Finite Group Databasebeta

← C83×#9312 #35237136

Subgroups — C83×#9312 #35237136

Order 21248.

492 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3116C211yesmaximal, characteristic
2C83×#2875C211yesmaximal, characteristic
3C83×#1531C211yesmaximal, characteristic, complemented
4C83×#1488C211yesmaximal, characteristic
5C83×#1485C211yesmaximal, characteristic, complemented
6C83×#1302C211yesmaximal, characteristic, complemented
7C83×#1291C211yesmaximal, characteristic, complemented
8C2×#131×C83—21no—
9#533×C83C2211yescharacteristic
10C4×#41×C83C2211yescharacteristic
11#485×C83—21no—
12#477×C83C411yescharacteristic
13#476×C83C411yescharacteristic
14#462×C83C411yescharacteristic
15#461×C83C411yescharacteristic
16#452×C83—21no—
17#441×C83—21no—
18#440×C83—21no—
19#432×C83C2211yescharacteristic
20#432×C83C2211yescharacteristic
21#431×C83C2211yescharacteristic
22#405×C83—21no—
23#405×C83—22no—
24C4×M16×C83—21no—
25C4×M16×C83C2211yescharacteristic
26C2×#97×C83—21no—
27C2×#97×C83C2211yescharacteristic
28#330×C83—21no—
29#330×C83—21no—
30Q8⋊C8×C83—21no—
31Q8⋊C8×C83—22no—
32D4⋊C8×C83—21no—
33C2×#41×C83D411yescharacteristic
34C2×Q16×C83—21no—
35C2×SD16×C83—21no—
36#131×C83—24no—
37#131×C83—24no—
38C2×M16×C83—22no—
39C2×M16×C83—21no—
40C22×C8×C83—22no—
41C22×C8×C83—21no—
42C4⋊Q8×C83C2×C411yescharacteristic
43C4⋊Q8×C83C2×C411yescharacteristic
44#125×C83—21no—
45C22⋊Q8×C83—21no—
46C22⋊Q8×C83—21no—
47C4×Q8×C83—21no—
48C4×Q8×C83—21no—
49C4×Q8×C83C2×C411yescharacteristic
50C4×Q8×C83D411yescharacteristic
51C4×D4×C83—21no—
52C4×D4×C83—21no—
53C4×D4×C83C2×C411yescharacteristic
54C4×D4×C83D411yescharacteristic
55#117×C83—21no—
56#117×C83—21no—
57#117×C83D411yescharacteristic
58C2×C42×C83—21no—
59C2×C42×C83C2311yescharacteristic
60#107×C83—21no—
61#106×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83C2×C411yescharacteristic
65C4⋊C8×C83C2×C411yescharacteristic
66#103×C83—21no—
67#103×C83—21no—
68#103×C83—21no—
69#102×C83—21no—
70C22⋊C8×C83—21no—
71C22⋊C8×C83—21no—
72C22⋊C8×C83—21no—
73#97×C83—22no—
74#97×C83—21no—
75#97×C83—21no—
76#97×C83—21no—
77#97×C83—21no—
78#97×C83D411yescharacteristic
79#97×C83D411yescharacteristic
80C4×C8×C83—22no—
81C4×C8×C83—21no—
82C4×C8×C83D412yes—
83#41×C83—22no—
84#41×C83—22no—
85C2×Q8×C83—21no—
86C2×Q8×C83—21no—
87C2×Q8×C83C22⋊C411yescharacteristic
88C2×D4×C83—21no—
89C2×D4×C83C22⋊C411yescharacteristic
90C22×C4×C83—21no—
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83#4111yescharacteristic
95C22×C4×C83#4111yescharacteristic
96Q16×C83—42no—
97SD16×C83—42no—
98M16×C83—28no—
99M16×C83—22no—
100M16×C83—22no—
101C2×C8×C83—28no—
102C2×C8×C83—22no—
103C2×C8×C83—22no—
104C2×C8×C83—21no—
105C2×C8×C83—22no—
106C2×C8×C83—21no—
107C2×C8×C83—21no—
108C2×C8×C83—21no—
109C2×C8×C83—41no—
110C2×C8×C83—41no—
111C4⋊C4×C83—21no—
112C4⋊C4×C83—21no—
113C4⋊C4×C83—21no—
114C4⋊C4×C83—21no—
115C4⋊C4×C83—21no—
116C4⋊C4×C83—21no—
117C4⋊C4×C83—41no—
118C4⋊C4×C83—41no—
119C4⋊C4×C83C22⋊C411yescharacteristic
120C4⋊C4×C83C22⋊C411yescharacteristic
121C22⋊C4×C83—21no—
122C22⋊C4×C83—21no—
123C22⋊C4×C83—21no—
124C22⋊C4×C83—41no—
125C42×C83—21no—
126C42×C83—21no—
127C42×C83—21no—
128C42×C83—41no—
129C42×C83C22×C411yescharacteristic
130C42×C83C2×D411yescharacteristic
131C42×C83C2×D411yescharacteristic
132C42×C83C2×D411yescharacteristic
133C23×C83—21no—
134C23×C83#12311yescharacteristic
135Q8×C83—21no—
136Q8×C83—42no—
137Q8×C83—41no—
138Q8×C83—41no—
139Q8×C83#10412yes—
140D4×C83—24no—
141D4×C83—21no—
142D4×C83#10412yes—
143C2×C4×C83—24no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—81no—
154C2×C4×C83—81no—
155C2×C4×C83C2×C22⋊C411yescharacteristic
156C2×C4×C83C4×D411yescharacteristic
157C2×C4×C83C4×D411yescharacteristic
158C2×C4×C83#12011yescharacteristic
159C2×C4×C83#12111yescharacteristic
160C2×C4×C83#12111yescharacteristic
161C8×C83—22no—
162C8×C83—22no—
163C8×C83—24no—
164C8×C83—41no—
165C8×C83—41no—
166C8×C83—81no—
167C8×C83—81no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—22no—
171C22×C83—21no—
172C22×C83—41no—
173C22×C83—41no—
174C22×C83#38611yescharacteristic
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—22no—
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83—81no—
184C4×C83—81no—
185C4×C83C2×#10411yescharacteristic
186C4×C83C2×#10411yescharacteristic
187#9312C8311yesmaximal, characteristic, direct product factor, Sylow
188C2×C83—41no—
189C2×C83—42no—
190C2×C83#188111yescharacteristic
191C2×C83#189611yescharacteristic
192C2×C83#190211yescharacteristic
193#3116C2×C8311yescharacteristic
194#2875C2×C8311yescharacteristic
195#1531C2×C8311yescharacteristic, complemented
196#1488C2×C8311yescharacteristic
197#1485C2×C8311yescharacteristic, complemented
198#1302C2×C8311yescharacteristic, complemented
199#1291C2×C8311yescharacteristic, complemented
200C83#931211yescharacteristic, direct product factor, Sylow
201C2×#131—21no—
202#533C22×C8311yescharacteristic
203C4×#41C22×C8311yescharacteristic
204#485—21no—
205#477C4×C8311yescharacteristic
206#476C4×C8311yescharacteristic
207#462C4×C8311yescharacteristic
208#461C4×C8311yescharacteristic
209#452—21no—
210#441—21no—
211#440—21no—
212#432C22×C8311yescharacteristic
213#432C22×C8311yescharacteristic
214#431C22×C8311yescharacteristic
215#405—21no—
216#405—22no—
217C4×M16—21no—
218C4×M16C22×C8311yescharacteristic
219C2×#97—21no—
220C2×#97C22×C8311yescharacteristic
221#330—21no—
222#330—21no—
223Q8⋊C8—21no—
224Q8⋊C8—22no—
225D4⋊C8—21no—
226C2×#41D4×C8311yescharacteristic
227C2×Q16—21no—
228C2×SD16—21no—
229#131—24no—
230#131—24no—
231C2×M16—22no—
232C2×M16—21no—
233C22×C8—22no—
234C22×C8—21no—
235C4⋊Q8C2×C4×C8311yescharacteristic
236C4⋊Q8C2×C4×C8311yescharacteristic
237#125—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8C2×C4×C8311yescharacteristic
243C4×Q8D4×C8311yescharacteristic
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C2×C4×C8311yescharacteristic
247C4×D4D4×C8311yescharacteristic
248#117—21no—
249#117—21no—
250#117D4×C8311yescharacteristic
251C2×C42—21no—
252C2×C42C23×C8311yescharacteristic
253#107—21no—
254#106—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8C2×C4×C8311yescharacteristic
258C4⋊C8C2×C4×C8311yescharacteristic
259#103—21no—
260#103—21no—
261#103—21no—
262#102—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266#97—22no—
267#97—21no—
268#97—21no—
269#97—21no—
270#97—21no—
271#97D4×C8311yescharacteristic
272#97D4×C8311yescharacteristic
273C4×C8—22no—
274C4×C8—21no—
275C4×C8D4×C8312yes—
276#41—22no—
277#41—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8C22⋊C4×C8311yescharacteristic
281C2×D4—21no—
282C2×D4C22⋊C4×C8311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4C2×D4×C8311yescharacteristic
287C22×C4#41×C8311yescharacteristic
288C22×C4#41×C8311yescharacteristic
289Q16—42no—
290SD16—42no—
291M16—28no—
292M16—22no—
293M16—22no—
294C2×C8—28no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—41no—
303C2×C8—41no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4C22⋊C4×C8311yescharacteristic
313C4⋊C4C22⋊C4×C8311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—41no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—41no—
322C42C22×C4×C8311yescharacteristic
323C42C2×D4×C8311yescharacteristic
324C42C2×D4×C8311yescharacteristic
325C42C2×D4×C8311yescharacteristic
326C23—21no—
327C23#123×C8311yescharacteristic
328Q8—21no—
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8#104×C8312yes—
333D4—24no—
334D4—21no—
335D4#104×C8312yes—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4C2×C22⋊C4×C8311yescharacteristic
349C2×C4C4×D4×C8311yescharacteristic
350C2×C4C4×D4×C8311yescharacteristic
351C2×C4#120×C8311yescharacteristic
352C2×C4#121×C8311yescharacteristic
353C2×C4#121×C8311yescharacteristic
354C8—22no—
355C8—22no—
356C8—24no—
357C8—41no—
358C8—41no—
359C8—81no—
360C8—81no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22#386×C8311yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4C2×#104×C8311yescharacteristic
379C4C2×#104×C8311yescharacteristic
380C2—41no—
381C2—42no—
382C2C83×#188111yescharacteristic
383C2C83×#189611yescharacteristic
384C2C83×#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×#1531C2semidirect
5. C83×#1485C2semidirect
6. C83×#1302C2semidirect
7. C83×#1291C2semidirect
187. #9312C83direct (N × H)
195. #1531C2×C83semidirect
197. #1485C2×C83semidirect
198. #1302C2×C83semidirect
199. #1291C2×C83semidirect
200. C83#9312direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#31162
C83×#28752
C83×#15312
C83×#14882
C83×#14852
C83×#13022
C83×#12912
#931283