Finite Group Databasebeta

← C83×#9186 #35237010

Subgroups — C83×#9186 #35237010

Order 21248.

580 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2975C211yesmaximal, characteristic, complemented
2C83×#2635C211yesmaximal, characteristic
3C4×#99×C83C211yesmaximal, characteristic, complemented
4C83×#1507C212yesmaximal, complemented
5C83×#1284C212yesmaximal, complemented
6C2×#121×C83C2211yescharacteristic
7C2×C4×D4×C83C2211yescharacteristic
8#436×C83—24no—
9#431×C83—21no—
10C2×#99×C83—21no—
11#407×C83—21no—
12#407×C83C2212yes—
13#400×C83C411yescharacteristic, complemented
14#399×C83C411yescharacteristic, complemented
15#394×C83—21no—
16#394×C83C411yescharacteristic, complemented
17#392×C83—21no—
18#392×C83C411yescharacteristic, complemented
19#388×C83—21no—
20#386×C83—21no—
21#381×C83C2211yescharacteristic
22C4×C22⋊C4×C83C2212yes—
23#342×C83—21no—
24#327×C83—24no—
25C22×D4×C83—21no—
26C22×D4×C83—21no—
27C22×D4×C83C2×C411yescharacteristic
28C22×D4×C83D411yescharacteristic
29C23×C4×C83D411yescharacteristic
30C23×C4×C83C2311yescharacteristic
31C2×M16×C83—22no—
32C2×M16×C83—22no—
33#121×C83—22no—
34#121×C83—22no—
35C4×D4×C83—22no—
36C4×D4×C83—22no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83C2×C411yescharacteristic
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83—22no—
50C2×C42×C83D411yescharacteristic
51C4⋊C8×C83—22no—
52#99×C83—24no—
53C22⋊C8×C83—22no—
54C22⋊C8×C83—22no—
55C22⋊C8×C83D414yescomplemented
56C4×C8×C83—22no—
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60#95×C83—22no—
61#95×C83C2×C411yescharacteristic
62#95×C83C2×C411yescharacteristic
63C24×C83—21no—
64C24×C83C22⋊C411yescharacteristic
65C24×C83C2×D411yescharacteristic
66C2×D4×C83—22no—
67C2×D4×C83—22no—
68C2×D4×C83—22no—
69C2×D4×C83—41no—
70C2×D4×C83—42no—
71C2×D4×C83—42no—
72C2×D4×C83—81no—
73C2×D4×C83—81no—
74C2×D4×C83—81no—
75C2×D4×C83—81no—
76C2×D4×C83—81no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—42no—
83C22×C4×C83—81no—
84C22×C4×C83—81no—
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D412yes—
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83#4112yes—
92M16×C83—82no—
93M16×C83—82no—
94C2×C8×C83—24no—
95C4⋊C4×C83—22no—
96C4⋊C4×C83—22no—
97C4⋊C4×C83—81no—
98C22⋊C4×C83—22no—
99C22⋊C4×C83—22no—
100C22⋊C4×C83—24no—
101C22⋊C4×C83—42no—
102C22⋊C4×C83—42no—
103C22⋊C4×C83—41no—
104C22⋊C4×C83—41no—
105C22⋊C4×C83—81no—
106C22⋊C4×C83—81no—
107C22⋊C4×C83—81no—
108C22⋊C4×C83—81no—
109C42×C83—21no—
110C42×C83—21no—
111C42×C83—82no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83—41no—
120C23×C83—41no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—41no—
124C23×C83—41no—
125C23×C83—41no—
126C23×C83—41no—
127C23×C83—41no—
128C23×C83C2×C22⋊C411yescharacteristic
129C23×C83#12112yes—
130D4×C83—44no—
131D4×C83—82no—
132D4×C83—84no—
133D4×C83—84no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—42no—
139C2×C4×C83—42no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—42no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—44no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—41no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—41no—
154C2×C4×C83—41no—
155C2×C4×C83—81no—
156C2×C4×C83—82no—
157C2×C4×C83—82no—
158C2×C4×C83#10412yes—
159C2×C4×C83C4×D412yes—
160C2×C4×C83#12011yescharacteristic
161C2×C4×C83#12311yescharacteristic
162C8×C83—44no—
163C22×C83—21no—
164C22×C83—21no—
165C22×C83—22no—
166C22×C83—22no—
167C22×C83—22no—
168C22×C83—42no—
169C22×C83—41no—
170C22×C83—41no—
171C22×C83—41no—
172C22×C83—81no—
173C22×C83—81no—
174C22×C83—81no—
175C22×C83—81no—
176C22×C83—81no—
177C22×C83—81no—
178C22×C83—81no—
179C22×C83—81no—
180C22×C83—81no—
181C22×C83—81no—
182C22×C83#38611yescharacteristic
183C22×C83C2×#10411yescharacteristic
184C22×C83#42111yescharacteristic
185C4×C83—22no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—22no—
189C4×C83—41no—
190C4×C83—41no—
191C4×C83—81no—
192C4×C83—81no—
193C4×C83—84no—
194C4×C83—81no—
195#9186C8311yesmaximal, characteristic, direct product factor, Sylow
196C2×C83—21no—
197C2×C83—21no—
198C2×C83—42no—
199C2×C83—81no—
200C2×C83—161no—
201C2×C83#187111yescharacteristic
202C2×C83#191011yescharacteristic
203C2×C83#191311yescharacteristic
204#2975C2×C8311yescharacteristic, complemented
205#2635C2×C8311yescharacteristic
206C4×#99C2×C8311yescharacteristic, complemented
207#1507C2×C8312yescomplemented
208#1284C2×C8312yescomplemented
209C83#918611yescharacteristic, direct product factor, Sylow
210C2×#121C22×C8311yescharacteristic
211C2×C4×D4C22×C8311yescharacteristic
212#436—24no—
213#431—21no—
214C2×#99—21no—
215#407—21no—
216#407C22×C8312yes—
217#400C4×C8311yescharacteristic, complemented
218#399C4×C8311yescharacteristic, complemented
219#394—21no—
220#394C4×C8311yescharacteristic, complemented
221#392—21no—
222#392C4×C8311yescharacteristic, complemented
223#388—21no—
224#386—21no—
225#381C22×C8311yescharacteristic
226C4×C22⋊C4C22×C8312yes—
227#342—21no—
228#327—24no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×C4×C8311yescharacteristic
232C22×D4D4×C8311yescharacteristic
233C23×C4D4×C8311yescharacteristic
234C23×C4C23×C8311yescharacteristic
235C2×M16—22no—
236C2×M16—22no—
237#121—22no—
238#121—22no—
239C4×D4—22no—
240C4×D4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×C4×C8311yescharacteristic
243C2×C4⋊C4C2×C4×C8311yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4C2×C4×C8311yescharacteristic
252C2×C22⋊C4D4×C8311yescharacteristic
253C2×C42—22no—
254C2×C42D4×C8311yescharacteristic
255C4⋊C8—22no—
256#99—24no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8D4×C8314yescomplemented
260C4×C8—22no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95C2×C4×C8311yescharacteristic
266#95C2×C4×C8311yescharacteristic
267C24—21no—
268C24C22⋊C4×C8311yescharacteristic
269C24C2×D4×C8311yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C22⋊C4×C8311yescharacteristic
290C22×C4C22⋊C4×C8311yescharacteristic
291C22×C4C22⋊C4×C8311yescharacteristic
292C22×C4C22×C4×C8311yescharacteristic
293C22×C4C2×D4×C8312yes—
294C22×C4C2×D4×C8311yescharacteristic
295C22×C4#41×C8312yes—
296M16—82no—
297M16—82no—
298C2×C8—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—81no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C42—21no—
314C42—21no—
315C42—82no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23C2×C22⋊C4×C8311yescharacteristic
333C23#121×C8312yes—
334D4—44no—
335D4—82no—
336D4—84no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#104×C8312yes—
363C2×C4C4×D4×C8312yes—
364C2×C4#120×C8311yescharacteristic
365C2×C4#123×C8311yescharacteristic
366C8—44no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#386×C8311yescharacteristic
387C22C2×#104×C8311yescharacteristic
388C22#421×C8311yescharacteristic
389C4—22no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—84no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—81no—
403C2—161no—
404C2C83×#187111yescharacteristic
405C2C83×#191011yescharacteristic
406C2C83×#191311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC83×#2635C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#2975C2semidirect
3. C4×#99×C83C2semidirect
4. C83×#1507C2semidirect
5. C83×#1284C2semidirect
13. #400×C83C4semidirect
14. #399×C83C4semidirect
16. #394×C83C4semidirect
18. #392×C83C4semidirect
55. C22⋊C8×C83D4semidirect
195. #9186C83direct (N × H)
204. #2975C2×C83semidirect
206. C4×#99C2×C83semidirect
207. #1507C2×C83semidirect
208. #1284C2×C83semidirect
209. C83#9186direct (N × H)
217. #400C4×C83semidirect
218. #399C4×C83semidirect
220. #394C4×C83semidirect
222. #392C4×C83semidirect
259. C22⋊C8D4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#29752
C83×#26352
C4×#99×C832
C83×#15072
C83×#12842
#918683