no—
343#125—22no—
344#125—24no—
345#125—24no—
346#125—24no—
347#125—24no—
348#125—24no—
349#125—24no—
350#125—24no—
351#125—24no—
352#125—24no—
353#125—21no—
354#125—21no—
355#125—22no—
356#125D4×C8311yescharacteristic
357#125D4×C8311yescharacteristic
358#125C23×C8314yes—
359#125C23×C8314yes—
360#125C23×C8314yes—
361C22⋊Q8—24no—
362C22⋊Q8—22no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366C22⋊Q8D4×C8314yes—
367C22⋊Q8C23×C8314yes—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×Q8—24no—
371C4×Q8—22no—
372C4×Q8C23×C8312yes—
373C4×Q8C23×C8314yes—
374C4×Q8C23×C8314yes—
375C4×Q8C23×C8314yes—
376C4×Q8C23×C8314yes—
377C4×Q8C23×C8314yes—
378C4×D4—22no—
379C4×D4—24no—
380C4×D4C23×C8312yes—
381C4×D4C23×C8314yes—
382#117—22no—
383#117—22no—
384#117—21no—
385#117C23×C8311yescharacteristic
386#117C23×C8312yes—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4C23×C8312yes—
391C2×C42—21no—
392C2×C42C23×C8311yescharacteristic
393#107C23×C8314yes—
394#107C23×C8314yes—
395#107C23×C8314yes—
396#106C23×C8314yes—
397#106C23×C8314yes—
398#106C23×C8314yes—
399C4⋊C8C23×C8312yes—
400C4⋊C8C23×C8312yes—
401C4⋊C8C23×C8314yes—
402C4⋊C8C23×C8314yes—
403#103C23×C8314yes—
404#103C23×C8314yes—
405#103C23×C8314yes—
406#103C23×C8314yes—
407#103C23×C8314yes—
408#103C23×C8314yes—
409#103C23×C8314yes—
410#102C23×C8314yes—
411C22⋊C8C23×C8314yes—
412#97C23×C8314yes—
413#97C23×C8314yes—
414C4×C8C23×C8314yes—
415#41—22no—
416#41—22no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8C2×D4×C8314yes—
420C2×Q8C24×C8314yes—
421C2×Q8C24×C8311yescharacteristic
422C2×D4—21no—
423C2×D4C24×C8311yescharacteristic
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4C2×D4×C8312yes—
429C22×C4C24×C8311yescharacteristic
430Q16—28no—
431SD16—28no—
432C2×C8C24×C8314yes—
433C2×C8C24×C8314yes—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4C2×D4×C8312yes—
454C4⋊C4C2×D4×C8314yes—
455C4⋊C4C2×D4×C8312yes—
456C4⋊C4C2×D4×C8314yes—
457C4⋊C4C2×D4×C8314yes—
458C4⋊C4C24×C8314yes—
459C4⋊C4C24×C8314yes—
460C4⋊C4C24×C8312yes—
461C22⋊C4—24no—
462C22⋊C4—21no—
463C22⋊C4—22no—
464C22⋊C4—24no—
465C22⋊C4C2×D4×C8314yes—
466C42—21no—
467C42—22no—
468C42—24no—
469C42—24no—
470C42—24no—
471C42—24no—
472C42C2×D4×C8311yescharacteristic
473C42C2×D4×C8311yescharacteristic
474C42C24×C8314yes—
475C42C24×C8312yes—
476C23—21no—
477C23C22×D4×C8311yescharacteristic
478Q8—21no—
479Q8—24no—
480Q8—24no—
481Q8—44no—
482Q8—44no—
483Q8—84no—
484Q82−1+4×C8312yes—
485D4—24no—
486D4—21no—
487D42−1+4×C8312yes—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4C22×D4×C8312yes—
501C2×C4C22×D4×C8314yes—
502C2×C4C22×D4×C8311yescharacteristic
503C2×C4C22×D4×C8312yes—
504C2×C4C22×D4×C8314yes—
505C2×C4C25×C8311yescharacteristic
506C8—24no—
507C8—24no—
508C22—22no—
509C22—21no—
510C22—22no—
511C22—42no—
512C22C23×D4×C8311yescharacteristic
513C4—22no—
514C4—22no—
515C4—24no—
516C4—24no—
517C4—21no—
518C4—22no—
519C4—44no—
520C4—42no—
521C4—44no—
522C4—44no—
523C4C2×2−1+4×C8311yescharacteristic
524C4C2×2−1+4×C8311yescharacteristic
525C2—41no—
526C2—42no—
527C2C83×#348611yescharacteristic
528C2C83×#358612yes—
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC2×#41×C83C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ #533×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3556C2semidirect
3. C83×#3390C2semidirect
4. C83×#3388C2semidirect
5. C83×#3374C2semidirect
6. C83×#3372C2semidirect
7. C83×#3242C2semidirect
8. C83×#3236C2semidirect
11. C83×#2987C2semidirect
260. #61877C83direct (N × H)
265. #3556C2×C83semidirect
267. #3390C2×C83semidirect
268. #3388C2×C83semidirect
269. #3374C2×C83semidirect
270. #3372C2×C83semidirect
271. #3242C2×C83semidirect
272. #3236C2×C83semidirect
275. #2987C2×C83semidirect
276. C83#61877direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35562
C83×#35042
C83×#33902
C83×#33882
C83×#33742
C83×#33722
C83×#32422
C83×#32362
C83×#31202
C83×#31162
C83×#29872
#6187783