Finite Group Databasebeta

← C83×#9517 #35237341

Subgroups — C83×#9517 #35237341

Order 21248.

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

normal subgroups only

This group has 520 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
2C83×#2579C211yesmaximal, characteristic
3C83×#1465C211yesmaximal, characteristic
4C83×#1463C212yesmaximal, complemented
5C83×#1305C212yesmaximal, complemented
6C2×C4×D4×C83C2211yescharacteristic
7C22×C4⋊C4×C83C2211yescharacteristic
8#435×C83—24no—
9#432×C83—21no—
10#407×C83C2212yes—
11#407×C83C2212yes—
12C2×C22⋊C8×C83—21no—
13#398×C83—21no—
14#396×C83—21no—
15#387×C83—21no—
16#386×C83C411yescharacteristic
17#386×C83C411yescharacteristic
18#385×C83—21no—
19#385×C83C411yescharacteristic
20#385×C83C411yescharacteristic
21#380×C83C2211yescharacteristic
22#324×C83—21no—
23#324×C83—24no—
24#323×C83—21no—
25C22×D4×C83C2×C411yescharacteristic
26C23×C4×C83C2×C411yescharacteristic
27C23×C4×C83C2×C411yescharacteristic
28C23×C4×C83D411yescharacteristic
29C23×C4×C83C2311yescharacteristic
30C2×M16×C83—22no—
31C2×M16×C83—22no—
32C22×C8×C83—22no—
33C4×D4×C83—22no—
34C4×D4×C83—22no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50C4⋊C8×C83—22no—
51C22⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53C22⋊C8×C83—22no—
54C22⋊C8×C83—22no—
55C22⋊C8×C83D414yes—
56#97×C83—22no—
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60#95×C83—21no—
61#95×C83—21no—
62C24×C83C22⋊C411yescharacteristic
63C24×C83C22×C411yescharacteristic
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83—41no—
67C22×C4×C83—21no—
68C22×C4×C83—22no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—22no—
75C22×C4×C83—42no—
76C22×C4×C83—81no—
77C22×C4×C83—81no—
78C22×C4×C83—81no—
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C2×D412yes—
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83#4112yes—
86M16×C83—82no—
87M16×C83—82no—
88C2×C8×C83—24no—
89C2×C8×C83—42no—
90C2×C8×C83—82no—
91C4⋊C4×C83—22no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—42no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—81no—
97C4⋊C4×C83—81no—
98C22⋊C4×C83—22no—
99C22⋊C4×C83—22no—
100C22⋊C4×C83—42no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C22⋊C4×C83—81no—
104C42×C83—21no—
105C42×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83—21no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83—41no—
113C23×C83—41no—
114C23×C83#10112yes—
115C23×C83C2×C22⋊C411yescharacteristic
116C23×C83C4×D412yes—
117D4×C83—44no—
118C2×C4×C83—22no—
119C2×C4×C83—24no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—41no—
126C2×C4×C83—42no—
127C2×C4×C83—42no—
128C2×C4×C83—41no—
129C2×C4×C83—42no—
130C2×C4×C83—41no—
131C2×C4×C83—42no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—81no—
145C2×C4×C83—82no—
146C2×C4×C83#12011yescharacteristic
147C2×C4×C83#12112yes—
148C2×C4×C83#12311yescharacteristic
149C8×C83—44no—
150C8×C83—82no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—22no—
154C22×C83—22no—
155C22×C83—22no—
156C22×C83—22no—
157C22×C83—42no—
158C22×C83—42no—
159C22×C83—42no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—81no—
163C22×C83#38611yescharacteristic
164C22×C83C2×#10111yescharacteristic
165C22×C83#41311yescharacteristic
166C4×C83—21no—
167C4×C83—22no—
168C4×C83—21no—
169C4×C83—42no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—41no—
173C4×C83—81no—
174C4×C83—81no—
175C4×C83—81no—
176#9517C8311yesmaximal, characteristic, direct product factor, Sylow
177C2×C83—21no—
178C2×C83—21no—
179C2×C83—42no—
180C2×C83—42no—
181C2×C83#189111yescharacteristic
182C2×C83#190111yescharacteristic
183C2×C83#191311yescharacteristic
184#2973C2×C8311yescharacteristic
185#2579C2×C8311yescharacteristic
186#1465C2×C8311yescharacteristic
187#1463C2×C8312yescomplemented
188#1305C2×C8312yescomplemented
189C83#951711yescharacteristic, direct product factor, Sylow
190C2×C4×D4C22×C8311yescharacteristic
191C22×C4⋊C4C22×C8311yescharacteristic
192#435—24no—
193#432—21no—
194#407C22×C8312yes—
195#407C22×C8312yes—
196C2×C22⋊C8—21no—
197#398—21no—
198#396—21no—
199#387—21no—
200#386C4×C8311yescharacteristic
201#386C4×C8311yescharacteristic
202#385—21no—
203#385C4×C8311yescharacteristic
204#385C4×C8311yescharacteristic
205#380C22×C8311yescharacteristic
206#324—21no—
207#324—24no—
208#323—21no—
209C22×D4C2×C4×C8311yescharacteristic
210C23×C4C2×C4×C8311yescharacteristic
211C23×C4C2×C4×C8311yescharacteristic
212C23×C4D4×C8311yescharacteristic
213C23×C4C23×C8311yescharacteristic
214C2×M16—22no—
215C2×M16—22no—
216C22×C8—22no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4D4×C8311yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4C2×C4×C8311yescharacteristic
230C2×C22⋊C4C2×C4×C8311yescharacteristic
231C2×C22⋊C4C2×C4×C8311yescharacteristic
232C2×C22⋊C4D4×C8311yescharacteristic
233C2×C42D4×C8311yescharacteristic
234C4⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8D4×C8314yes—
240#97—22no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24C22⋊C4×C8311yescharacteristic
247C24C22×C4×C8311yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊C4×C8311yescharacteristic
264C22×C4C22⋊C4×C8311yescharacteristic
265C22×C4C22⋊C4×C8311yescharacteristic
266C22×C4C2×D4×C8312yes—
267C22×C4C2×D4×C8311yescharacteristic
268C22×C4C2×D4×C8311yescharacteristic
269C22×C4#41×C8312yes—
270M16—82no—
271M16—82no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—82no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—81no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#101×C8312yes—
299C23C2×C22⋊C4×C8311yescharacteristic
300C23C4×D4×C8312yes—
301D4—44no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4#120×C8311yescharacteristic
331C2×C4#121×C8312yes—
332C2×C4#123×C8311yescharacteristic
333C8—44no—
334C8—82no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#386×C8311yescharacteristic
348C22C2×#101×C8311yescharacteristic
349C22#413×C8311yescharacteristic
350C4—21no—
351C4—22no—
352C4—21no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2C83×#189111yescharacteristic
365C2C83×#190111yescharacteristic
366C2C83×#191311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC24C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC22×D4×C83C2×C4
LCSG ▷ C24 ▷ 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×#1463C2semidirect
5. C83×#1305C2semidirect
176. #9517C83direct (N × H)
187. #1463C2×C83semidirect
188. #1305C2×C83semidirect
189. C83#9517direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#29732
C83×#25792
C83×#14652
C83×#14632
C83×#13052
#951783