Finite Group Databasebeta

← C83×#11481 #35239305

Subgroups — C83×#11481 #35239305

Order 21248.

460 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 130 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
2C83×#1459C212yesmaximal
3C83×#1325C212yesmaximal, complemented
4C83×#1319C212yesmaximal, complemented
5C2×#124×C83C411yescharacteristic
6C2×C4×Q8×C83C411yescharacteristic
7C2×C4×D4×C83C2211yescharacteristic
8C2×C22⋊C8×C83—22no—
9#391×C83C2211yescharacteristic
10#390×C83C411yescharacteristic
11#388×C83—22no—
12#385×C83—22no—
13#384×C83C2211yescharacteristic
14#376×C83C411yescharacteristic
15#324×C83C2214yes—
16#323×C83—22no—
17C22×Q8×C83C2×C411yescharacteristic
18C22×D4×C83C2×C411yescharacteristic
19C23×C4×C83D412yes—
20C22×C8×C83—42no—
21#124×C83—22no—
22#124×C83C814yescomplemented
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×C83C2311yescharacteristic
29C2×C22⋊C4×C83—22no—
30C2×C22⋊C4×C83—22no—
31C2×C22⋊C4×C83D412yes—
32C2×C42×C83—21no—
33C2×C42×C83C2×C411yescharacteristic
34C2×C42×C83C2×C411yescharacteristic
35C2×C42×C83C2×C411yescharacteristic
36C22⋊C8×C83—24no—
37C22⋊C8×C83—42no—
38#95×C83—21no—
39#95×C83—21no—
40#95×C83—22no—
41C24×C83C22⋊C412yes—
42C2×Q8×C83—21no—
43C2×Q8×C83—41no—
44C2×Q8×C83—41no—
45C2×Q8×C83C2×C812yescomplemented
46C2×D4×C83—41no—
47C2×D4×C83—41no—
48C2×D4×C83C2×C812yes—
49C2×D4×C83M1612yes—
50C22×C4×C83—21no—
51C22×C4×C83—21no—
52C22×C4×C83—22no—
53C22×C4×C83—22no—
54C22×C4×C83—21no—
55C22×C4×C83—21no—
56C22×C4×C83—82no—
57C22×C4×C83C22⋊C411yescharacteristic
58C22×C4×C83C22⋊C411yescharacteristic
59C22×C4×C83C22×C411yescharacteristic
60C22×C4×C83C2×D412yes—
61C2×C8×C83—44no—
62C2×C8×C83—84no—
63C4⋊C4×C83—41no—
64C4⋊C4×C83—41no—
65C4⋊C4×C83—42no—
66C4⋊C4×C83—82no—
67C22⋊C4×C83—28no—
68C22⋊C4×C83—42no—
69C22⋊C4×C83—82no—
70C42×C83—22no—
71C42×C83—21no—
72C42×C83—41no—
73C42×C83—42no—
74C42×C83C2×C812yescomplemented
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—42no—
78C23×C83—42no—
79C23×C83C22⋊C814yes—
80C23×C83C2×C22⋊C411yescharacteristic
81Q8×C83—42no—
82Q8×C83—42no—
83D4×C83—42no—
84D4×C83—42no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—41no—
92C2×C4×C83—44no—
93C2×C4×C83—41no—
94C2×C4×C83—41no—
95C2×C4×C83—41no—
96C2×C4×C83—41no—
97C2×C4×C83—42no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83—42no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—82no—
105C2×C4×C83#9912yes—
106C2×C4×C83C22×C811yescharacteristic
107C2×C4×C83C2×M1611yescharacteristic
108C8×C83—84no—
109C22×C83—21no—
110C22×C83—21no—
111C22×C83—24no—
112C22×C83—44no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—82no—
116C22×C83C2×C22⋊C811yescharacteristic
117C22×C83C2×#9911yescharacteristic
118C22×C83#41311yescharacteristic
119C4×C83—22no—
120C4×C83—21no—
121C4×C83—21no—
122C4×C83—42no—
123C4×C83—42no—
124C4×C83—41no—
125C4×C83—42no—
126C4×C83—81no—
127C4×C83—81no—
128#11481C8311yesmaximal, characteristic, direct product factor, Sylow
129C2×C83—21no—
130C2×C83—21no—
131C2×C83—44no—
132C2×C83#145911yescharacteristic
133C2×C83#212711yescharacteristic
134C2×C83#213511yescharacteristic
135#2326C2×C8311yescharacteristic
136#1459C2×C8312yes—
137#1325C2×C8312yescomplemented
138#1319C2×C8312yescomplemented
139C83#1148111yescharacteristic, direct product factor, Sylow
140C2×#124C4×C8311yescharacteristic
141C2×C4×Q8C4×C8311yescharacteristic
142C2×C4×D4C22×C8311yescharacteristic
143C2×C22⋊C8—22no—
144#391C22×C8311yescharacteristic
145#390C4×C8311yescharacteristic
146#388—22no—
147#385—22no—
148#384C22×C8311yescharacteristic
149#376C4×C8311yescharacteristic
150#324C22×C8314yes—
151#323—22no—
152C22×Q8C2×C4×C8311yescharacteristic
153C22×D4C2×C4×C8311yescharacteristic
154C23×C4D4×C8312yes—
155C22×C8—42no—
156#124—22no—
157#124C8×C8314yescomplemented
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×C4×C8311yescharacteristic
163C2×C4⋊C4C23×C8311yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4D4×C8312yes—
167C2×C42—21no—
168C2×C42C2×C4×C8311yescharacteristic
169C2×C42C2×C4×C8311yescharacteristic
170C2×C42C2×C4×C8311yescharacteristic
171C22⋊C8—24no—
172C22⋊C8—42no—
173#95—21no—
174#95—21no—
175#95—22no—
176C24C22⋊C4×C8312yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C2×C8×C8312yescomplemented
181C2×D4—41no—
182C2×D4—41no—
183C2×D4C2×C8×C8312yes—
184C2×D4M16×C8312yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊C4×C8311yescharacteristic
193C22×C4C22⋊C4×C8311yescharacteristic
194C22×C4C22×C4×C8311yescharacteristic
195C22×C4C2×D4×C8312yes—
196C2×C8—44no—
197C2×C8—84no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42C2×C8×C8312yescomplemented
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23C22⋊C8×C8314yes—
215C23C2×C22⋊C4×C8311yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#99×C8312yes—
241C2×C4C22×C8×C8311yescharacteristic
242C2×C4C2×M16×C8311yescharacteristic
243C8—84no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C2×C22⋊C8×C8311yescharacteristic
252C22C2×#99×C8311yescharacteristic
253C22#413×C8311yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—41no—
260C4—42no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—44no—
266C2C83×#145911yescharacteristic
267C2C83×#212711yescharacteristic
268C2C83×#213511yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC2×C4C22×C8×C83
Frattini, cosocleC2×C4⋊C4C23×C83
prime radicalC22×D4×C83C2×C4
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
3. C83×#1325C2semidirect
4. C83×#1319C2semidirect
22. #124×C83C8semidirect
45. C2×Q8×C83C2×C8semidirect
74. C42×C83C2×C8semidirect
128. #11481C83direct (N × H)
137. #1325C2×C83semidirect
138. #1319C2×C83semidirect
139. C83#11481direct (N × H)
157. #124C8×C83semidirect
180. C2×Q8C2×C8×C83semidirect
209. C42C2×C8×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#23262
C83×#14592
C83×#13252
C83×#13192
#1148183