Finite Group Databasebeta

← C83×#9321 #35237145

Subgroups — C83×#9321 #35237145

Order 21248.

474 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3079C211yesmaximal, characteristic
2C83×#2989C211yesmaximal, characteristic
3C83×#1981C211yesmaximal, characteristic, complemented
4C83×#1806C211yesmaximal, characteristic
5C83×#1549C211yesmaximal, characteristic, complemented
6C83×#1303C211yesmaximal, characteristic, complemented
7C83×#1292C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C83C2211yescharacteristic
9C4×#41×C83C2211yescharacteristic
10#477×C83C412yescomplemented
11#476×C83C412yescomplemented
12#475×C83—21no—
13#474×C83—21no—
14#446×C83—21no—
15#436×C83—21no—
16#435×C83—22no—
17#432×C83—21no—
18#431×C83C2211yescharacteristic
19#428×C83—21no—
20#423×C83—21no—
21#423×C83C2211yescharacteristic
22#423×C83C2211yescharacteristic
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×#104×C83—21no—
25#417×C83—21no—
26C2×#103×C83—22no—
27C4×C4⋊C4×C83C2211yescharacteristic
28#339×C83—21no—
29#332×C83—21no—
30#332×C83—21no—
31C2×#41×C83D411yescharacteristic
32C22×Q8×C83—21no—
33C2×M16×C83—21no—
34C2×M16×C83—22no—
35C2×M16×C83—21no—
36C22×C8×C83D412yes—
37C4⋊Q8×C83—21no—
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83—21no—
40C4⋊Q8×C83C2×C412yes—
41C4×Q8×C83C2×C411yescharacteristic
42C4×Q8×C83D411yescharacteristic
43C4×D4×C83—21no—
44C4×D4×C83—21no—
45C4×D4×C83C2×C411yescharacteristic
46C4×D4×C83D411yescharacteristic
47#117×C83—21no—
48#117×C83D411yescharacteristic
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83—21no—
52C2×C42×C83—21no—
53C2×C42×C83—21no—
54C2×C42×C83C2311yescharacteristic
55#107×C83—22no—
56#106×C83—22no—
57C4⋊C8×C83—22no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83C2×C412yes—
64C4⋊C8×C83D411yescharacteristic
65C4⋊C8×C83D411yescharacteristic
66#104×C83—42no—
67#103×C83—22no—
68#103×C83—44no—
69#102×C83—22no—
70C22⋊C8×C83—22no—
71C22⋊C8×C83—21no—
72#97×C83—22no—
73C4×C8×C83—21no—
74#95×C83—21no—
75#41×C83—21no—
76#41×C83—21no—
77#41×C83—21no—
78#41×C83—21no—
79C2×Q8×C83—22no—
80C2×Q8×C83—42no—
81C2×Q8×C83—42no—
82C2×Q8×C83—41no—
83C2×Q8×C83C22⋊C411yescharacteristic
84C2×D4×C83—21no—
85C2×D4×C83C22⋊C411yescharacteristic
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—41no—
90C22×C4×C83—41no—
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83#4111yescharacteristic
94M16×C83—41no—
95M16×C83—41no—
96M16×C83—41no—
97M16×C83—41no—
98M16×C83—82no—
99C2×C8×C83—22no—
100C2×C8×C83—21no—
101C2×C8×C83—21no—
102C2×C8×C83—22no—
103C2×C8×C83—42no—
104C2×C8×C83—41no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—22no—
108C4⋊C4×C83—22no—
109C4⋊C4×C83—22no—
110C4⋊C4×C83—41no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C4⋊C4×C83C22⋊C411yescharacteristic
114C4⋊C4×C83C22⋊C411yescharacteristic
115C22⋊C4×C83—21no—
116C22⋊C4×C83—21no—
117C22⋊C4×C83—21no—
118C42×C83—21no—
119C42×C83—21no—
120C42×C83—21no—
121C42×C83—41no—
122C42×C83—41no—
123C42×C83C22×C411yescharacteristic
124C42×C83C2×D411yescharacteristic
125C42×C83C2×D411yescharacteristic
126C42×C83#4111yescharacteristic
127C23×C83—21no—
128C23×C83#12111yescharacteristic
129Q8×C83—21no—
130Q8×C83—21no—
131Q8×C83—44no—
132Q8×C83—81no—
133Q8×C83—81no—
134D4×C83—21no—
135D4×C83—21no—
136D4×C83—41no—
137D4×C83—41no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—42no—
149C2×C4×C83—42no—
150C2×C4×C83—81no—
151C2×C4×C83—81no—
152C2×C4×C83—82no—
153C2×C4×C83—81no—
154C2×C4×C83C2×C22⋊C411yescharacteristic
155C2×C4×C83C4×D411yescharacteristic
156C2×C4×C83C4×D411yescharacteristic
157C2×C4×C83#12011yescharacteristic
158C2×C4×C83#12111yescharacteristic
159C2×C4×C83#12311yescharacteristic
160C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C83#13711yescharacteristic
162C2×C4×C83#13711yescharacteristic
163C2×C4×C83#13711yescharacteristic
164C8×C83—42no—
165C8×C83—42no—
166C8×C83—41no—
167C8×C83—41no—
168C8×C83—81no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83—41no—
174C22×C83#38611yescharacteristic
175C22×C83#46811yescharacteristic
176C22×C83#47011yescharacteristic
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—21no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83—82no—
184C4×C83—82no—
185C4×C83#41711yescharacteristic
186C4×C83#41911yescharacteristic
187C4×C83#45311yescharacteristic
188C4×C83#45411yescharacteristic
189C4×C83#46811yescharacteristic
190C4×C83#47011yescharacteristic
191#9321C8311yesmaximal, characteristic, direct product factor, Sylow
192C2×C83—21no—
193C2×C83—21no—
194C2×C83—81no—
195C2×C83#188311yescharacteristic
196C2×C83#189411yescharacteristic
197C2×C83#190211yescharacteristic
198#3079C2×C8311yescharacteristic
199#2989C2×C8311yescharacteristic
200#1981C2×C8311yescharacteristic, complemented
201#1806C2×C8311yescharacteristic
202#1549C2×C8311yescharacteristic, complemented
203#1303C2×C8311yescharacteristic, complemented
204#1292C2×C8311yescharacteristic, complemented
205C83#932111yescharacteristic, direct product factor, Sylow
206C2×C4⋊Q8C22×C8311yescharacteristic
207C4×#41C22×C8311yescharacteristic
208#477C4×C8312yescomplemented
209#476C4×C8312yescomplemented
210#475—21no—
211#474—21no—
212#446—21no—
213#436—21no—
214#435—22no—
215#432—21no—
216#431C22×C8311yescharacteristic
217#428—21no—
218#423—21no—
219#423C22×C8311yescharacteristic
220#423C22×C8311yescharacteristic
221C2×C4⋊C8C22×C8311yescharacteristic
222C2×#104—21no—
223#417—21no—
224C2×#103—22no—
225C4×C4⋊C4C22×C8311yescharacteristic
226#339—21no—
227#332—21no—
228#332—21no—
229C2×#41D4×C8311yescharacteristic
230C22×Q8—21no—
231C2×M16—21no—
232C2×M16—22no—
233C2×M16—21no—
234C22×C8D4×C8312yes—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C2×C4×C8312yes—
239C4×Q8C2×C4×C8311yescharacteristic
240C4×Q8D4×C8311yescharacteristic
241C4×D4—21no—
242C4×D4—21no—
243C4×D4C2×C4×C8311yescharacteristic
244C4×D4D4×C8311yescharacteristic
245#117—21no—
246#117D4×C8311yescharacteristic
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42C23×C8311yescharacteristic
253#107—22no—
254#106—22no—
255C4⋊C8—22no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8C2×C4×C8312yes—
262C4⋊C8D4×C8311yescharacteristic
263C4⋊C8D4×C8311yescharacteristic
264#104—42no—
265#103—22no—
266#103—44no—
267#102—22no—
268C22⋊C8—22no—
269C22⋊C8—21no—
270#97—22no—
271C4×C8—21no—
272#95—21no—
273#41—21no—
274#41—21no—
275#41—21no—
276#41—21no—
277C2×Q8—22no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8C22⋊C4×C8311yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊C4×C8311yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×D4×C8311yescharacteristic
290C22×C4C2×D4×C8311yescharacteristic
291C22×C4#41×C8311yescharacteristic
292M16—41no—
293M16—41no—
294M16—41no—
295M16—41no—
296M16—82no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊C4×C8311yescharacteristic
312C4⋊C4C22⋊C4×C8311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—41no—
321C42C22×C4×C8311yescharacteristic
322C42C2×D4×C8311yescharacteristic
323C42C2×D4×C8311yescharacteristic
324C42#41×C8311yescharacteristic
325C23—21no—
326C23#121×C8311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—44no—
330Q8—81no—
331Q8—81no—
332D4—21no—
333D4—21no—
334D4—41no—
335D4—41no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C2×C22⋊C4×C8311yescharacteristic
353C2×C4C4×D4×C8311yescharacteristic
354C2×C4C4×D4×C8311yescharacteristic
355C2×C4#120×C8311yescharacteristic
356C2×C4#121×C8311yescharacteristic
357C2×C4#123×C8311yescharacteristic
358C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
359C2×C4#137×C8311yescharacteristic
360C2×C4#137×C8311yescharacteristic
361C2×C4#137×C8311yescharacteristic
362C8—42no—
363C8—42no—
364C8—41no—
365C8—41no—
366C8—81no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22#386×C8311yescharacteristic
373C22#468×C8311yescharacteristic
374C22#470×C8311yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—82no—
382C4—82no—
383C4#417×C8311yescharacteristic
384C4#419×C8311yescharacteristic
385C4#453×C8311yescharacteristic
386C4#454×C8311yescharacteristic
387C4#468×C8311yescharacteristic
388C4#470×C8311yescharacteristic
389C2—21no—
390C2—21no—
391C2—81no—
392C2C83×#188311yescharacteristic
393C2C83×#189411yescharacteristic
394C2C83×#190211yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#41×C83D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1981C2semidirect
5. C83×#1549C2semidirect
6. C83×#1303C2semidirect
7. C83×#1292C2semidirect
10. #477×C83C4semidirect
11. #476×C83C4semidirect
191. #9321C83direct (N × H)
200. #1981C2×C83semidirect
202. #1549C2×C83semidirect
203. #1303C2×C83semidirect
204. #1292C2×C83semidirect
205. C83#9321direct (N × H)
208. #477C4×C83semidirect
209. #476C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#30792
C83×#29892
C83×#19812
C83×#18062
C83×#15492
C83×#13032
C83×#12922
#932183