Finite Group Databasebeta

← C83×#9522 #35237346

Subgroups — C83×#9522 #35237346

Order 21248.

472 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×#3092C211yesmaximal, characteristic
2C83×#2976C211yesmaximal, characteristic
3C83×#1823C211yesmaximal, characteristic, complemented
4C83×#1821C211yesmaximal, characteristic, complemented
5C83×#1513C211yesmaximal, characteristic
6C83×#1305C211yesmaximal, characteristic, complemented
7C83×#1293C211yesmaximal, characteristic, complemented
8#525×C83C2211yescharacteristic
9C22×C4⋊C4×C83C2211yescharacteristic
10#484×C83C412yescomplemented
11#483×C83—21no—
12#481×C83C412yescomplemented
13#480×C83—21no—
14#436×C83—21no—
15#435×C83—21no—
16#435×C83—21no—
17C8×D4×C83—21no—
18#433×C83—21no—
19#428×C83—21no—
20#417×C83—21no—
21#408×C83—21no—
22#407×C83C2211yescharacteristic
23#407×C83C2211yescharacteristic
24#407×C83C2211yescharacteristic
25C2×C22⋊C8×C83C2211yescharacteristic
26#380×C83C2211yescharacteristic
27#343×C83—21no—
28#343×C83—21no—
29#340×C83—21no—
30#340×C83—21no—
31#328×C83—21no—
32#327×C83—21no—
33C2×#41×C83D411yescharacteristic
34C23×C4×C83—21no—
35C23×C4×C83C2311yescharacteristic
36C2×M16×C83—21no—
37C2×M16×C83—21no—
38C2×M16×C83—21no—
39C2×M16×C83D411yescharacteristic
40C2×M16×C83D411yescharacteristic
41C22×C8×C83—21no—
42C22×C8×C83D411yescharacteristic
43C22×C8×C83D411yescharacteristic
44#123×C83—21no—
45C22⋊Q8×C83C2×C411yescharacteristic
46C22⋊Q8×C83D411yescharacteristic
47#121×C83C2×C411yescharacteristic
48#121×C83D411yescharacteristic
49#120×C83—21no—
50C4×D4×C83—21no—
51C4×D4×C83—21no—
52#117×C83D411yescharacteristic
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C4⋊C4×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83—21no—
58C2×C4⋊C4×C83—21no—
59C2×C4⋊C4×C83C2×C412yes—
60C2×C22⋊C4×C83—21no—
61#107×C83—22no—
62#106×C83—22no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65#103×C83—22no—
66#102×C83—22no—
67C22⋊C8×C83—21no—
68C22⋊C8×C83—21no—
69C22⋊C8×C83—21no—
70C22⋊C8×C83—21no—
71C22⋊C8×C83—21no—
72C22⋊C8×C83—21no—
73C22⋊C8×C83—21no—
74C22⋊C8×C83—21no—
75C22⋊C8×C83—21no—
76C22⋊C8×C83C2×C412yes—
77#97×C83—21no—
78C4×C8×C83—21no—
79#95×C83—21no—
80C24×C83#4111yescharacteristic
81#41×C83—41no—
82#41×C83—41no—
83C2×Q8×C83C22⋊C411yescharacteristic
84C2×D4×C83—21no—
85C2×D4×C83—21no—
86C2×D4×C83—21no—
87C2×D4×C83C22⋊C411yescharacteristic
88C22×C4×C83—21no—
89C22×C4×C83—22no—
90C22×C4×C83—41no—
91C22×C4×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83—41no—
95C22×C4×C83—81no—
96C22×C4×C83—81no—
97C22×C4×C83C22×C411yescharacteristic
98C22×C4×C83C2×D411yescharacteristic
99C22×C4×C83C2×D411yescharacteristic
100C22×C4×C83C2×D411yescharacteristic
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83#4111yescharacteristic
103M16×C83—41no—
104M16×C83—41no—
105M16×C83—41no—
106M16×C83—41no—
107M16×C83—41no—
108M16×C83—41no—
109M16×C83—81no—
110C2×C8×C83—22no—
111C2×C8×C83—21no—
112C2×C8×C83—21no—
113C2×C8×C83—21no—
114C2×C8×C83—21no—
115C2×C8×C83—41no—
116C2×C8×C83—41no—
117C2×C8×C83—41no—
118C2×C8×C83—81no—
119C4⋊C4×C83—21no—
120C4⋊C4×C83—21no—
121C4⋊C4×C83—22no—
122C4⋊C4×C83—22no—
123C4⋊C4×C83—41no—
124C4⋊C4×C83—41no—
125C4⋊C4×C83—41no—
126C4⋊C4×C83—41no—
127C4⋊C4×C83C22⋊C411yescharacteristic
128C4⋊C4×C83C22⋊C411yescharacteristic
129C22⋊C4×C83—21no—
130C22⋊C4×C83—21no—
131C22⋊C4×C83—21no—
132C22⋊C4×C83—21no—
133C22⋊C4×C83—21no—
134C22⋊C4×C83—81no—
135C42×C83—21no—
136C23×C83—21no—
137C23×C83—21no—
138C23×C83—21no—
139C23×C83—41no—
140C23×C83—41no—
141C23×C83C4×D411yescharacteristic
142C23×C83#12111yescharacteristic
143C23×C83#12311yescharacteristic
144Q8×C83—41no—
145D4×C83—41no—
146D4×C83—41no—
147D4×C83—41no—
148D4×C83—81no—
149D4×C83—81no—
150C2×C4×C83—21no—
151C2×C4×C83—21no—
152C2×C4×C83—21no—
153C2×C4×C83—21no—
154C2×C4×C83—21no—
155C2×C4×C83—21no—
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—42no—
165C2×C4×C83—82no—
166C2×C4×C83—81no—
167C2×C4×C83—81no—
168C2×C4×C83—82no—
169C2×C4×C83—81no—
170C2×C4×C83—81no—
171C2×C4×C83—81no—
172C2×C4×C83C2×C22⋊C411yescharacteristic
173C2×C4×C83C4×D411yescharacteristic
174C2×C4×C83#12011yescharacteristic
175C2×C4×C83#12111yescharacteristic
176C8×C83—42no—
177C8×C83—41no—
178C8×C83—41no—
179C8×C83—41no—
180C8×C83—41no—
181C8×C83—81no—
182C22×C83—21no—
183C22×C83—21no—
184C22×C83—21no—
185C22×C83—21no—
186C22×C83—21no—
187C22×C83—21no—
188C22×C83—21no—
189C22×C83—21no—
190C22×C83—42no—
191C22×C83—42no—
192C22×C83—41no—
193C22×C83—41no—
194C22×C83—41no—
195C22×C83#38611yescharacteristic
196C22×C83#47111yescharacteristic
197C22×C83#47211yescharacteristic
198C4×C83—21no—
199C4×C83—21no—
200C4×C83—21no—
201C4×C83—41no—
202C4×C83—41no—
203C4×C83—41no—
204C4×C83—82no—
205C4×C83—161no—
206C4×C83#45811yescharacteristic
207C4×C83#45811yescharacteristic
208#9522C8311yesmaximal, characteristic, direct product factor, Sylow
209C2×C83—21no—
210C2×C83—21no—
211C2×C83—41no—
212C2×C83—41no—
213C2×C83—81no—
214C2×C83#189011yescharacteristic
215C2×C83#190211yescharacteristic
216C2×C83#190711yescharacteristic
217#3092C2×C8311yescharacteristic
218#2976C2×C8311yescharacteristic
219#1823C2×C8311yescharacteristic, complemented
220#1821C2×C8311yescharacteristic, complemented
221#1513C2×C8311yescharacteristic
222#1305C2×C8311yescharacteristic, complemented
223#1293C2×C8311yescharacteristic, complemented
224C83#952211yescharacteristic, direct product factor, Sylow
225#525C22×C8311yescharacteristic
226C22×C4⋊C4C22×C8311yescharacteristic
227#484C4×C8312yescomplemented
228#483—21no—
229#481C4×C8312yescomplemented
230#480—21no—
231#436—21no—
232#435—21no—
233#435—21no—
234C8×D4—21no—
235#433—21no—
236#428—21no—
237#417—21no—
238#408—21no—
239#407C22×C8311yescharacteristic
240#407C22×C8311yescharacteristic
241#407C22×C8311yescharacteristic
242C2×C22⋊C8C22×C8311yescharacteristic
243#380C22×C8311yescharacteristic
244#343—21no—
245#343—21no—
246#340—21no—
247#340—21no—
248#328—21no—
249#327—21no—
250C2×#41D4×C8311yescharacteristic
251C23×C4—21no—
252C23×C4C23×C8311yescharacteristic
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16D4×C8311yescharacteristic
257C2×M16D4×C8311yescharacteristic
258C22×C8—21no—
259C22×C8D4×C8311yescharacteristic
260C22×C8D4×C8311yescharacteristic
261#123—21no—
262C22⋊Q8C2×C4×C8311yescharacteristic
263C22⋊Q8D4×C8311yescharacteristic
264#121C2×C4×C8311yescharacteristic
265#121D4×C8311yescharacteristic
266#120—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117D4×C8311yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C2×C4×C8312yes—
277C2×C22⋊C4—21no—
278#107—22no—
279#106—22no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282#103—22no—
283#102—22no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8C2×C4×C8312yes—
294#97—21no—
295C4×C8—21no—
296#95—21no—
297C24#41×C8311yescharacteristic
298#41—41no—
299#41—41no—
300C2×Q8C22⋊C4×C8311yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C22⋊C4×C8311yescharacteristic
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4C22×C4×C8311yescharacteristic
315C22×C4C2×D4×C8311yescharacteristic
316C22×C4C2×D4×C8311yescharacteristic
317C22×C4C2×D4×C8311yescharacteristic
318C22×C4C2×D4×C8311yescharacteristic
319C22×C4#41×C8311yescharacteristic
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—81no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—81no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4C22⋊C4×C8311yescharacteristic
345C4⋊C4C22⋊C4×C8311yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—81no—
352C42—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23C4×D4×C8311yescharacteristic
359C23#121×C8311yescharacteristic
360C23#123×C8311yescharacteristic
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—81no—
366D4—81no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
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—42no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4C2×C22⋊C4×C8311yescharacteristic
390C2×C4C4×D4×C8311yescharacteristic
391C2×C4#120×C8311yescharacteristic
392C2×C4#121×C8311yescharacteristic
393C8—42no—
394C8—41no—
395C8—41no—
396C8—41no—
397C8—41no—
398C8—81no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22#386×C8311yescharacteristic
413C22#471×C8311yescharacteristic
414C22#472×C8311yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—161no—
423C4#458×C8311yescharacteristic
424C4#458×C8311yescharacteristic
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—81no—
430C2C83×#189011yescharacteristic
431C2C83×#190211yescharacteristic
432C2C83×#190711yescharacteristic
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
3. C83×#1823C2semidirect
4. C83×#1821C2semidirect
6. C83×#1305C2semidirect
7. C83×#1293C2semidirect
10. #484×C83C4semidirect
12. #481×C83C4semidirect
208. #9522C83direct (N × H)
219. #1823C2×C83semidirect
220. #1821C2×C83semidirect
222. #1305C2×C83semidirect
223. #1293C2×C83semidirect
224. C83#9522direct (N × H)
227. #484C4×C83semidirect
229. #481C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#30922
C83×#29762
C83×#18232
C83×#18212
C83×#15132
C83×#13052
C83×#12932
#952283