Finite Group Databasebeta

← C83×#8518 #35236342

Subgroups — C83×#8518 #35236342

Order 21248.

818 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#540×C83C211yesmaximal, characteristic
2C83×#1822C211yesmaximal, characteristic, complemented
3C83×#1820C211yesmaximal, characteristic, complemented
4C83×#1510C214yesmaximal, complemented
5#540×C83—28no—
6C2×#127×C83C411yescharacteristic, complemented
7C2×#125×C83C411yescharacteristic, complemented
8C2×#123×C83—22no—
9C2×#121×C83—22no—
10C2×#121×C83C2212yes—
11C2×C4×D4×C83C412yescomplemented
12C2×#117×C83C2211yescharacteristic
13#407×C83C2212yes—
14#381×C83C2212yes—
15#343×C83—22no—
16#342×C83—22no—
17#327×C83—28no—
18C22×D4×C83—22no—
19C22×D4×C83—22no—
20C22×D4×C83C2×C412yes—
21C23×C4×C83—22no—
22C23×C4×C83C2311yescharacteristic
23C2×M16×C83—22no—
24#127×C83—24no—
25#125×C83—24no—
26#123×C83—216no—
27#121×C83—216no—
28#121×C83—24no—
29#121×C83D418yes—
30C4×D4×C83—24no—
31C4×D4×C83—24no—
32#117×C83—24no—
33C2×C4⋊C4×C83—22no—
34C2×C4⋊C4×C83C2×C412yes—
35C2×C4⋊C4×C83D412yes—
36C2×C22⋊C4×C83—22no—
37C2×C22⋊C4×C83—22no—
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83D412yes—
40C2×C42×C83C2×C411yescharacteristic
41C2×C42×C83C2×C411yescharacteristic
42C22⋊C8×C83—24no—
43#95×C83—24no—
44C24×C83—22no—
45C24×C83C2×D411yescharacteristic
46C2×D4×C83—28no—
47C2×D4×C83—28no—
48C2×D4×C83—22no—
49C2×D4×C83—44no—
50C2×D4×C83—84no—
51C2×D4×C83—84no—
52C2×D4×C83C22⋊C414yes—
53C22×C4×C83—22no—
54C22×C4×C83—28no—
55C22×C4×C83—42no—
56C22×C4×C83—42no—
57C22×C4×C83—84no—
58C22×C4×C83C22⋊C412yes—
59C22×C4×C83C22⋊C412yes—
60C22×C4×C83C22×C411yescharacteristic
61C22×C4×C83C2×D411yescharacteristic
62C22×C4×C83C2×D414yes—
63M16×C83—44no—
64C2×C8×C83—42no—
65C4⋊C4×C83—28no—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—24no—
68C4⋊C4×C83C22⋊C414yes—
69C22⋊C4×C83—28no—
70C22⋊C4×C83—24no—
71C22⋊C4×C83—28no—
72C22⋊C4×C83—44no—
73C42×C83—22no—
74C42×C83—22no—
75C23×C83—22no—
76C23×C83—24no—
77C23×C83—42no—
78C23×C83—44no—
79C23×C83—44no—
80C23×C83—44no—
81C23×C83C2×C22⋊C411yescharacteristic
82C23×C83#12012yes—
83D4×C83—44no—
84D4×C83—44no—
85D4×C83—88no—
86D4×C83—88no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—24no—
92C2×C4×C83—44no—
93C2×C4×C83—44no—
94C2×C4×C83—42no—
95C2×C4×C83—42no—
96C2×C4×C83—88no—
97C2×C4×C83—82no—
98C2×C4×C83—84no—
99C2×C4×C83C2×C22⋊C412yes—
100C2×C4×C83#12012yes—
101C8×C83—82no—
102C22×C83—22no—
103C22×C83—24no—
104C22×C83—22no—
105C22×C83—44no—
106C22×C83—44no—
107C22×C83—44no—
108C22×C83—82no—
109C22×C83—88no—
110C22×C83#37911yescharacteristic
111C22×C83#41012yes—
112C4×C83—22no—
113C4×C83—22no—
114C4×C83—44no—
115C4×C83—42no—
116C4×C83—42no—
117C4×C83—162no—
118#8518C8311yesmaximal, characteristic, direct product factor, Sylow
119C2×C83—22no—
120C2×C83—42no—
121C2×C83—84no—
122C2×C83#178211yescharacteristic
123C2×C83#179512yes—
124C2×#540C2×C8311yescharacteristic
125#1822C2×C8311yescharacteristic, complemented
126#1820C2×C8311yescharacteristic, complemented
127#1510C2×C8314yescomplemented
128C83#851811yescharacteristic, direct product factor, Sylow
129#540—28no—
130C2×#127C4×C8311yescharacteristic, complemented
131C2×#125C4×C8311yescharacteristic, complemented
132C2×#123—22no—
133C2×#121—22no—
134C2×#121C22×C8312yes—
135C2×C4×D4C4×C8312yescomplemented
136C2×#117C22×C8311yescharacteristic
137#407C22×C8312yes—
138#381C22×C8312yes—
139#343—22no—
140#342—22no—
141#327—28no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4C2×C4×C8312yes—
145C23×C4—22no—
146C23×C4C23×C8311yescharacteristic
147C2×M16—22no—
148#127—24no—
149#125—24no—
150#123—216no—
151#121—216no—
152#121—24no—
153#121D4×C8318yes—
154C4×D4—24no—
155C4×D4—24no—
156#117—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×C4×C8312yes—
159C2×C4⋊C4D4×C8312yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4D4×C8312yes—
164C2×C42C2×C4×C8311yescharacteristic
165C2×C42C2×C4×C8311yescharacteristic
166C22⋊C8—24no—
167#95—24no—
168C24—22no—
169C24C2×D4×C8311yescharacteristic
170C2×D4—28no—
171C2×D4—28no—
172C2×D4—22no—
173C2×D4—44no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4C22⋊C4×C8314yes—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—84no—
182C22×C4C22⋊C4×C8312yes—
183C22×C4C22⋊C4×C8312yes—
184C22×C4C22×C4×C8311yescharacteristic
185C22×C4C2×D4×C8311yescharacteristic
186C22×C4C2×D4×C8314yes—
187M16—44no—
188C2×C8—42no—
189C4⋊C4—28no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4C22⋊C4×C8314yes—
193C22⋊C4—28no—
194C22⋊C4—24no—
195C22⋊C4—28no—
196C22⋊C4—44no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—24no—
201C23—42no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23C2×C22⋊C4×C8311yescharacteristic
206C23#120×C8312yes—
207D4—44no—
208D4—44no—
209D4—88no—
210D4—88no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—88no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4C2×C22⋊C4×C8312yes—
224C2×C4#120×C8312yes—
225C8—82no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—82no—
233C22—88no—
234C22#379×C8311yescharacteristic
235C22#410×C8312yes—
236C4—22no—
237C4—22no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—162no—
242C2—22no—
243C2—42no—
244C2—84no—
245C2C83×#178211yescharacteristic
246C2C83×#179512yes—
subgroupG/H
center, socleC22×C83#379
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC2×#540×C83C2
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
2. C83×#1822C2semidirect
3. C83×#1820C2semidirect
4. C83×#1510C2semidirect
6. C2×#127×C83C4semidirect
7. C2×#125×C83C4semidirect
11. C2×C4×D4×C83C4semidirect
118. #8518C83direct (N × H)
125. #1822C2×C83semidirect
126. #1820C2×C83semidirect
127. #1510C2×C83semidirect
128. C83#8518direct (N × H)
130. C2×#127C4×C83semidirect
131. C2×#125C4×C83semidirect
135. C2×C4×D4C4×C83semidirect

Maximal subgroups

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

subgroupindex
C2×#540×C832
C83×#18222
C83×#18202
C83×#15102
#851883