Finite Group Databasebeta

← C83×#9331 #35237155

Subgroups — C83×#9331 #35237155

Order 21248.

438 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3196C211yesmaximal, characteristic
2C83×#2978C211yesmaximal, characteristic
3C83×#1516C211yesmaximal, characteristic
4C83×#1511C211yesmaximal, characteristic, complemented
5C83×#1472C211yesmaximal, characteristic, complemented
6C83×#1306C211yesmaximal, characteristic, complemented
7C83×#1293C211yesmaximal, characteristic, complemented
8#525×C83C2211yescharacteristic
9C2×C22⋊Q8×C83C2211yescharacteristic
10#485×C83—21no—
11#484×C83C411yescharacteristic, complemented
12#481×C83C411yescharacteristic, complemented
13#470×C83—21no—
14#468×C83—21no—
15#452×C83—21no—
16#451×C83C411yescharacteristic, complemented
17#448×C83C411yescharacteristic, complemented
18#436×C83—21no—
19#436×C83—21no—
20#435×C83—21no—
21#435×C83—21no—
22#433×C83—21no—
23#408×C83—21no—
24#407×C83C2211yescharacteristic
25#407×C83C2211yescharacteristic
26#407×C83C2211yescharacteristic
27C2×C22⋊C8×C83C2211yescharacteristic
28#381×C83C2211yescharacteristic
29#328×C83—21no—
30#328×C83—21no—
31#328×C83—21no—
32#327×C83—21no—
33#324×C83—21no—
34#324×C83—21no—
35C2×#41×C83D411yescharacteristic
36C22×Q8×C83C2×C411yescharacteristic
37C23×C4×C83C2311yescharacteristic
38C2×Q16×C83—21no—
39C2×SD16×C83—21no—
40C2×M16×C83—21no—
41C2×M16×C83—21no—
42C2×M16×C83—21no—
43C2×M16×C83—21no—
44C22×C8×C83—21no—
45C22×C8×C83—21no—
46#123×C83—21no—
47C22⋊Q8×C83—21no—
48C22⋊Q8×C83—21no—
49C22⋊Q8×C83—21no—
50C22⋊Q8×C83—21no—
51C22⋊Q8×C83C2×C411yescharacteristic
52C22⋊Q8×C83D411yescharacteristic
53#121×C83C2×C411yescharacteristic
54#121×C83D411yescharacteristic
55#120×C83—21no—
56C4×D4×C83—21no—
57C4×D4×C83—21no—
58#117×C83D411yescharacteristic
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83C2×C411yescharacteristic
61C2×C22⋊C4×C83—21no—
62C2×C22⋊C4×C83—21no—
63#107×C83—21no—
64#106×C83—21no—
65C4⋊C8×C83—21no—
66C4⋊C8×C83—21no—
67#103×C83—21no—
68#103×C83—21no—
69#103×C83—21no—
70#102×C83—21no—
71C22⋊C8×C83—21no—
72C22⋊C8×C83—21no—
73C22⋊C8×C83—21no—
74C22⋊C8×C83—21no—
75C22⋊C8×C83—21no—
76C22⋊C8×C83—21no—
77C22⋊C8×C83—21no—
78C22⋊C8×C83C2×C411yescharacteristic
79C22⋊C8×C83C2×C411yescharacteristic
80C22⋊C8×C83D411yescharacteristic
81C22⋊C8×C83D411yescharacteristic
82C22⋊C8×C83D411yescharacteristic
83C22⋊C8×C83D411yescharacteristic
84#97×C83—21no—
85C4×C8×C83—21no—
86#95×C83—21no—
87#95×C83—21no—
88C24×C83#4111yescharacteristic
89#41×C83—81no—
90C2×Q8×C83—21no—
91C2×Q8×C83—21no—
92C2×Q8×C83—81no—
93C2×Q8×C83C22⋊C411yescharacteristic
94C2×D4×C83—21no—
95C2×D4×C83—21no—
96C2×D4×C83—21no—
97C2×D4×C83C22⋊C411yescharacteristic
98C22×C4×C83—21no—
99C22×C4×C83—21no—
100C22×C4×C83—21no—
101C22×C4×C83—41no—
102C22×C4×C83—81no—
103C22×C4×C83C22×C411yescharacteristic
104C22×C4×C83C2×D411yescharacteristic
105C22×C4×C83C2×D411yescharacteristic
106C22×C4×C83C2×D411yescharacteristic
107C22×C4×C83C2×D411yescharacteristic
108C22×C4×C83#4111yescharacteristic
109Q16×C83—81no—
110SD16×C83—81no—
111M16×C83—81no—
112M16×C83—81no—
113M16×C83—81no—
114M16×C83—81no—
115C2×C8×C83—21no—
116C2×C8×C83—21no—
117C2×C8×C83—21no—
118C2×C8×C83—21no—
119C2×C8×C83—21no—
120C2×C8×C83—21no—
121C2×C8×C83—41no—
122C2×C8×C83—81no—
123C2×C8×C83—81no—
124C4⋊C4×C83—21no—
125C4⋊C4×C83—21no—
126C4⋊C4×C83—21no—
127C4⋊C4×C83—21no—
128C4⋊C4×C83—41no—
129C4⋊C4×C83—41no—
130C4⋊C4×C83C22⋊C411yescharacteristic
131C4⋊C4×C83C22⋊C411yescharacteristic
132C22⋊C4×C83—21no—
133C22⋊C4×C83—21no—
134C22⋊C4×C83—21no—
135C22⋊C4×C83—21no—
136C22⋊C4×C83—21no—
137C22⋊C4×C83—41no—
138C22⋊C4×C83—41no—
139C22⋊C4×C83—81no—
140C42×C83—21no—
141C23×C83—21no—
142C23×C83—41no—
143C23×C83—41no—
144C23×C83—41no—
145C23×C83C4×D411yescharacteristic
146C23×C83#12111yescharacteristic
147C23×C83#12311yescharacteristic
148Q8×C83—41no—
149Q8×C83—41no—
150Q8×C83—41no—
151Q8×C83—81no—
152D4×C83—41no—
153D4×C83—81no—
154D4×C83—81no—
155D4×C83—81no—
156C2×C4×C83—21no—
157C2×C4×C83—21no—
158C2×C4×C83—21no—
159C2×C4×C83—41no—
160C2×C4×C83—41no—
161C2×C4×C83—41no—
162C2×C4×C83—41no—
163C2×C4×C83—41no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—42no—
168C2×C4×C83—41no—
169C2×C4×C83—41no—
170C2×C4×C83—81no—
171C2×C4×C83—81no—
172C2×C4×C83—81no—
173C2×C4×C83—81no—
174C2×C4×C83C2×C22⋊C411yescharacteristic
175C2×C4×C83C4×D411yescharacteristic
176C2×C4×C83#12011yescharacteristic
177C2×C4×C83#12111yescharacteristic
178C8×C83—41no—
179C8×C83—41no—
180C8×C83—41no—
181C8×C83—41no—
182C8×C83—41no—
183C8×C83—41no—
184C8×C83—81no—
185C22×C83—21no—
186C22×C83—21no—
187C22×C83—21no—
188C22×C83—21no—
189C22×C83—21no—
190C22×C83—21no—
191C22×C83—21no—
192C22×C83—21no—
193C22×C83—21no—
194C22×C83—41no—
195C22×C83—41no—
196C22×C83—41no—
197C22×C83—82no—
198C22×C83#38611yescharacteristic
199C4×C83—21no—
200C4×C83—21no—
201C4×C83—21no—
202C4×C83—21no—
203C4×C83—41no—
204C4×C83—41no—
205C4×C83—41no—
206C4×C83—81no—
207C4×C83—81no—
208C4×C83—161no—
209#9331C8311yesmaximal, characteristic, direct product factor, Sylow
210C2×C83—41no—
211C2×C83—41no—
212C2×C83—41no—
213C2×C83—81no—
214C2×C83#188111yescharacteristic
215C2×C83#190211yescharacteristic
216C2×C83#191311yescharacteristic
217#3196C2×C8311yescharacteristic
218#2978C2×C8311yescharacteristic
219#1516C2×C8311yescharacteristic
220#1511C2×C8311yescharacteristic, complemented
221#1472C2×C8311yescharacteristic, complemented
222#1306C2×C8311yescharacteristic, complemented
223#1293C2×C8311yescharacteristic, complemented
224C83#933111yescharacteristic, direct product factor, Sylow
225#525C22×C8311yescharacteristic
226C2×C22⋊Q8C22×C8311yescharacteristic
227#485—21no—
228#484C4×C8311yescharacteristic, complemented
229#481C4×C8311yescharacteristic, complemented
230#470—21no—
231#468—21no—
232#452—21no—
233#451C4×C8311yescharacteristic, complemented
234#448C4×C8311yescharacteristic, complemented
235#436—21no—
236#436—21no—
237#435—21no—
238#435—21no—
239#433—21no—
240#408—21no—
241#407C22×C8311yescharacteristic
242#407C22×C8311yescharacteristic
243#407C22×C8311yescharacteristic
244C2×C22⋊C8C22×C8311yescharacteristic
245#381C22×C8311yescharacteristic
246#328—21no—
247#328—21no—
248#328—21no—
249#327—21no—
250#324—21no—
251#324—21no—
252C2×#41D4×C8311yescharacteristic
253C22×Q8C2×C4×C8311yescharacteristic
254C23×C4C23×C8311yescharacteristic
255C2×Q16—21no—
256C2×SD16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—21no—
261C22×C8—21no—
262C22×C8—21no—
263#123—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8C2×C4×C8311yescharacteristic
269C22⋊Q8D4×C8311yescharacteristic
270#121C2×C4×C8311yescharacteristic
271#121D4×C8311yescharacteristic
272#120—21no—
273C4×D4—21no—
274C4×D4—21no—
275#117D4×C8311yescharacteristic
276C2×C4⋊C4—21no—
277C2×C4⋊C4C2×C4×C8311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280#107—21no—
281#106—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#102—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8C2×C4×C8311yescharacteristic
296C22⋊C8C2×C4×C8311yescharacteristic
297C22⋊C8D4×C8311yescharacteristic
298C22⋊C8D4×C8311yescharacteristic
299C22⋊C8D4×C8311yescharacteristic
300C22⋊C8D4×C8311yescharacteristic
301#97—21no—
302C4×C8—21no—
303#95—21no—
304#95—21no—
305C24#41×C8311yescharacteristic
306#41—81no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—81no—
310C2×Q8C22⋊C4×C8311yescharacteristic
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4C22⋊C4×C8311yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—81no—
320C22×C4C22×C4×C8311yescharacteristic
321C22×C4C2×D4×C8311yescharacteristic
322C22×C4C2×D4×C8311yescharacteristic
323C22×C4C2×D4×C8311yescharacteristic
324C22×C4C2×D4×C8311yescharacteristic
325C22×C4#41×C8311yescharacteristic
326Q16—81no—
327SD16—81no—
328M16—81no—
329M16—81no—
330M16—81no—
331M16—81no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—41no—
339C2×C8—81no—
340C2×C8—81no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4C22⋊C4×C8311yescharacteristic
348C4⋊C4C22⋊C4×C8311yescharacteristic
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—81no—
357C42—21no—
358C23—21no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23C4×D4×C8311yescharacteristic
363C23#121×C8311yescharacteristic
364C23#123×C8311yescharacteristic
365Q8—41no—
366Q8—41no—
367Q8—41no—
368Q8—81no—
369D4—41no—
370D4—81no—
371D4—81no—
372D4—81no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4C2×C22⋊C4×C8311yescharacteristic
392C2×C4C4×D4×C8311yescharacteristic
393C2×C4#120×C8311yescharacteristic
394C2×C4#121×C8311yescharacteristic
395C8—41no—
396C8—41no—
397C8—41no—
398C8—41no—
399C8—41no—
400C8—41no—
401C8—81no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—82no—
415C22#386×C8311yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—81no—
424C4—81no—
425C4—161no—
426C2—41no—
427C2—41no—
428C2—41no—
429C2—81no—
430C2C83×#188111yescharacteristic
431C2C83×#190211yescharacteristic
432C2C83×#191311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radical#525×C83C22
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
4. C83×#1511C2semidirect
5. C83×#1472C2semidirect
6. C83×#1306C2semidirect
7. C83×#1293C2semidirect
11. #484×C83C4semidirect
12. #481×C83C4semidirect
16. #451×C83C4semidirect
17. #448×C83C4semidirect
209. #9331C83direct (N × H)
220. #1511C2×C83semidirect
221. #1472C2×C83semidirect
222. #1306C2×C83semidirect
223. #1293C2×C83semidirect
224. C83#9331direct (N × H)
228. #484C4×C83semidirect
229. #481C4×C83semidirect
233. #451C4×C83semidirect
234. #448C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#31962
C83×#29782
C83×#15162
C83×#15112
C83×#14722
C83×#13062
C83×#12932
#933183