Finite Group Databasebeta

← C83×#12627 #35240451

Subgroups — C83×#12627 #35240451

Order 21248.

564 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 162 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×#2326C211yesmaximal, characteristic
2C2×#354×C83C212yesmaximal
3C4×#99×C83C212yesmaximal
4C83×#1393C212yesmaximal
5C2×#124×C83C2211yescharacteristic
6C2×C4×Q8×C83C411yescharacteristic, complemented
7C2×C4×D4×C83C2211yescharacteristic
8C2×#99×C83C2214yes—
9#391×C83C411yescharacteristic, complemented
10#390×C83C2211yescharacteristic
11#388×C83—22no—
12#385×C83—22no—
13#384×C83C411yescharacteristic, complemented
14C4×C22⋊C4×C83—22no—
15#376×C83C411yescharacteristic, complemented
16#354×C83C418yescomplemented
17#342×C83—22no—
18C22×Q8×C83C2×C411yescharacteristic
19C22×D4×C83C2311yescharacteristic
20C23×C4×C83D412yes—
21#124×C83—22no—
22#124×C83C2×C414yes—
23C4×Q8×C83—42no—
24C4×D4×C83—42no—
25C2×C4⋊C4×C83—21no—
26C2×C4⋊C4×C83—22no—
27C2×C4⋊C4×C83C2×C411yescharacteristic
28C2×C4⋊C4×C83C2×C411yescharacteristic
29C2×C22⋊C4×C83—22no—
30C2×C22⋊C4×C83—24no—
31C2×C22⋊C4×C83—22no—
32C2×C22⋊C4×C83D412yes—
33C2×C42×C83—21no—
34C2×C42×C83—42no—
35C2×C42×C83C2×C411yescharacteristic
36C2×C42×C83C2×C411yescharacteristic
37C2×C42×C83C2×C411yescharacteristic
38#99×C83—24no—
39#99×C83C2×C418yes—
40#95×C83—21no—
41#95×C83—22no—
42#95×C83—21no—
43#95×C83—42no—
44C24×C83C2×D412yes—
45C2×Q8×C83—21no—
46C2×Q8×C83—41no—
47C2×Q8×C83—41no—
48C2×Q8×C83C4212yescomplemented
49C2×D4×C83—41no—
50C2×D4×C83—41no—
51C2×D4×C83C22×C412yes—
52C2×D4×C83#4112yes—
53C22×C4×C83—21no—
54C22×C4×C83—21no—
55C22×C4×C83—22no—
56C22×C4×C83—22no—
57C22×C4×C83—21no—
58C22×C4×C83—21no—
59C22×C4×C83—44no—
60C22×C4×C83—82no—
61C22×C4×C83C22⋊C411yescharacteristic
62C22×C4×C83C22⋊C412yes—
63C22×C4×C83C22⋊C411yescharacteristic
64C22×C4×C83C22×C411yescharacteristic
65C4⋊C4×C83—41no—
66C4⋊C4×C83—41no—
67C4⋊C4×C83—42no—
68C4⋊C4×C83—82no—
69C22⋊C4×C83—28no—
70C22⋊C4×C83—28no—
71C22⋊C4×C83—42no—
72C22⋊C4×C83—44no—
73C22⋊C4×C83—82no—
74C42×C83—22no—
75C42×C83—21no—
76C42×C83—41no—
77C42×C83—42no—
78C42×C83—84no—
79C42×C83C4212yescomplemented
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—42no—
83C23×C83—42no—
84C23×C83C2×C22⋊C411yescharacteristic
85C23×C83C4×D414yes—
86Q8×C83—42no—
87Q8×C83—42no—
88D4×C83—42no—
89D4×C83—42no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—41no—
97C2×C4×C83—44no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83—41no—
101C2×C4×C83—41no—
102C2×C4×C83—42no—
103C2×C4×C83—41no—
104C2×C4×C83—48no—
105C2×C4×C83—41no—
106C2×C4×C83—42no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—82no—
111C2×C4×C83—84no—
112C2×C4×C83—84no—
113C2×C4×C83#9912yes—
114C2×C4×C83C2×C4211yescharacteristic
115C2×C4×C83#11711yescharacteristic
116C22×C83—21no—
117C22×C83—21no—
118C22×C83—24no—
119C22×C83—44no—
120C22×C83—42no—
121C22×C83—42no—
122C22×C83—82no—
123C22×C83C4×C22⋊C411yescharacteristic
124C22×C83C2×#9911yescharacteristic
125C22×C83#41011yescharacteristic
126C4×C83—22no—
127C4×C83—21no—
128C4×C83—21no—
129C4×C83—42no—
130C4×C83—42no—
131C4×C83—41no—
132C4×C83—42no—
133C4×C83—81no—
134C4×C83—88no—
135C4×C83—81no—
136#12627C8311yesmaximal, characteristic, direct product factor, Sylow
137C2×C83—21no—
138C2×C83—21no—
139C2×C83—44no—
140C2×C83C4×#9911yescharacteristic
141C2×C83#212711yescharacteristic
142C2×C83#212911yescharacteristic
143#2326C2×C8311yescharacteristic
144C2×#354C2×C8312yes—
145C4×#99C2×C8312yes—
146#1393C2×C8312yes—
147C83#1262711yescharacteristic, direct product factor, Sylow
148C2×#124C22×C8311yescharacteristic
149C2×C4×Q8C4×C8311yescharacteristic, complemented
150C2×C4×D4C22×C8311yescharacteristic
151C2×#99C22×C8314yes—
152#391C4×C8311yescharacteristic, complemented
153#390C22×C8311yescharacteristic
154#388—22no—
155#385—22no—
156#384C4×C8311yescharacteristic, complemented
157C4×C22⋊C4—22no—
158#376C4×C8311yescharacteristic, complemented
159#354C4×C8318yescomplemented
160#342—22no—
161C22×Q8C2×C4×C8311yescharacteristic
162C22×D4C23×C8311yescharacteristic
163C23×C4D4×C8312yes—
164#124—22no—
165#124C2×C4×C8314yes—
166C4×Q8—42no—
167C4×D4—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×C4×C8311yescharacteristic
171C2×C4⋊C4C2×C4×C8311yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4D4×C8312yes—
176C2×C42—21no—
177C2×C42—42no—
178C2×C42C2×C4×C8311yescharacteristic
179C2×C42C2×C4×C8311yescharacteristic
180C2×C42C2×C4×C8311yescharacteristic
181#99—24no—
182#99C2×C4×C8318yes—
183#95—21no—
184#95—21no—
185#95—22no—
186#95—42no—
187C24C2×D4×C8312yes—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C42×C8312yescomplemented
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C22×C4×C8312yes—
195C2×D4#41×C8312yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—44no—
203C22×C4—82no—
204C22×C4C22⋊C4×C8311yescharacteristic
205C22×C4C22⋊C4×C8312yes—
206C22×C4C22⋊C4×C8311yescharacteristic
207C22×C4C22×C4×C8311yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C22⋊C4—28no—
213C22⋊C4—28no—
214C22⋊C4—42no—
215C22⋊C4—44no—
216C22⋊C4—82no—
217C42—22no—
218C42—21no—
219C42—41no—
220C42—42no—
221C42—84no—
222C42C42×C8312yescomplemented
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C2×C22⋊C4×C8311yescharacteristic
228C23C4×D4×C8314yes—
229Q8—42no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—48no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4#99×C8312yes—
257C2×C4C2×C42×C8311yescharacteristic
258C2×C4#117×C8311yescharacteristic
259C22—21no—
260C22—21no—
261C22—24no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22C4×C22⋊C4×C8311yescharacteristic
267C22C2×#99×C8311yescharacteristic
268C22#410×C8311yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—41no—
276C4—81no—
277C4—88no—
278C4—81no—
279C2—21no—
280C2—21no—
281C2—44no—
282C2C4×#99×C8311yescharacteristic
283C2C83×#212711yescharacteristic
284C2C83×#212911yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC2×C4C2×C42×C83
Frattini, cosocleC22×D4C23×C83
prime radicalC22×D4×C83C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C22×C4×C83 ◁ C2×C4×D4×C83 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×C4×Q8×C83C4semidirect
9. #391×C83C4semidirect
13. #384×C83C4semidirect
15. #376×C83C4semidirect
16. #354×C83C4semidirect
48. C2×Q8×C83C42semidirect
79. C42×C83C42semidirect
136. #12627C83direct (N × H)
147. C83#12627direct (N × H)
149. C2×C4×Q8C4×C83semidirect
152. #391C4×C83semidirect
156. #384C4×C83semidirect
158. #376C4×C83semidirect
159. #354C4×C83semidirect
191. C2×Q8C42×C83semidirect
222. C42C42×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#23262
C2×#354×C832
C4×#99×C832
C83×#13932
#1262783