Finite Group Databasebeta

← C83×#9448 #35237272

Subgroups — C83×#9448 #35237272

Order 21248.

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

normal subgroups only

This group has 486 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×#2983C211yesmaximal, characteristic, complemented
2C83×#2545C211yesmaximal, characteristic
3C83×#1533C212yesmaximal
4C83×#1467C211yesmaximal, characteristic, complemented
5C83×#1285C212yesmaximal, complemented
6C2×C4⋊Q8×C83C411yescharacteristic
7C2×#124×C83C411yescharacteristic
8C2×#124×C83C2211yescharacteristic
9C2×C4×Q8×C83C2211yescharacteristic
10#446×C83—24no—
11#432×C83—21no—
12#432×C83C2212yes—
13#432×C83C2212yes—
14#431×C83—21no—
15#396×C83—21no—
16#396×C83—21no—
17#391×C83C411yescharacteristic
18#390×C83C411yescharacteristic
19#388×C83—21no—
20#388×C83—21no—
21C4×C4⋊C4×C83C2211yescharacteristic
22#329×C83—24no—
23#324×C83—21no—
24#324×C83—21no—
25C22×Q8×C83C2×C411yescharacteristic
26C22×Q8×C83C2×C411yescharacteristic
27C22×D4×C83C2×C411yescharacteristic
28C4⋊Q8×C83—42no—
29#124×C83—22no—
30#124×C83—22no—
31#124×C83—42no—
32C4×Q8×C83—22no—
33C4×Q8×C83—22no—
34C2×C4⋊C4×C83—21no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83C2×C411yescharacteristic
38C2×C4⋊C4×C83D411yescharacteristic
39C2×C4⋊C4×C83D411yescharacteristic
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C42×C83C2×C411yescharacteristic
47C2×C42×C83C2×C411yescharacteristic
48C2×C42×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50C2×C42×C83C2311yescharacteristic
51C4⋊C8×C83—22no—
52C4⋊C8×C83—22no—
53C4⋊C8×C83—22no—
54C4⋊C8×C83—22no—
55C22⋊C8×C83—22no—
56C22⋊C8×C83—22no—
57#97×C83—22no—
58#97×C83—22no—
59#97×C83D414yes—
60C4×C8×C83—22no—
61#95×C83—21no—
62#95×C83—21no—
63#95×C83—21no—
64C24×C83—21no—
65C2×Q8×C83—22no—
66C2×Q8×C83—22no—
67C2×Q8×C83—22no—
68C2×Q8×C83—41no—
69C2×Q8×C83—41no—
70C2×Q8×C83—41no—
71C2×D4×C83—22no—
72C2×D4×C83—41no—
73C2×D4×C83—41no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C2×C8×C83—24no—
85C2×C8×C83—42no—
86C4⋊C4×C83—22no—
87C4⋊C4×C83—22no—
88C4⋊C4×C83—22no—
89C4⋊C4×C83—42no—
90C4⋊C4×C83—42no—
91C4⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—42no—
96C22⋊C4×C83—41no—
97C22⋊C4×C83—42no—
98C22⋊C4×C83—41no—
99C42×C83—21no—
100C42×C83—21no—
101C42×C83—21no—
102C42×C83—21no—
103C42×C83—41no—
104C42×C83—41no—
105C42×C83C2×D412yes—
106C42×C83#4112yes—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83—41no—
113C23×C83—41no—
114C23×C83C2×C22⋊C411yescharacteristic
115Q8×C83—44no—
116Q8×C83—82no—
117D4×C83—82no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—22no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
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#10012yes—
141C2×C4×C83C4×D412yes—
142C2×C4×C83#12011yescharacteristic
143C2×C4×C83#12112yes—
144C2×C4×C83#12311yescharacteristic
145C8×C83—44no—
146C8×C83—82no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—81no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83#38611yescharacteristic
157C22×C83C2×#10011yescharacteristic
158C22×C83#41311yescharacteristic
159C4×C83—22no—
160C4×C83—22no—
161C4×C83—22no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—42no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—81no—
169C4×C83—81no—
170#9448C8311yesmaximal, characteristic, direct product factor, Sylow
171C2×C83—21no—
172C2×C83—21no—
173C2×C83—161no—
174C2×C83#188411yescharacteristic
175C2×C83#190011yescharacteristic
176C2×C83#191211yescharacteristic
177#2983C2×C8311yescharacteristic, complemented
178#2545C2×C8311yescharacteristic
179#1533C2×C8312yes—
180#1467C2×C8311yescharacteristic, complemented
181#1285C2×C8312yescomplemented
182C83#944811yescharacteristic, direct product factor, Sylow
183C2×C4⋊Q8C4×C8311yescharacteristic
184C2×#124C4×C8311yescharacteristic
185C2×#124C22×C8311yescharacteristic
186C2×C4×Q8C22×C8311yescharacteristic
187#446—24no—
188#432—21no—
189#432C22×C8312yes—
190#432C22×C8312yes—
191#431—21no—
192#396—21no—
193#396—21no—
194#391C4×C8311yescharacteristic
195#390C4×C8311yescharacteristic
196#388—21no—
197#388—21no—
198C4×C4⋊C4C22×C8311yescharacteristic
199#329—24no—
200#324—21no—
201#324—21no—
202C22×Q8C2×C4×C8311yescharacteristic
203C22×Q8C2×C4×C8311yescharacteristic
204C22×D4C2×C4×C8311yescharacteristic
205C4⋊Q8—42no—
206#124—22no—
207#124—22no—
208#124—42no—
209C4×Q8—22no—
210C4×Q8—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×C4×C8311yescharacteristic
215C2×C4⋊C4D4×C8311yescharacteristic
216C2×C4⋊C4D4×C8311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42C2×C4×C8311yescharacteristic
224C2×C42C2×C4×C8311yescharacteristic
225C2×C42D4×C8311yescharacteristic
226C2×C42D4×C8311yescharacteristic
227C2×C42C23×C8311yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234#97—22no—
235#97—22no—
236#97D4×C8314yes—
237C4×C8—22no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22⋊C4×C8311yescharacteristic
255C22×C4C22⋊C4×C8311yescharacteristic
256C22×C4C22⋊C4×C8311yescharacteristic
257C22×C4C22⋊C4×C8311yescharacteristic
258C22×C4C22×C4×C8311yescharacteristic
259C22×C4C2×D4×C8311yescharacteristic
260C22×C4C2×D4×C8311yescharacteristic
261C2×C8—24no—
262C2×C8—42no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—41no—
281C42—41no—
282C42C2×D4×C8312yes—
283C42#41×C8312yes—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊C4×C8311yescharacteristic
292Q8—44no—
293Q8—82no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4#100×C8312yes—
318C2×C4C4×D4×C8312yes—
319C2×C4#120×C8311yescharacteristic
320C2×C4#121×C8312yes—
321C2×C4#123×C8311yescharacteristic
322C8—44no—
323C8—82no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#386×C8311yescharacteristic
334C22C2×#100×C8311yescharacteristic
335C22#413×C8311yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2C83×#188411yescharacteristic
351C2C83×#190011yescharacteristic
352C2C83×#191211yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC22×D4×C83C2×C4
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×#2983C2semidirect
4. C83×#1467C2semidirect
5. C83×#1285C2semidirect
170. #9448C83direct (N × H)
177. #2983C2×C83semidirect
180. #1467C2×C83semidirect
181. #1285C2×C83semidirect
182. C83#9448direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#29832
C83×#25452
C83×#15332
C83×#14672
C83×#12852
#944883