Finite Group Databasebeta

← C83×#19657 #35247481

Subgroups — C83×#19657 #35247481

Order 21248.

700 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3230C211yesmaximal, characteristic, complemented
2C83×#3229C211yesmaximal, characteristic, complemented
3C83×#3226C211yesmaximal, characteristic
4C83×#3206C211yesmaximal, characteristic
5C83×#3131C211yesmaximal, characteristic
6C2×#487×C83C211yesmaximal, characteristic, complemented
7C83×#1719C212yesmaximal, complemented
8C83×#1718C212yesmaximal, complemented
9C83×#1710C212yesmaximal, complemented
10C83×#1708C212yesmaximal, complemented
11C83×#1450C211yesmaximal, characteristic, complemented
12#533×C83C2211yescharacteristic
13#532×C83C2211yescharacteristic
14C2×C4⋊Q8×C83C2211yescharacteristic
15C2×C4⋊Q8×C83C2211yescharacteristic
16#500×C83C2212yes—
17#500×C83C2212yes—
18#498×C83C2212yes—
19#492×C83C2212yes—
20#488×C83—22no—
21#487×C83—22no—
22#487×C83C2212yes—
23#487×C83C2212yes—
24#487×C83C2211yescharacteristic
25#487×C83C2211yescharacteristic
26#487×C83C2212yes—
27#487×C83C2212yes—
28#487×C83C2212yes—
29#486×C83C2211yescharacteristic
30#486×C83C2211yescharacteristic
31#486×C83C2212yes—
32#484×C83—22no—
33#483×C83—22no—
34#480×C83—22no—
35#477×C83—22no—
36#475×C83—22no—
37#474×C83—22no—
38#461×C83—22no—
39#448×C83—22no—
40#431×C83C2211yescharacteristic
41#431×C83C2211yescharacteristic
42#416×C83—22no—
43C2×#103×C83—22no—
44C2×C4×C8×C83C2211yescharacteristic
45#322×C83C2212yes—
46C82×C83C2212yes—
47C2×#41×C83—21no—
48C22×Q8×C83—21no—
49C22×Q8×C83—21no—
50C2×SD16×C83—24no—
51C22×C8×C83D412yes—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83—21no—
54C4⋊Q8×C83—21no—
55C4⋊Q8×C83—21no—
56C4⋊Q8×C83—21no—
57C4⋊Q8×C83—21no—
58C4⋊Q8×C83C2311yescharacteristic
59C4⋊Q8×C83C2312yes—
60C4⋊Q8×C83C2311yescharacteristic
61C4⋊Q8×C83C2312yes—
62C4⋊Q8×C83C2311yescharacteristic
63#127×C83C2311yescharacteristic
64#125×C83—21no—
65#124×C83—21no—
66C22⋊Q8×C83—21no—
67C22⋊Q8×C83—21no—
68#121×C83—21no—
69#121×C83—21no—
70C4×Q8×C83—21no—
71C4×Q8×C83—21no—
72C4×D4×C83—21no—
73C4×D4×C83—21no—
74#117×C83—21no—
75C2×C4⋊C4×C83—21no—
76C2×C4⋊C4×C83—21no—
77C2×C4⋊C4×C83—21no—
78C2×C4⋊C4×C83—21no—
79C2×C42×C83C2311yescharacteristic
80#107×C83—24no—
81#107×C83—24no—
82#106×C83—24no—
83C4⋊C8×C83D412yes—
84C4⋊C8×C83D412yes—
85#103×C83—24no—
86#103×C83—24no—
87#103×C83—24no—
88#103×C83—24no—
89#103×C83—22no—
90#103×C83—22no—
91#103×C83—24no—
92#102×C83—24no—
93#102×C83—22no—
94#102×C83—22no—
95C22⋊C8×C83D412yes—
96C22⋊C8×C83D412yes—
97C4×C8×C83D412yes—
98C4×C8×C83C2311yescharacteristic
99C4×C8×C83C2311yescharacteristic
100C4×C8×C83C2312yes—
101C4×C8×C83C2312yes—
102#41×C83—44no—
103C2×Q8×C83—21no—
104C2×Q8×C83—21no—
105C2×Q8×C83—21no—
106C2×Q8×C83—22no—
107C2×Q8×C83—22no—
108C2×Q8×C83—41no—
109C2×Q8×C83—41no—
110C2×Q8×C83—44no—
111C2×Q8×C83—82no—
112C2×D4×C83—21no—
113C2×D4×C83—21no—
114C2×D4×C83—21no—
115C2×D4×C83—41no—
116C22×C4×C83—41no—
117C22×C4×C83—41no—
118C22×C4×C83—41no—
119C22×C4×C83C2×D411yescharacteristic
120C22×C4×C83C2×D411yescharacteristic
121C22×C4×C83C2×D411yescharacteristic
122SD16×C83—84no—
123C2×C8×C83—24no—
124C2×C8×C83C2×D412yes—
125C2×C8×C83C2×D412yes—
126C2×C8×C83C2×D414yes—
127C2×C8×C83C2×D414yes—
128C4⋊C4×C83—21no—
129C4⋊C4×C83—21no—
130C4⋊C4×C83—22no—
131C4⋊C4×C83—22no—
132C4⋊C4×C83—21no—
133C4⋊C4×C83—22no—
134C4⋊C4×C83—22no—
135C4⋊C4×C83—21no—
136C4⋊C4×C83—21no—
137C4⋊C4×C83—21no—
138C4⋊C4×C83—41no—
139C4⋊C4×C83—41no—
140C4⋊C4×C83—41no—
141C4⋊C4×C83—41no—
142C4⋊C4×C83—41no—
143C4⋊C4×C83—41no—
144C22⋊C4×C83—41no—
145C22⋊C4×C83—41no—
146C22⋊C4×C83—41no—
147C42×C83—41no—
148C42×C83C2×D411yescharacteristic
149C42×C83C2×D411yescharacteristic
150C42×C83C2×D411yescharacteristic
151C42×C83C2411yescharacteristic
152C23×C83—41no—
153C23×C83#12711yescharacteristic
154Q8×C83—44no—
155Q8×C83—42no—
156Q8×C83—44no—
157Q8×C83—42no—
158Q8×C83—42no—
159Q8×C83—82no—
160Q8×C83—82no—
161D4×C83—42no—
162D4×C83—42no—
163D4×C83—42no—
164D4×C83—82no—
165C2×C4×C83—22no—
166C2×C4×C83—22no—
167C2×C4×C83—41no—
168C2×C4×C83—41no—
169C2×C4×C83—41no—
170C2×C4×C83—42no—
171C2×C4×C83—42no—
172C2×C4×C83—82no—
173C2×C4×C83—82no—
174C2×C4×C83—82no—
175C2×C4×C83#12711yescharacteristic
176C2×C4×C83#12711yescharacteristic
177C2×C4×C83#12711yescharacteristic
178C2×C4×C83#13514yes—
179C2×C4×C83C22×D411yescharacteristic
180C2×C4×C83C22×D411yescharacteristic
181C2×C4×C83C22×D411yescharacteristic
182C8×C83—22no—
183C8×C83—22no—
184C8×C83—24no—
185C8×C83—24no—
186C22×C83—21no—
187C22×C83—21no—
188C22×C83—81no—
189C22×C83—81no—
190C22×C83—81no—
191C22×C83#49512yes—
192C22×C83C2×#12711yescharacteristic
193C4×C83—21no—
194C4×C83—21no—
195C4×C83—81no—
196C4×C83—82no—
197C4×C83—82no—
198C4×C83—81no—
199C4×C83—81no—
200C4×C83#49512yes—
201C4×C83C2×#13512yes—
202C4×C83#57712yes—
203C4×C83#57812yes—
204#19657C8311yesmaximal, characteristic, direct product factor, Sylow
205C2×C83—22no—
206C2×C83—161no—
207C2×C83C2×#49511yescharacteristic
208C2×C83#315411yescharacteristic
209C2×C83#315611yescharacteristic
210#3230C2×C8311yescharacteristic, complemented
211#3229C2×C8311yescharacteristic, complemented
212#3226C2×C8311yescharacteristic
213#3206C2×C8311yescharacteristic
214#3131C2×C8311yescharacteristic
215C2×#487C2×C8311yescharacteristic, complemented
216#1719C2×C8312yescomplemented
217#1718C2×C8312yescomplemented
218#1710C2×C8312yescomplemented
219#1708C2×C8312yescomplemented
220#1450C2×C8311yescharacteristic, complemented
221C83#1965711yescharacteristic, direct product factor, Sylow
222#533C22×C8311yescharacteristic
223#532C22×C8311yescharacteristic
224C2×C4⋊Q8C22×C8311yescharacteristic
225C2×C4⋊Q8C22×C8311yescharacteristic
226#500C22×C8312yes—
227#500C22×C8312yes—
228#498C22×C8312yes—
229#492C22×C8312yes—
230#488—22no—
231#487—22no—
232#487C22×C8312yes—
233#487C22×C8312yes—
234#487C22×C8311yescharacteristic
235#487C22×C8311yescharacteristic
236#487C22×C8312yes—
237#487C22×C8312yes—
238#487C22×C8312yes—
239#486C22×C8311yescharacteristic
240#486C22×C8311yescharacteristic
241#486C22×C8312yes—
242#484—22no—
243#483—22no—
244#480—22no—
245#477—22no—
246#475—22no—
247#474—22no—
248#461—22no—
249#448—22no—
250#431C22×C8311yescharacteristic
251#431C22×C8311yescharacteristic
252#416—22no—
253C2×#103—22no—
254C2×C4×C8C22×C8311yescharacteristic
255#322C22×C8312yes—
256C82C22×C8312yes—
257C2×#41—21no—
258C22×Q8—21no—
259C22×Q8—21no—
260C2×SD16—24no—
261C22×C8D4×C8312yes—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8C23×C8312yes—
269C4⋊Q8C23×C8312yes—
270C4⋊Q8C23×C8311yescharacteristic
271C4⋊Q8C23×C8311yescharacteristic
272C4⋊Q8C23×C8311yescharacteristic
273#127C23×C8311yescharacteristic
274#125—21no—
275#124—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278#121—21no—
279#121—21no—
280C4×Q8—21no—
281C4×Q8—21no—
282C4×D4—21no—
283C4×D4—21no—
284#117—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C42C23×C8311yescharacteristic
290#107—24no—
291#107—24no—
292#106—24no—
293C4⋊C8D4×C8312yes—
294C4⋊C8D4×C8312yes—
295#103—24no—
296#103—24no—
297#103—22no—
298#103—22no—
299#103—24no—
300#103—24no—
301#103—24no—
302#102—22no—
303#102—22no—
304#102—24no—
305C22⋊C8D4×C8312yes—
306C22⋊C8D4×C8312yes—
307C4×C8D4×C8312yes—
308C4×C8C23×C8311yescharacteristic
309C4×C8C23×C8312yes—
310C4×C8C23×C8312yes—
311C4×C8C23×C8311yescharacteristic
312#41—44no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—44no—
321C2×Q8—82no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4C2×D4×C8311yescharacteristic
330C22×C4C2×D4×C8311yescharacteristic
331C22×C4C2×D4×C8311yescharacteristic
332SD16—84no—
333C2×C8—24no—
334C2×C8C2×D4×C8312yes—
335C2×C8C2×D4×C8312yes—
336C2×C8C2×D4×C8314yes—
337C2×C8C2×D4×C8314yes—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C42—41no—
358C42C2×D4×C8311yescharacteristic
359C42C2×D4×C8311yescharacteristic
360C42C2×D4×C8311yescharacteristic
361C42C24×C8311yescharacteristic
362C23—41no—
363C23#127×C8311yescharacteristic
364Q8—44no—
365Q8—42no—
366Q8—44no—
367Q8—42no—
368Q8—42no—
369Q8—82no—
370Q8—82no—
371D4—42no—
372D4—42no—
373D4—42no—
374D4—82no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4#127×C8311yescharacteristic
386C2×C4#127×C8311yescharacteristic
387C2×C4#127×C8311yescharacteristic
388C2×C4#135×C8314yes—
389C2×C4C22×D4×C8311yescharacteristic
390C2×C4C22×D4×C8311yescharacteristic
391C2×C4C22×D4×C8311yescharacteristic
392C8—22no—
393C8—22no—
394C8—24no—
395C8—24no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#495×C8312yes—
402C22C2×#127×C8311yescharacteristic
403C4—21no—
404C4—21no—
405C4—81no—
406C4—82no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4#495×C8312yes—
411C4C2×#135×C8312yes—
412C4#577×C8312yes—
413C4#578×C8312yes—
414C2—22no—
415C2—161no—
416C2C2×#495×C8311yescharacteristic
417C2C83×#315411yescharacteristic
418C2C83×#315611yescharacteristic
subgroupG/H
center, socleC22×C83C2×#127
commutator, Frattini, cosocleC42C24×C83
prime radical#532×C83C22
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
1. C83×#3230C2semidirect
2. C83×#3229C2semidirect
6. C2×#487×C83C2semidirect
7. C83×#1719C2semidirect
8. C83×#1718C2semidirect
9. C83×#1710C2semidirect
10. C83×#1708C2semidirect
11. C83×#1450C2semidirect
204. #19657C83direct (N × H)
210. #3230C2×C83semidirect
211. #3229C2×C83semidirect
215. C2×#487C2×C83semidirect
216. #1719C2×C83semidirect
217. #1718C2×C83semidirect
218. #1710C2×C83semidirect
219. #1708C2×C83semidirect
220. #1450C2×C83semidirect
221. C83#19657direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#32302
C83×#32292
C83×#32262
C83×#32062
C83×#31312
C2×#487×C832
C83×#17192
C83×#17182
C83×#17102
C83×#17082
C83×#14502
#1965783