Finite Group Databasebeta

← C83×#34372 #35262196

Subgroups — C83×#34372 #35262196

Order 21248.

1910 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125×C83C211yesmaximal, characteristic
2C83×#2382C214yesmaximal, complemented
3C83×#2371C214yesmaximal, complemented
4C83×#2345C218yesmaximal
5C83×#2336C218yesmaximal, complemented
6C83×#2313C214yesmaximal
7C83×#2283C212yesmaximal
8C2×#125×C83C4116yescomplemented
9C2×#125×C83C2214yes—
10C2×#125×C83C2212yes—
11C2×#125×C83C2212yes—
12C22×C4⋊C4×C83C2212yes—
13C22×C4⋊C4×C83C2214yes—
14C22×C42×C83C2211yescharacteristic
15#389×C83C22116yes—
16#387×C83C22116yes—
17#387×C83C22116yes—
18#387×C83C22116yes—
19#387×C83C22116yes—
20#383×C83C2214yes—
21#382×C83C2214yes—
22#381×C83C2214yes—
23#380×C83C2214yes—
24#380×C83C2218yes—
25C4×C4⋊C4×C83C2214yes—
26C4×C4⋊C4×C83C22116yes—
27C4×C22⋊C4×C83C2214yes—
28C4×C22⋊C4×C83C2218yes—
29#376×C83C2214yes—
30C23×C4×C83C2311yescharacteristic
31C23×C4×C83C2312yes—
32C23×C4×C83C2314yes—
33#125×C83C2×C4132yes—
34#125×C83C2×C4116yes—
35#125×C83C2×C4116yes—
36C2×C4⋊C4×C83C2×C4116yes—
37C2×C4⋊C4×C83C2×C4132yes—
38C2×C4⋊C4×C83C2318yes—
39C2×C4⋊C4×C83C23116yes—
40C2×C4⋊C4×C83C2318yes—
41C2×C4⋊C4×C83C2314yes—
42C2×C4⋊C4×C83C2314yes—
43C2×C4⋊C4×C83C2318yes—
44C2×C22⋊C4×C83C2318yes—
45C2×C42×C83C2×C418yes—
46C2×C42×C83C2318yes—
47C2×C42×C83C23116yes—
48C2×C42×C83C2311yescharacteristic
49C2×C42×C83C2312yes—
50C2×C42×C83C2311yescharacteristic
51#95×C83C2318yes—
52#95×C83C2318yes—
53#95×C83C2318yes—
54#95×C83C23116yes—
55#95×C83C23116yes—
56#95×C83C2318yes—
57C24×C83C2411yescharacteristic
58C22×C4×C83—24no—
59C22×C4×C83—44no—
60C22×C4×C83—48no—
61C22×C4×C83C22×C414yes—
62C22×C4×C83C22×C418yes—
63C22×C4×C83C22×C4116yes—
64C22×C4×C83C24116yes—
65C22×C4×C83C2411yescharacteristic
66C22×C4×C83C2411yescharacteristic
67C22×C4×C83C2412yes—
68C22×C4×C83C2418yes—
69C22×C4×C83C2412yes—
70C4⋊C4×C83—416no—
71C4⋊C4×C83—48no—
72C4⋊C4×C83C22×C4132yes—
73C4⋊C4×C83C22×C4116yes—
74C4⋊C4×C83C22×C4116yes—
75C4⋊C4×C83C22×C4132yes—
76C22⋊C4×C83—216no—
77C42×C83—48no—
78C42×C83—416no—
79C42×C83C22×C414yes—
80C42×C83C22×C418yes—
81C42×C83C22×C414yes—
82C23×C83—24no—
83C23×C83C23×C412yes—
84C23×C832−1+414yes—
85C23×C83C2511yescharacteristic
86C2×C4×C83—28no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—24no—
90C2×C4×C83—216no—
91C2×C4×C83—24no—
92C2×C4×C83—216no—
93C2×C4×C83—216no—
94C2×C4×C83—216no—
95C2×C4×C83—48no—
96C2×C4×C83—416no—
97C2×C4×C83C23×C412yes—
98C2×C4×C83C23×C412yes—
99C2×C4×C83C23×C414yes—
100C2×C4×C83C23×C414yes—
101C2×C4×C83C23×C4116yes—
102C22×C83—24no—
103C22×C83—28no—
104C22×C83#51914yes—
105C22×C83C24×C411yescharacteristic
106C22×C83C2×2−1+412yes—
107C22×C83#58514yes—
108C4×C83—22no—
109C4×C83—22no—
110C4×C83—24no—
111C4×C83—216no—
112C4×C83—24no—
113C4×C83—416no—
114#34372C8311yesmaximal, characteristic, direct product factor, Sylow
115C2×C83—24no—
116C2×C83C2×#51911yescharacteristic
117C2×C83#342812yes—
118C2×C83#355314yes—
119C22×#125C2×C8311yescharacteristic
120#2382C2×C8314yescomplemented
121#2371C2×C8314yescomplemented
122#2345C2×C8318yes—
123#2336C2×C8318yescomplemented
124#2313C2×C8314yes—
125#2283C2×C8312yes—
126C83#3437211yescharacteristic, direct product factor, Sylow
127C2×#125C4×C83116yescomplemented
128C2×#125C22×C8314yes—
129C2×#125C22×C8312yes—
130C2×#125C22×C8312yes—
131C22×C4⋊C4C22×C8312yes—
132C22×C4⋊C4C22×C8314yes—
133C22×C42C22×C8311yescharacteristic
134#389C22×C83116yes—
135#387C22×C83116yes—
136#387C22×C83116yes—
137#387C22×C83116yes—
138#387C22×C83116yes—
139#383C22×C8314yes—
140#382C22×C8314yes—
141#381C22×C8314yes—
142#380C22×C8318yes—
143#380C22×C8314yes—
144C4×C4⋊C4C22×C8314yes—
145C4×C4⋊C4C22×C83116yes—
146C4×C22⋊C4C22×C8318yes—
147C4×C22⋊C4C22×C8314yes—
148#376C22×C8314yes—
149C23×C4C23×C8311yescharacteristic
150C23×C4C23×C8314yes—
151C23×C4C23×C8312yes—
152#125C2×C4×C83132yes—
153#125C2×C4×C83116yes—
154#125C2×C4×C83116yes—
155C2×C4⋊C4C2×C4×C83116yes—
156C2×C4⋊C4C2×C4×C83132yes—
157C2×C4⋊C4C23×C8318yes—
158C2×C4⋊C4C23×C83116yes—
159C2×C4⋊C4C23×C8318yes—
160C2×C4⋊C4C23×C8314yes—
161C2×C4⋊C4C23×C8314yes—
162C2×C4⋊C4C23×C8318yes—
163C2×C22⋊C4C23×C8318yes—
164C2×C42C2×C4×C8318yes—
165C2×C42C23×C8318yes—
166C2×C42C23×C83116yes—
167C2×C42C23×C8311yescharacteristic
168C2×C42C23×C8312yes—
169C2×C42C23×C8311yescharacteristic
170#95C23×C8318yes—
171#95C23×C8318yes—
172#95C23×C8318yes—
173#95C23×C83116yes—
174#95C23×C83116yes—
175#95C23×C8318yes—
176C24C24×C8311yescharacteristic
177C22×C4—24no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4C22×C4×C8314yes—
181C22×C4C22×C4×C8318yes—
182C22×C4C22×C4×C83116yes—
183C22×C4C24×C83116yes—
184C22×C4C24×C8311yescharacteristic
185C22×C4C24×C8311yescharacteristic
186C22×C4C24×C8312yes—
187C22×C4C24×C8318yes—
188C22×C4C24×C8312yes—
189C4⋊C4—416no—
190C4⋊C4—48no—
191C4⋊C4C22×C4×C83132yes—
192C4⋊C4C22×C4×C83116yes—
193C4⋊C4C22×C4×C83116yes—
194C4⋊C4C22×C4×C83132yes—
195C22⋊C4—216no—
196C42—48no—
197C42—416no—
198C42C22×C4×C8314yes—
199C42C22×C4×C8318yes—
200C42C22×C4×C8314yes—
201C23—24no—
202C23C23×C4×C8312yes—
203C232−1+4×C8314yes—
204C23C25×C8311yescharacteristic
205C2×C4—28no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—216no—
210C2×C4—24no—
211C2×C4—216no—
212C2×C4—216no—
213C2×C4—216no—
214C2×C4—48no—
215C2×C4—416no—
216C2×C4C23×C4×C8312yes—
217C2×C4C23×C4×C8312yes—
218C2×C4C23×C4×C8314yes—
219C2×C4C23×C4×C8314yes—
220C2×C4C23×C4×C83116yes—
221C22—28no—
222C22—24no—
223C22#519×C8314yes—
224C22C24×C4×C8311yescharacteristic
225C22C2×2−1+4×C8312yes—
226C22#585×C8314yes—
227C4—22no—
228C4—22no—
229C4—24no—
230C4—216no—
231C4—24no—
232C4—416no—
233C2—24no—
234C2C2×#519×C8311yescharacteristic
235C2C83×#342812yes—
236C2C83×#355314yes—
subgroupG/H
center, socleC23×C83C25
commutatorC22C24×C4×C83
Frattini, cosocleC23C25×C83
prime radicalC24×C83C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2382C2semidirect
3. C83×#2371C2semidirect
5. C83×#2336C2semidirect
8. C2×#125×C83C4semidirect
114. #34372C83direct (N × H)
120. #2382C2×C83semidirect
121. #2371C2×C83semidirect
123. #2336C2×C83semidirect
126. C83#34372direct (N × H)
127. C2×#125C4×C83semidirect

Maximal subgroups

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

subgroupindex
C22×#125×C832
C83×#23822
C83×#23712
C83×#23452
C83×#23362
C83×#23132
C83×#22832
#3437283