Finite Group Databasebeta

← C83×#9313 #35237137

Subgroups — C83×#9313 #35237137

Order 21248.

428 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3115C211yesmaximal, characteristic
2C83×#2923C211yesmaximal, characteristic, complemented
3C83×#1530C211yesmaximal, characteristic, complemented
4C83×#1488C211yesmaximal, characteristic, complemented
5C83×#1484C211yesmaximal, characteristic, complemented
6C83×#1302C211yesmaximal, characteristic, complemented
7C83×#1291C211yesmaximal, characteristic, complemented
8#532×C83C2211yescharacteristic
9C4×#41×C83C2211yescharacteristic
10#482×C83—21no—
11#475×C83C411yescharacteristic
12#474×C83C411yescharacteristic
13#460×C83C411yescharacteristic
14#459×C83C411yescharacteristic
15#449×C83—21no—
16#442×C83—21no—
17#440×C83—21no—
18#433×C83—22no—
19#432×C83—22no—
20#432×C83C2211yescharacteristic
21#432×C83C2211yescharacteristic
22#431×C83C2211yescharacteristic
23#424×C83—21no—
24#408×C83—21no—
25C4×M16×C83C2211yescharacteristic
26C2×#97×C83C2211yescharacteristic
27#329×C83—22no—
28Q8⋊C8×C83—21no—
29D4⋊C8×C83—21no—
30D4⋊C8×C83—22no—
31C2×#41×C83—21no—
32C2×#41×C83D411yescharacteristic
33C2×SD16×C83—21no—
34C2×D8×C83—21no—
35C2×M16×C83—21no—
36C22×C8×C83—21no—
37C4⋊Q8×C83C2×C411yescharacteristic
38#127×C83C2×C411yescharacteristic
39#124×C83—21no—
40#121×C83—21no—
41#121×C83—21no—
42C4×Q8×C83C2×C411yescharacteristic
43C4×Q8×C83D411yescharacteristic
44C4×D4×C83—21no—
45C4×D4×C83—21no—
46C4×D4×C83—21no—
47C4×D4×C83—21no—
48C4×D4×C83C2×C411yescharacteristic
49C4×D4×C83D411yescharacteristic
50#117×C83—21no—
51#117×C83D411yescharacteristic
52C2×C42×C83—21no—
53C2×C42×C83C2311yescharacteristic
54#107×C83—21no—
55#106×C83—21no—
56C4⋊C8×C83—24no—
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83C2×C411yescharacteristic
60C4⋊C8×C83C2×C411yescharacteristic
61#103×C83—21no—
62#102×C83—21no—
63#102×C83—21no—
64#102×C83—21no—
65C22⋊C8×C83—24no—
66C22⋊C8×C83—21no—
67C22⋊C8×C83—21no—
68C22⋊C8×C83—21no—
69#97×C83—21no—
70#97×C83—21no—
71#97×C83—21no—
72#97×C83—21no—
73#97×C83D412yes—
74C4×C8×C83—21no—
75C4×C8×C83D412yes—
76#41×C83—41no—
77#41×C83—41no—
78#41×C83—82no—
79C2×Q8×C83—21no—
80C2×Q8×C83C22⋊C411yescharacteristic
81C2×D4×C83—21no—
82C2×D4×C83—21no—
83C2×D4×C83—21no—
84C2×D4×C83—21no—
85C2×D4×C83—41no—
86C2×D4×C83C22⋊C411yescharacteristic
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83—41no—
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83#4111yescharacteristic
93C22×C4×C83#4111yescharacteristic
94SD16×C83—42no—
95D8×C83—42no—
96M16×C83—41no—
97M16×C83—41no—
98C2×C8×C83—21no—
99C2×C8×C83—21no—
100C2×C8×C83—22no—
101C2×C8×C83—22no—
102C2×C8×C83—41no—
103C2×C8×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—21no—
108C4⋊C4×C83—21no—
109C4⋊C4×C83—21no—
110C4⋊C4×C83C22⋊C411yescharacteristic
111C4⋊C4×C83C22⋊C411yescharacteristic
112C22⋊C4×C83—21no—
113C22⋊C4×C83—21no—
114C22⋊C4×C83—21no—
115C22⋊C4×C83—41no—
116C22⋊C4×C83—41no—
117C42×C83—21no—
118C42×C83—21no—
119C42×C83—21no—
120C42×C83C22×C411yescharacteristic
121C42×C83C2×D411yescharacteristic
122C42×C83C2×D411yescharacteristic
123C42×C83C2×D411yescharacteristic
124C23×C83—21no—
125C23×C83—41no—
126C23×C83#12311yescharacteristic
127Q8×C83—21no—
128Q8×C83—21no—
129Q8×C83—81no—
130D4×C83—21no—
131D4×C83—21no—
132D4×C83—42no—
133D4×C83—42no—
134D4×C83—41no—
135D4×C83—41no—
136D4×C83—81no—
137D4×C83—82no—
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—42no—
146C2×C4×C83—41no—
147C2×C4×C83—81no—
148C2×C4×C83—81no—
149C2×C4×C83—82no—
150C2×C4×C83C2×C22⋊C411yescharacteristic
151C2×C4×C83C4×D411yescharacteristic
152C2×C4×C83C4×D411yescharacteristic
153C2×C4×C83#12011yescharacteristic
154C2×C4×C83#12111yescharacteristic
155C2×C4×C83#12111yescharacteristic
156C8×C83—42no—
157C8×C83—42no—
158C8×C83—41no—
159C8×C83—41no—
160C8×C83—81no—
161C8×C83—81no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—21no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—81no—
169C22×C83—81no—
170C22×C83—81no—
171C22×C83#38611yescharacteristic
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83—81no—
181C4×C83#42111yescharacteristic
182C4×C83#42111yescharacteristic
183#9313C8311yesmaximal, characteristic, direct product factor, Sylow
184C2×C83—41no—
185C2×C83—81no—
186C2×C83—161no—
187C2×C83#188111yescharacteristic
188C2×C83#189711yescharacteristic
189C2×C83#190311yescharacteristic
190#3115C2×C8311yescharacteristic
191#2923C2×C8311yescharacteristic, complemented
192#1530C2×C8311yescharacteristic, complemented
193#1488C2×C8311yescharacteristic, complemented
194#1484C2×C8311yescharacteristic, complemented
195#1302C2×C8311yescharacteristic, complemented
196#1291C2×C8311yescharacteristic, complemented
197C83#931311yescharacteristic, direct product factor, Sylow
198#532C22×C8311yescharacteristic
199C4×#41C22×C8311yescharacteristic
200#482—21no—
201#475C4×C8311yescharacteristic
202#474C4×C8311yescharacteristic
203#460C4×C8311yescharacteristic
204#459C4×C8311yescharacteristic
205#449—21no—
206#442—21no—
207#440—21no—
208#433—22no—
209#432—22no—
210#432C22×C8311yescharacteristic
211#432C22×C8311yescharacteristic
212#431C22×C8311yescharacteristic
213#424—21no—
214#408—21no—
215C4×M16C22×C8311yescharacteristic
216C2×#97C22×C8311yescharacteristic
217#329—22no—
218Q8⋊C8—21no—
219D4⋊C8—21no—
220D4⋊C8—22no—
221C2×#41—21no—
222C2×#41D4×C8311yescharacteristic
223C2×SD16—21no—
224C2×D8—21no—
225C2×M16—21no—
226C22×C8—21no—
227C4⋊Q8C2×C4×C8311yescharacteristic
228#127C2×C4×C8311yescharacteristic
229#124—21no—
230#121—21no—
231#121—21no—
232C4×Q8C2×C4×C8311yescharacteristic
233C4×Q8D4×C8311yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C2×C4×C8311yescharacteristic
239C4×D4D4×C8311yescharacteristic
240#117—21no—
241#117D4×C8311yescharacteristic
242C2×C42—21no—
243C2×C42C23×C8311yescharacteristic
244#107—21no—
245#106—21no—
246C4⋊C8—24no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8C2×C4×C8311yescharacteristic
250C4⋊C8C2×C4×C8311yescharacteristic
251#103—21no—
252#102—21no—
253#102—21no—
254#102—21no—
255C22⋊C8—24no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259#97—21no—
260#97—21no—
261#97—21no—
262#97—21no—
263#97D4×C8312yes—
264C4×C8—21no—
265C4×C8D4×C8312yes—
266#41—41no—
267#41—41no—
268#41—82no—
269C2×Q8—21no—
270C2×Q8C22⋊C4×C8311yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4C22⋊C4×C8311yescharacteristic
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4C2×D4×C8311yescharacteristic
282C22×C4#41×C8311yescharacteristic
283C22×C4#41×C8311yescharacteristic
284SD16—42no—
285D8—42no—
286M16—41no—
287M16—41no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4C22⋊C4×C8311yescharacteristic
301C4⋊C4C22⋊C4×C8311yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42C22×C4×C8311yescharacteristic
311C42C2×D4×C8311yescharacteristic
312C42C2×D4×C8311yescharacteristic
313C42C2×D4×C8311yescharacteristic
314C23—21no—
315C23—41no—
316C23#123×C8311yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—81no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—41no—
325D4—41no—
326D4—81no—
327D4—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4C2×C22⋊C4×C8311yescharacteristic
341C2×C4C4×D4×C8311yescharacteristic
342C2×C4C4×D4×C8311yescharacteristic
343C2×C4#120×C8311yescharacteristic
344C2×C4#121×C8311yescharacteristic
345C2×C4#121×C8311yescharacteristic
346C8—42no—
347C8—42no—
348C8—41no—
349C8—41no—
350C8—81no—
351C8—81no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#386×C8311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4#421×C8311yescharacteristic
372C4#421×C8311yescharacteristic
373C2—41no—
374C2—81no—
375C2—161no—
376C2C83×#188111yescharacteristic
377C2C83×#189711yescharacteristic
378C2C83×#190311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC83×#3115C2
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
2. C83×#2923C2semidirect
3. C83×#1530C2semidirect
4. C83×#1488C2semidirect
5. C83×#1484C2semidirect
6. C83×#1302C2semidirect
7. C83×#1291C2semidirect
183. #9313C83direct (N × H)
191. #2923C2×C83semidirect
192. #1530C2×C83semidirect
193. #1488C2×C83semidirect
194. #1484C2×C83semidirect
195. #1302C2×C83semidirect
196. #1291C2×C83semidirect
197. C83#9313direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#31152
C83×#29232
C83×#15302
C83×#14882
C83×#14842
C83×#13022
C83×#12912
#931383