Finite Group Databasebeta

← C83×#9447 #35237271

Subgroups — C83×#9447 #35237271

Order 21248.

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

normal subgroups only

This group has 568 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×#2973C211yesmaximal, characteristic, complemented
2C83×#2544C211yesmaximal, characteristic
3C83×#1533C212yesmaximal, complemented
4C83×#1466C211yesmaximal, characteristic, complemented
5C83×#1285C212yesmaximal, complemented
6C2×#127×C83C411yescharacteristic
7C2×#124×C83C411yescharacteristic
8C2×#124×C83C2211yescharacteristic
9C2×C4×D4×C83C2211yescharacteristic
10#435×C83—24no—
11#432×C83C2212yes—
12#432×C83C2212yes—
13#407×C83—21no—
14C2×C22⋊C8×C83—21no—
15#394×C83—21no—
16#394×C83—21no—
17#391×C83C411yescharacteristic
18#390×C83C411yescharacteristic
19#386×C83—21no—
20#386×C83—21no—
21C4×C4⋊C4×C83C2211yescharacteristic
22#329×C83—24no—
23#323×C83—21no—
24#323×C83—21no—
25C22×Q8×C83C2×C411yescharacteristic
26C22×D4×C83—21no—
27C22×D4×C83—21no—
28C22×D4×C83C2×C411yescharacteristic
29C22×D4×C83C2×C411yescharacteristic
30C23×C4×C83D411yescharacteristic
31C23×C4×C83D411yescharacteristic
32C2×M16×C83—22no—
33C22×C8×C83—22no—
34#127×C83—42no—
35#124×C83—22no—
36#124×C83—22no—
37#124×C83—42no—
38C4×D4×C83—22no—
39C4×D4×C83—22no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83C2×C411yescharacteristic
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×C83—21no—
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83D411yescharacteristic
50C2×C42×C83C2×C411yescharacteristic
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83C2311yescharacteristic
53C4⋊C8×C83—22no—
54C4⋊C8×C83—22no—
55C22⋊C8×C83—22no—
56C22⋊C8×C83—22no—
57C22⋊C8×C83—22no—
58C22⋊C8×C83—22no—
59#97×C83—22no—
60#97×C83D414yescomplemented
61#95×C83—21no—
62#95×C83—21no—
63#95×C83—21no—
64C24×C83—21no—
65C24×C83C22⋊C411yescharacteristic
66C24×C83C22⋊C411yescharacteristic
67C2×Q8×C83—22no—
68C2×Q8×C83—41no—
69C2×Q8×C83—41no—
70C2×D4×C83—22no—
71C2×D4×C83—22no—
72C2×D4×C83—22no—
73C2×D4×C83—41no—
74C2×D4×C83—41no—
75C2×D4×C83—41no—
76C2×D4×C83—42no—
77C2×D4×C83—42no—
78C2×D4×C83—42no—
79C2×D4×C83—42no—
80C2×D4×C83—81no—
81C2×D4×C83—81no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—81no—
90C22×C4×C83—81no—
91C22×C4×C83C22⋊C411yescharacteristic
92C22×C4×C83C22⋊C411yescharacteristic
93C22×C4×C83C22×C411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96M16×C83—82no—
97C2×C8×C83—24no—
98C2×C8×C83—42no—
99C2×C8×C83—82no—
100C4⋊C4×C83—22no—
101C4⋊C4×C83—42no—
102C22⋊C4×C83—22no—
103C22⋊C4×C83—22no—
104C22⋊C4×C83—42no—
105C22⋊C4×C83—42no—
106C22⋊C4×C83—41no—
107C22⋊C4×C83—42no—
108C22⋊C4×C83—41no—
109C22⋊C4×C83—81no—
110C22⋊C4×C83—81no—
111C42×C83—41no—
112C42×C83—41no—
113C42×C83C2×D412yes—
114C42×C83#4112yes—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83—21no—
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×C83—41no—
129C23×C83—41no—
130C23×C83—41no—
131C23×C83#10012yes—
132C23×C83C2×C22⋊C411yescharacteristic
133Q8×C83—82no—
134D4×C83—44no—
135D4×C83—82no—
136D4×C83—84no—
137D4×C83—84no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—22no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—42no—
146C2×C4×C83—42no—
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—41no—
156C2×C4×C83—41no—
157C2×C4×C83—81no—
158C2×C4×C83—81no—
159C2×C4×C83C4×D412yes—
160C2×C4×C83#12011yescharacteristic
161C2×C4×C83#12112yes—
162C2×C4×C83#12311yescharacteristic
163C8×C83—44no—
164C8×C83—82no—
165C22×C83—21no—
166C22×C83—21no—
167C22×C83—22no—
168C22×C83—42no—
169C22×C83—42no—
170C22×C83—41no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83—41no—
174C22×C83—41no—
175C22×C83—81no—
176C22×C83—81no—
177C22×C83—81no—
178C22×C83—81no—
179C22×C83—81no—
180C22×C83—81no—
181C22×C83—81no—
182C22×C83—81no—
183C22×C83—81no—
184C22×C83#38611yescharacteristic
185C22×C83C2×#10011yescharacteristic
186C22×C83#41311yescharacteristic
187C4×C83—22no—
188C4×C83—22no—
189C4×C83—21no—
190C4×C83—21no—
191C4×C83—42no—
192C4×C83—41no—
193C4×C83—41no—
194C4×C83—81no—
195C4×C83—81no—
196#9447C8311yesmaximal, characteristic, direct product factor, Sylow
197C2×C83—21no—
198C2×C83—21no—
199C2×C83—42no—
200C2×C83—81no—
201C2×C83—161no—
202C2×C83#188411yescharacteristic
203C2×C83#190011yescharacteristic
204C2×C83#191111yescharacteristic
205#2973C2×C8311yescharacteristic, complemented
206#2544C2×C8311yescharacteristic
207#1533C2×C8312yescomplemented
208#1466C2×C8311yescharacteristic, complemented
209#1285C2×C8312yescomplemented
210C83#944711yescharacteristic, direct product factor, Sylow
211C2×#127C4×C8311yescharacteristic
212C2×#124C4×C8311yescharacteristic
213C2×#124C22×C8311yescharacteristic
214C2×C4×D4C22×C8311yescharacteristic
215#435—24no—
216#432C22×C8312yes—
217#432C22×C8312yes—
218#407—21no—
219C2×C22⋊C8—21no—
220#394—21no—
221#394—21no—
222#391C4×C8311yescharacteristic
223#390C4×C8311yescharacteristic
224#386—21no—
225#386—21no—
226C4×C4⋊C4C22×C8311yescharacteristic
227#329—24no—
228#323—21no—
229#323—21no—
230C22×Q8C2×C4×C8311yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C2×C4×C8311yescharacteristic
234C22×D4C2×C4×C8311yescharacteristic
235C23×C4D4×C8311yescharacteristic
236C23×C4D4×C8311yescharacteristic
237C2×M16—22no—
238C22×C8—22no—
239#127—42no—
240#124—22no—
241#124—22no—
242#124—42no—
243C4×D4—22no—
244C4×D4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C2×C4×C8311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4D4×C8311yescharacteristic
254C2×C22⋊C4D4×C8311yescharacteristic
255C2×C42C2×C4×C8311yescharacteristic
256C2×C42C2×C4×C8311yescharacteristic
257C2×C42C23×C8311yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264#97—22no—
265#97D4×C8314yescomplemented
266#95—21no—
267#95—21no—
268#95—21no—
269C24—21no—
270C24C22⋊C4×C8311yescharacteristic
271C24C22⋊C4×C8311yescharacteristic
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—81no—
286C2×D4—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C22⋊C4×C8311yescharacteristic
297C22×C4C22⋊C4×C8311yescharacteristic
298C22×C4C22×C4×C8311yescharacteristic
299C22×C4C2×D4×C8311yescharacteristic
300C22×C4C2×D4×C8311yescharacteristic
301M16—82no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—82no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C42—41no—
317C42—41no—
318C42C2×D4×C8312yes—
319C42#41×C8312yes—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#100×C8312yes—
337C23C2×C22⋊C4×C8311yescharacteristic
338Q8—82no—
339D4—44no—
340D4—82no—
341D4—84no—
342D4—84no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4C4×D4×C8312yes—
365C2×C4#120×C8311yescharacteristic
366C2×C4#121×C8312yes—
367C2×C4#123×C8311yescharacteristic
368C8—44no—
369C8—82no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#386×C8311yescharacteristic
390C22C2×#100×C8311yescharacteristic
391C22#413×C8311yescharacteristic
392C4—22no—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—161no—
406C2C83×#188411yescharacteristic
407C2C83×#190011yescharacteristic
408C2C83×#191111yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#127×C83C4
LCSG ▷ C22×C4 ▷ 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×#2973C2semidirect
3. C83×#1533C2semidirect
4. C83×#1466C2semidirect
5. C83×#1285C2semidirect
60. #97×C83D4semidirect
196. #9447C83direct (N × H)
205. #2973C2×C83semidirect
207. #1533C2×C83semidirect
208. #1466C2×C83semidirect
209. #1285C2×C83semidirect
210. C83#9447direct (N × H)
265. #97D4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#29732
C83×#25442
C83×#15332
C83×#14662
C83×#12852
#944783