Finite Group Databasebeta

← C83×#61900 #35289724

Subgroups — C83×#61900 #35289724

Order 21248.

1692 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3523C211yesmaximal, characteristic
2C2×#557×C83C211yesmaximal, characteristic, complemented
3C83×#3398C216yesmaximal, complemented
4C83×#3395C212yesmaximal, complemented
5C83×#3374C216yesmaximal, complemented
6C83×#3359C212yesmaximal, complemented
7C83×#3218C213yesmaximal
8C83×#3209C213yesmaximal, complemented
9C83×#3113C213yesmaximal, complemented
10C83×#3108C213yesmaximal
11C83×#2984C211yesmaximal, characteristic, complemented
12#557×C83—21no—
13#557×C83—23no—
14#557×C83C2212yes—
15#557×C83C2216yes—
16#557×C83C2211yescharacteristic
17#557×C83C2213yes—
18#556×C83C2213yes—
19#552×C83—23no—
20D4×Q8×C83—21no—
21D4×Q8×C83C2211yescharacteristic
22#541×C83—23no—
23#533×C83C2213yes—
24C2×C4⋊Q8×C83C2213yes—
25#529×C83C2213yes—
26C2×#125×C83—23no—
27#519×C83—21no—
28C2×C4×Q8×C83—21no—
29C2×C4×Q8×C83C2213yes—
30C2×C4×Q8×C83C2211yescharacteristic
31#497×C83C2216yes—
32#494×C83C2216yes—
33#491×C83C2216yes—
34#487×C83C2216yes—
35#483×C83C2213yes—
36#480×C83C2211yescharacteristic
37#479×C83C2216yes—
38#475×C83C2216yes—
39#464×C83C2216yes—
40#462×C83C2216yes—
41#462×C83C2216yes—
42#462×C83C22112yes—
43#461×C83C2216yes—
44#451×C83C2213yes—
45#451×C83C2216yes—
46#448×C83C2213yes—
47#446×C83C2216yes—
48C8×Q8×C83C2212yes—
49#441×C83C22112yes—
50C4×Q16×C83C2216yes—
51C4×SD16×C83C2216yes—
52#432×C83C2213yes—
53#431×C83C2213yes—
54C22×Q8×C83—21no—
55C22×Q8×C83D411yescharacteristic
56C22×Q8×C83C2313yes—
57C2×Q16×C83C2316yes—
58C2×Q16×C83C23112yes—
59C2×SD16×C83C2316yes—
60C4⋊Q8×C83—23no—
61C4⋊Q8×C83—23no—
62C4⋊Q8×C83—23no—
63C4⋊Q8×C83C2316yes—
64C4⋊Q8×C83C2316yes—
65C4⋊Q8×C83C2313yes—
66C4⋊Q8×C83C2313yes—
67#126×C83—26no—
68#126×C83—23no—
69#125×C83—26no—
70#125×C83—26no—
71#125×C83—212no—
72#125×C83—26no—
73#125×C83—23no—
74#125×C83—23no—
75#125×C83C2313yes—
76#124×C83C2313yes—
77#123×C83—23no—
78C22⋊Q8×C83—23no—
79C22⋊Q8×C83—23no—
80C22⋊Q8×C83—23no—
81C22⋊Q8×C83C2313yes—
82#121×C83C2311yescharacteristic
83C4×Q8×C83—26no—
84C4×Q8×C83—22no—
85C4×Q8×C83—21no—
86C4×Q8×C83—23no—
87C4×Q8×C83—23no—
88C4×Q8×C83—23no—
89C4×Q8×C83—23no—
90C4×Q8×C83D411yescharacteristic
91C4×Q8×C83D413yes—
92C4×Q8×C83C2316yes—
93C4×Q8×C83C2316yes—
94C4×Q8×C83C2313yes—
95C4×Q8×C83C2316yes—
96C4×Q8×C83C2311yescharacteristic
97C4×Q8×C83C2313yes—
98C4×D4×C83—23no—
99C4×D4×C83C2313yes—
100#117×C83—23no—
101#117×C83—23no—
102#117×C83—23no—
103C2×C4⋊C4×C83—23no—
104C2×C4⋊C4×C83—23no—
105C2×C4⋊C4×C83—23no—
106C2×C4⋊C4×C83D413yes—
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C42×C83—23no—
109C2×C42×C83C2313yes—
110#107×C83C2312yes—
111#106×C83C2316yes—
112C4⋊C8×C83C2316yes—
113C4⋊C8×C83C2316yes—
114#103×C83C2316yes—
115#103×C83C23112yes—
116#103×C83C2316yes—
117#103×C83C2316yes—
118#102×C83C2312yes—
119C22⋊C8×C83C2311yescharacteristic
120C22⋊C8×C83C2313yes—
121#97×C83C2316yes—
122C4×C8×C83C2316yes—
123C2×Q8×C83—24no—
124C2×Q8×C83—23no—
125C2×Q8×C83—23no—
126C2×Q8×C83—46no—
127C2×Q8×C83—44no—
128C2×Q8×C83C2×D413yes—
129C2×Q8×C83C2×D411yescharacteristic
130C2×Q8×C83C2416yes—
131C2×Q8×C83C2413yes—
132C2×D4×C83—21no—
133C2×D4×C83C2411yescharacteristic
134C22×C4×C83—21no—
135C22×C4×C83—23no—
136C22×C4×C83—23no—
137C22×C4×C83C2×D413yes—
138C22×C4×C83C2×D413yes—
139C22×C4×C83C2411yescharacteristic
140Q16×C83—412no—
141Q16×C83—46no—
142SD16×C83—46no—
143C2×C8×C83C2412yes—
144C2×C8×C83C2416yes—
145C4⋊C4×C83—26no—
146C4⋊C4×C83—26no—
147C4⋊C4×C83—26no—
148C4⋊C4×C83—26no—
149C4⋊C4×C83—26no—
150C4⋊C4×C83—26no—
151C4⋊C4×C83—21no—
152C4⋊C4×C83—23no—
153C4⋊C4×C83—23no—
154C4⋊C4×C83—23no—
155C4⋊C4×C83—23no—
156C4⋊C4×C83—23no—
157C4⋊C4×C83—23no—
158C4⋊C4×C83—26no—
159C4⋊C4×C83—23no—
160C4⋊C4×C83C2×D413yes—
161C4⋊C4×C83C2×D413yes—
162C4⋊C4×C83C2×D416yes—
163C4⋊C4×C83C2412yes—
164C4⋊C4×C83C2413yes—
165C4⋊C4×C83C2411yescharacteristic
166C22⋊C4×C83—23no—
167C22⋊C4×C83—21no—
168C22⋊C4×C83—23no—
169C22⋊C4×C83—23no—
170C22⋊C4×C83—23no—
171C42×C83—26no—
172C42×C83—26no—
173C42×C83—26no—
174C42×C83—23no—
175C42×C83C2×D413yes—
176C42×C83C2×D413yes—
177C42×C83C2413yes—
178C42×C83C2413yes—
179C23×C83—21no—
180C23×C83C22×D411yescharacteristic
181Q8×C83—26no—
182Q8×C83—26no—
183Q8×C83—26no—
184Q8×C83—26no—
185Q8×C83—46no—
186Q8×C83—46no—
187Q8×C83#13714yes—
188D4×C83—22no—
189D4×C83—42no—
190C2×C4×C83—26no—
191C2×C4×C83—26no—
192C2×C4×C83—22no—
193C2×C4×C83—22no—
194C2×C4×C83—23no—
195C2×C4×C83—21no—
196C2×C4×C83—23no—
197C2×C4×C83—26no—
198C2×C4×C83—46no—
199C2×C4×C83—42no—
200C2×C4×C83—46no—
201C2×C4×C83C22×D413yes—
202C2×C4×C83C22×D413yes—
203C2×C4×C83C22×D411yescharacteristic
204C2×C4×C83C22×D413yes—
205C2×C4×C83C22×D413yes—
206C2×C4×C83C2511yescharacteristic
207C8×C83—26no—
208C8×C83—22no—
209C22×C83—21no—
210C22×C83—21no—
211C22×C83—41no—
212C22×C83—41no—
213C22×C83—41no—
214C22×C83#57812yes—
215C22×C83C23×D411yescharacteristic
216C4×C83—21no—
217C4×C83—23no—
218C4×C83—23no—
219C4×C83—23no—
220C4×C83—46no—
221C4×C83—43no—
222C4×C83—43no—
223C4×C83—42no—
224C4×C83—41no—
225C4×C83C2×#13716yes—
226C4×C83#58512yes—
227#61900C8311yesmaximal, characteristic, direct product factor, Sylow
228C2×C83—22no—
229C2×C83—81no—
230C2×C83#348711yescharacteristic
231C2×C83C22×#13711yescharacteristic
232C2×C83C2×#57811yescharacteristic
233#3523C2×C8311yescharacteristic
234C2×#557C2×C8311yescharacteristic, complemented
235#3398C2×C8316yescomplemented
236#3395C2×C8312yescomplemented
237#3374C2×C8316yescomplemented
238#3359C2×C8312yescomplemented
239#3218C2×C8313yes—
240#3209C2×C8313yescomplemented
241#3113C2×C8313yescomplemented
242#3108C2×C8313yes—
243#2984C2×C8311yescharacteristic, complemented
244C83#6190011yescharacteristic, direct product factor, Sylow
245#557—21no—
246#557—23no—
247#557C22×C8312yes—
248#557C22×C8316yes—
249#557C22×C8311yescharacteristic
250#557C22×C8313yes—
251#556C22×C8313yes—
252#552—23no—
253D4×Q8—21no—
254D4×Q8C22×C8311yescharacteristic
255#541—23no—
256#533C22×C8313yes—
257C2×C4⋊Q8C22×C8313yes—
258#529C22×C8313yes—
259C2×#125—23no—
260#519—21no—
261C2×C4×Q8—21no—
262C2×C4×Q8C22×C8313yes—
263C2×C4×Q8C22×C8311yescharacteristic
264#497C22×C8316yes—
265#494C22×C8316yes—
266#491C22×C8316yes—
267#487C22×C8316yes—
268#483C22×C8313yes—
269#480C22×C8311yescharacteristic
270#479C22×C8316yes—
271#475C22×C8316yes—
272#464C22×C8316yes—
273#462C22×C8316yes—
274#462C22×C8316yes—
275#462C22×C83112yes—
276#461C22×C8316yes—
277#451C22×C8313yes—
278#451C22×C8316yes—
279#448C22×C8313yes—
280#446C22×C8316yes—
281C8×Q8C22×C8312yes—
282#441C22×C83112yes—
283C4×Q16C22×C8316yes—
284C4×SD16C22×C8316yes—
285#432C22×C8313yes—
286#431C22×C8313yes—
287C22×Q8—21no—
288C22×Q8D4×C8311yescharacteristic
289C22×Q8C23×C8313yes—
290C2×Q16C23×C8316yes—
291C2×Q16C23×C83112yes—
292C2×SD16C23×C8316yes—
293C4⋊Q8—23no—
294C4⋊Q8—23no—
295C4⋊Q8—23no—
296C4⋊Q8C23×C8316yes—
297C4⋊Q8C23×C8316yes—
298C4⋊Q8C23×C8313yes—
299C4⋊Q8C23×C8313yes—
300#126—26no—
301#126—23no—
302#125—26no—
303#125—26no—
304#125—212no—
305#125—26no—
306#125—23no—
307#125—23no—
308#125C23×C8313yes—
309#124C23×C8313yes—
310#123—23no—
311C22⋊Q8—23no—
312C22⋊Q8—23no—
313C22⋊Q8—23no—
314C22⋊Q8C23×C8313yes—
315#121C23×C8311yescharacteristic
316C4×Q8—26no—
317C4×Q8—22no—
318C4×Q8—21no—
319C4×Q8—23no—
320C4×Q8—23no—
321C4×Q8—23no—
322C4×Q8—23no—
323C4×Q8D4×C8311yescharacteristic
324C4×Q8D4×C8313yes—
325C4×Q8C23×C8316yes—
326C4×Q8C23×C8316yes—
327C4×Q8C23×C8313yes—
328C4×Q8C23×C8316yes—
329C4×Q8C23×C8311yescharacteristic
330C4×Q8C23×C8313yes—
331C4×D4—23no—
332C4×D4C23×C8313yes—
333#117—23no—
334#117—23no—
335#117—23no—
336C2×C4⋊C4—23no—
337C2×C4⋊C4—23no—
338C2×C4⋊C4—23no—
339C2×C4⋊C4D4×C8313yes—
340C2×C4⋊C4C23×C8311yescharacteristic
341C2×C42—23no—
342C2×C42C23×C8313yes—
343#107C23×C8312yes—
344#106C23×C8316yes—
345C4⋊C8C23×C8316yes—
346C4⋊C8C23×C8316yes—
347#103C23×C8316yes—
348#103C23×C8316yes—
349#103C23×C8316yes—
350#103C23×C83112yes—
351#102C23×C8312yes—
352C22⋊C8C23×C8311yescharacteristic
353C22⋊C8C23×C8313yes—
354#97C23×C8316yes—
355C4×C8C23×C8316yes—
356C2×Q8—24no—
357C2×Q8—23no—
358C2×Q8—23no—
359C2×Q8—46no—
360C2×Q8—44no—
361C2×Q8C2×D4×C8313yes—
362C2×Q8C2×D4×C8311yescharacteristic
363C2×Q8C24×C8316yes—
364C2×Q8C24×C8313yes—
365C2×D4—21no—
366C2×D4C24×C8311yescharacteristic
367C22×C4—21no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4C2×D4×C8313yes—
371C22×C4C2×D4×C8313yes—
372C22×C4C24×C8311yescharacteristic
373Q16—412no—
374Q16—46no—
375SD16—46no—
376C2×C8C24×C8312yes—
377C2×C8C24×C8316yes—
378C4⋊C4—26no—
379C4⋊C4—26no—
380C4⋊C4—26no—
381C4⋊C4—26no—
382C4⋊C4—26no—
383C4⋊C4—26no—
384C4⋊C4—21no—
385C4⋊C4—23no—
386C4⋊C4—23no—
387C4⋊C4—23no—
388C4⋊C4—23no—
389C4⋊C4—23no—
390C4⋊C4—23no—
391C4⋊C4—26no—
392C4⋊C4—23no—
393C4⋊C4C2×D4×C8313yes—
394C4⋊C4C2×D4×C8313yes—
395C4⋊C4C2×D4×C8316yes—
396C4⋊C4C24×C8312yes—
397C4⋊C4C24×C8313yes—
398C4⋊C4C24×C8311yescharacteristic
399C22⋊C4—23no—
400C22⋊C4—21no—
401C22⋊C4—23no—
402C22⋊C4—23no—
403C22⋊C4—23no—
404C42—26no—
405C42—26no—
406C42—26no—
407C42—23no—
408C42C2×D4×C8313yes—
409C42C2×D4×C8313yes—
410C42C24×C8313yes—
411C42C24×C8313yes—
412C23—21no—
413C23C22×D4×C8311yescharacteristic
414Q8—26no—
415Q8—26no—
416Q8—26no—
417Q8—26no—
418Q8—46no—
419Q8—46no—
420Q8#137×C8314yes—
421D4—22no—
422D4—42no—
423C2×C4—26no—
424C2×C4—26no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—23no—
428C2×C4—21no—
429C2×C4—23no—
430C2×C4—26no—
431C2×C4—46no—
432C2×C4—42no—
433C2×C4—46no—
434C2×C4C22×D4×C8313yes—
435C2×C4C22×D4×C8313yes—
436C2×C4C22×D4×C8311yescharacteristic
437C2×C4C22×D4×C8313yes—
438C2×C4C22×D4×C8313yes—
439C2×C4C25×C8311yescharacteristic
440C8—26no—
441C8—22no—
442C22—21no—
443C22—21no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22#578×C8312yes—
448C22C23×D4×C8311yescharacteristic
449C4—23no—
450C4—23no—
451C4—23no—
452C4—21no—
453C4—43no—
454C4—42no—
455C4—41no—
456C4—46no—
457C4—43no—
458C4C2×#137×C8316yes—
459C4#585×C8312yes—
460C2—22no—
461C2—81no—
462C2C83×#348711yescharacteristic
463C2C22×#137×C8311yescharacteristic
464C2C2×#578×C8311yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radical#121×C83C23
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
2. C2×#557×C83C2semidirect
3. C83×#3398C2semidirect
4. C83×#3395C2semidirect
5. C83×#3374C2semidirect
6. C83×#3359C2semidirect
8. C83×#3209C2semidirect
9. C83×#3113C2semidirect
11. C83×#2984C2semidirect
227. #61900C83direct (N × H)
234. C2×#557C2×C83semidirect
235. #3398C2×C83semidirect
236. #3395C2×C83semidirect
237. #3374C2×C83semidirect
238. #3359C2×C83semidirect
240. #3209C2×C83semidirect
241. #3113C2×C83semidirect
243. #2984C2×C83semidirect
244. C83#61900direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35232
C2×#557×C832
C83×#33982
C83×#33952
C83×#33742
C83×#33592
C83×#32182
C83×#32092
C83×#31132
C83×#31082
C83×#29842
#6190083