Finite Group Databasebeta

← C83×#9411 #35237235

Subgroups — C83×#9411 #35237235

Order 21248.

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

normal subgroups only

This group has 606 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×#2561C211yesmaximal, characteristic
2C83×#2312C211yesmaximal, characteristic
3C83×#1508C212yesmaximal, complemented
4C83×#1462C211yesmaximal, characteristic
5C83×#1297C212yesmaximal, complemented
6C2×C22⋊Q8×C83C2211yescharacteristic
7C22×C22⋊C4×C83C2211yescharacteristic
8#407×C83C2212yes—
9#397×C83C412yescomplemented
10#393×C83C412yescomplemented
11#388×C83—22no—
12#388×C83—24no—
13#381×C83—21no—
14#380×C83—21no—
15#379×C83—21no—
16#379×C83—21no—
17C4×C22⋊C4×C83C2211yescharacteristic
18C4×C22⋊C4×C83C2212yes—
19#328×C83—24no—
20#323×C83—22no—
21C25×C83D411yescharacteristic
22C22×Q8×C83C2×C411yescharacteristic
23C23×C4×C83D411yescharacteristic
24C23×C4×C83C2311yescharacteristic
25C2×M16×C83—22no—
26C22⋊Q8×C83—22no—
27C22⋊Q8×C83—22no—
28C2×C4⋊C4×C83—21no—
29C2×C4⋊C4×C83—22no—
30C2×C4⋊C4×C83C2×C411yescharacteristic
31C2×C22⋊C4×C83—22no—
32C2×C22⋊C4×C83—22no—
33C2×C22⋊C4×C83—22no—
34C2×C22⋊C4×C83—21no—
35C2×C22⋊C4×C83—22no—
36C2×C22⋊C4×C83—21no—
37C2×C22⋊C4×C83—22no—
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83—22no—
40C2×C22⋊C4×C83C2×C412yes—
41C2×C22⋊C4×C83D412yes—
42C2×C42×C83—21no—
43C2×C42×C83—22no—
44C22⋊C8×C83—22no—
45C22⋊C8×C83—22no—
46#95×C83—22no—
47#95×C83—22no—
48#95×C83—22no—
49#95×C83C2×C412yes—
50#95×C83D414yes—
51C24×C83—21no—
52C24×C83—21no—
53C24×C83—42no—
54C24×C83—42no—
55C24×C83—81no—
56C24×C83C22⋊C412yes—
57C24×C83C2×D411yescharacteristic
58C2×Q8×C83—22no—
59C2×Q8×C83—81no—
60C22×C4×C83—21no—
61C22×C4×C83—21no—
62C22×C4×C83—22no—
63C22×C4×C83—21no—
64C22×C4×C83—21no—
65C22×C4×C83—24no—
66C22×C4×C83—42no—
67C22×C4×C83—81no—
68C22×C4×C83C22⋊C412yes—
69C22×C4×C83C22×C411yescharacteristic
70C22×C4×C83C2×D412yes—
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83#4112yes—
73M16×C83—82no—
74C2×C8×C83—42no—
75C4⋊C4×C83—22no—
76C4⋊C4×C83—42no—
77C4⋊C4×C83—82no—
78C22⋊C4×C83—24no—
79C22⋊C4×C83—24no—
80C22⋊C4×C83—44no—
81C22⋊C4×C83—42no—
82C22⋊C4×C83—44no—
83C22⋊C4×C83—41no—
84C22⋊C4×C83—41no—
85C22⋊C4×C83—82no—
86C22⋊C4×C83—82no—
87C22⋊C4×C83—82no—
88C42×C83—81no—
89C42×C83—82no—
90C23×C83—22no—
91C23×C83—22no—
92C23×C83—22no—
93C23×C83—22no—
94C23×C83—22no—
95C23×C83—41no—
96C23×C83—42no—
97C23×C83—42no—
98C23×C83—42no—
99C23×C83—42no—
100C23×C83—42no—
101C23×C83—42no—
102C23×C83—81no—
103C23×C83—81no—
104C23×C83—88no—
105C23×C83C2×C22⋊C411yescharacteristic
106C23×C83#12011yescharacteristic
107C23×C83#12311yescharacteristic
108Q8×C83—82no—
109C2×C4×C83—22no—
110C2×C4×C83—41no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83—41no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—44no—
118C2×C4×C83—42no—
119C2×C4×C83—44no—
120C2×C4×C83—42no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—44no—
126C2×C4×C83—81no—
127C2×C4×C83C4×D412yes—
128C2×C4×C83#12112yes—
129C8×C83—82no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—44no—
138C22×C83—44no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83—82no—
143C22×C83—84no—
144C22×C83—84no—
145C22×C83#38611yescharacteristic
146C22×C83#42112yes—
147C22×C83#45512yes—
148C4×C83—22no—
149C4×C83—22no—
150C4×C83—42no—
151C4×C83—84no—
152C4×C83—82no—
153C4×C83—81no—
154C4×C83—81no—
155#9411C8311yesmaximal, characteristic, direct product factor, Sylow
156C2×C83—22no—
157C2×C83—22no—
158C2×C83—41no—
159C2×C83—82no—
160C2×C83#188511yescharacteristic
161C2×C83#189711yescharacteristic
162C2×C83#191311yescharacteristic
163#2561C2×C8311yescharacteristic
164#2312C2×C8311yescharacteristic
165#1508C2×C8312yescomplemented
166#1462C2×C8311yescharacteristic
167#1297C2×C8312yescomplemented
168C83#941111yescharacteristic, direct product factor, Sylow
169C2×C22⋊Q8C22×C8311yescharacteristic
170C22×C22⋊C4C22×C8311yescharacteristic
171#407C22×C8312yes—
172#397C4×C8312yescomplemented
173#393C4×C8312yescomplemented
174#388—22no—
175#388—24no—
176#381—21no—
177#380—21no—
178#379—21no—
179#379—21no—
180C4×C22⋊C4C22×C8311yescharacteristic
181C4×C22⋊C4C22×C8312yes—
182#328—24no—
183#323—22no—
184C25D4×C8311yescharacteristic
185C22×Q8C2×C4×C8311yescharacteristic
186C23×C4D4×C8311yescharacteristic
187C23×C4C23×C8311yescharacteristic
188C2×M16—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C2×C4×C8311yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C2×C4×C8312yes—
204C2×C22⋊C4D4×C8312yes—
205C2×C42—21no—
206C2×C42—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95C2×C4×C8312yes—
213#95D4×C8314yes—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24C22⋊C4×C8312yes—
220C24C2×D4×C8311yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—81no—
231C22×C4C22⋊C4×C8312yes—
232C22×C4C22×C4×C8311yescharacteristic
233C22×C4C2×D4×C8312yes—
234C22×C4C2×D4×C8311yescharacteristic
235C22×C4#41×C8312yes—
236M16—82no—
237C2×C8—42no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—82no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—81no—
252C42—82no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—88no—
268C23C2×C22⋊C4×C8311yescharacteristic
269C23#120×C8311yescharacteristic
270C23#123×C8311yescharacteristic
271Q8—82no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4C4×D4×C8312yes—
291C2×C4#121×C8312yes—
292C8—82no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22#386×C8311yescharacteristic
309C22#421×C8312yes—
310C22#455×C8312yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—84no—
315C4—82no—
316C4—81no—
317C4—81no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2C83×#188511yescharacteristic
323C2C83×#189711yescharacteristic
324C2C83×#191311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC25×C83D4
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×#1508C2semidirect
5. C83×#1297C2semidirect
9. #397×C83C4semidirect
10. #393×C83C4semidirect
155. #9411C83direct (N × H)
165. #1508C2×C83semidirect
167. #1297C2×C83semidirect
168. C83#9411direct (N × H)
172. #397C4×C83semidirect
173. #393C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#25612
C83×#23122
C83×#15082
C83×#14622
C83×#12972
#941183