Finite Group Databasebeta

← C23×#34372 #19181998

Subgroups — C23×#34372 #19181998

Order 5888.

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×C23×#125C211yesmaximal, characteristic
2C23×#2382C214yesmaximal, complemented
3C23×#2371C214yesmaximal, complemented
4C23×#2345C218yesmaximal
5C23×#2336C218yesmaximal, complemented
6C23×#2313C214yesmaximal
7C23×#2283C212yesmaximal
8C2×C23×#125C4116yescomplemented
9C2×C23×#125C2214yes—
10C2×C23×#125C2212yes—
11C2×C23×#125C2212yes—
12C22×C4⋊C4×C23C2212yes—
13C22×C4⋊C4×C23C2214yes—
14C22×C42×C23C2211yescharacteristic
15C23×#389C22116yes—
16C23×#387C22116yes—
17C23×#387C22116yes—
18C23×#387C22116yes—
19C23×#387C22116yes—
20C23×#383C2214yes—
21C23×#382C2214yes—
22C23×#381C2214yes—
23C23×#380C2214yes—
24C23×#380C2218yes—
25C4×C4⋊C4×C23C2214yes—
26C4×C4⋊C4×C23C22116yes—
27C4×C22⋊C4×C23C2214yes—
28C4×C22⋊C4×C23C2218yes—
29C23×#376C2214yes—
30C23×C4×C23C2311yescharacteristic
31C23×C4×C23C2312yes—
32C23×C4×C23C2314yes—
33C23×#125C2×C4132yes—
34C23×#125C2×C4116yes—
35C23×#125C2×C4116yes—
36C2×C4⋊C4×C23C2×C4116yes—
37C2×C4⋊C4×C23C2×C4132yes—
38C2×C4⋊C4×C23C2318yes—
39C2×C4⋊C4×C23C23116yes—
40C2×C4⋊C4×C23C2318yes—
41C2×C4⋊C4×C23C2314yes—
42C2×C4⋊C4×C23C2314yes—
43C2×C4⋊C4×C23C2318yes—
44C2×C22⋊C4×C23C2318yes—
45C2×C42×C23C2×C418yes—
46C2×C42×C23C2318yes—
47C2×C42×C23C23116yes—
48C2×C42×C23C2311yescharacteristic
49C2×C42×C23C2312yes—
50C2×C42×C23C2311yescharacteristic
51C23×#95C2318yes—
52C23×#95C2318yes—
53C23×#95C2318yes—
54C23×#95C23116yes—
55C23×#95C23116yes—
56C23×#95C2318yes—
57C24×C23C2411yescharacteristic
58C22×C4×C23—24no—
59C22×C4×C23—44no—
60C22×C4×C23—48no—
61C22×C4×C23C22×C414yes—
62C22×C4×C23C22×C418yes—
63C22×C4×C23C22×C4116yes—
64C22×C4×C23C24116yes—
65C22×C4×C23C2412yes—
66C22×C4×C23C2418yes—
67C22×C4×C23C2412yes—
68C22×C4×C23C2411yescharacteristic
69C22×C4×C23C2411yescharacteristic
70C4⋊C4×C23—416no—
71C4⋊C4×C23—48no—
72C4⋊C4×C23C22×C4132yes—
73C4⋊C4×C23C22×C4116yes—
74C4⋊C4×C23C22×C4116yes—
75C4⋊C4×C23C22×C4132yes—
76C22⋊C4×C23—216no—
77C42×C23—48no—
78C42×C23—416no—
79C42×C23C22×C414yes—
80C42×C23C22×C418yes—
81C42×C23C22×C414yes—
82#34372C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—24no—
84C23×C23C23×C412yes—
85C23×C232−1+414yes—
86C23×C23C2511yescharacteristic
87C2×C4×C23—28no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—24no—
91C2×C4×C23—216no—
92C2×C4×C23—24no—
93C2×C4×C23—216no—
94C2×C4×C23—216no—
95C2×C4×C23—216no—
96C2×C4×C23—48no—
97C2×C4×C23—416no—
98C2×C4×C23C23×C412yes—
99C2×C4×C23C23×C412yes—
100C2×C4×C23C23×C414yes—
101C2×C4×C23C23×C414yes—
102C2×C4×C23C23×C4116yes—
103C22×#125C2×C2311yescharacteristic
104#2382C2×C2314yescomplemented
105#2371C2×C2314yescomplemented
106#2345C2×C2318yes—
107#2336C2×C2318yescomplemented
108#2313C2×C2314yes—
109#2283C2×C2312yes—
110C22×C23—24no—
111C22×C23—28no—
112C22×C23#51914yes—
113C22×C23C24×C411yescharacteristic
114C22×C23C2×2−1+412yes—
115C22×C23#58514yes—
116C4×C23—22no—
117C4×C23—22no—
118C4×C23—24no—
119C4×C23—216no—
120C4×C23—24no—
121C4×C23—416no—
122C2×#125C4×C23116yescomplemented
123C2×#125C22×C2314yes—
124C2×#125C22×C2312yes—
125C2×#125C22×C2312yes—
126C22×C4⋊C4C22×C2312yes—
127C22×C4⋊C4C22×C2314yes—
128C22×C42C22×C2311yescharacteristic
129#389C22×C23116yes—
130#387C22×C23116yes—
131#387C22×C23116yes—
132#387C22×C23116yes—
133#387C22×C23116yes—
134#383C22×C2314yes—
135#382C22×C2314yes—
136#381C22×C2314yes—
137#380C22×C2318yes—
138#380C22×C2314yes—
139C4×C4⋊C4C22×C2314yes—
140C4×C4⋊C4C22×C23116yes—
141C4×C22⋊C4C22×C2318yes—
142C4×C22⋊C4C22×C2314yes—
143#376C22×C2314yes—
144C2×C23—24no—
145C2×C23C2×#51911yescharacteristic
146C2×C23#342812yes—
147C2×C23#355314yes—
148C23×C4C23×C2311yescharacteristic
149C23×C4C23×C2314yes—
150C23×C4C23×C2312yes—
151#125C2×C4×C23132yes—
152#125C2×C4×C23116yes—
153#125C2×C4×C23116yes—
154C2×C4⋊C4C2×C4×C23116yes—
155C2×C4⋊C4C2×C4×C23132yes—
156C2×C4⋊C4C23×C2318yes—
157C2×C4⋊C4C23×C23116yes—
158C2×C4⋊C4C23×C2318yes—
159C2×C4⋊C4C23×C2314yes—
160C2×C4⋊C4C23×C2314yes—
161C2×C4⋊C4C23×C2318yes—
162C2×C22⋊C4C23×C2318yes—
163C2×C42C2×C4×C2318yes—
164C2×C42C23×C2318yes—
165C2×C42C23×C23116yes—
166C2×C42C23×C2311yescharacteristic
167C2×C42C23×C2312yes—
168C2×C42C23×C2311yescharacteristic
169#95C23×C2318yes—
170#95C23×C2318yes—
171#95C23×C2318yes—
172#95C23×C23116yes—
173#95C23×C23116yes—
174#95C23×C2318yes—
175C23#3437211yescharacteristic, direct product factor, Sylow
176C24C24×C2311yescharacteristic
177C22×C4—24no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4C22×C4×C2314yes—
181C22×C4C22×C4×C2318yes—
182C22×C4C22×C4×C23116yes—
183C22×C4C24×C23116yes—
184C22×C4C24×C2311yescharacteristic
185C22×C4C24×C2311yescharacteristic
186C22×C4C24×C2312yes—
187C22×C4C24×C2318yes—
188C22×C4C24×C2312yes—
189C4⋊C4—416no—
190C4⋊C4—48no—
191C4⋊C4C22×C4×C23132yes—
192C4⋊C4C22×C4×C23116yes—
193C4⋊C4C22×C4×C23116yes—
194C4⋊C4C22×C4×C23132yes—
195C22⋊C4—216no—
196C42—48no—
197C42—416no—
198C42C22×C4×C2314yes—
199C42C22×C4×C2318yes—
200C42C22×C4×C2314yes—
201C23—24no—
202C23C23×C4×C2312yes—
203C23C23×2−1+414yes—
204C23C25×C2311yescharacteristic
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—216no—
209C2×C4—24no—
210C2×C4—28no—
211C2×C4—216no—
212C2×C4—216no—
213C2×C4—216no—
214C2×C4—48no—
215C2×C4—416no—
216C2×C4C23×C4×C2312yes—
217C2×C4C23×C4×C2312yes—
218C2×C4C23×C4×C2314yes—
219C2×C4C23×C4×C2314yes—
220C2×C4C23×C4×C23116yes—
221C22—24no—
222C22—28no—
223C22C23×#51914yes—
224C22C24×C4×C2311yescharacteristic
225C22C2×C23×2−1+412yes—
226C22C23×#58514yes—
227C4—22no—
228C4—22no—
229C4—24no—
230C4—216no—
231C4—24no—
232C4—416no—
233C2—24no—
234C2C2×C23×#51911yescharacteristic
235C2C23×#342812yes—
236C2C23×#355314yes—
subgroupG/H
center, socleC23×C23C25
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
prime radicalC24×C23C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2382C2semidirect
3. C23×#2371C2semidirect
5. C23×#2336C2semidirect
8. C2×C23×#125C4semidirect
82. #34372C23direct (N × H)
104. #2382C2×C23semidirect
105. #2371C2×C23semidirect
107. #2336C2×C23semidirect
122. C2×#125C4×C23semidirect
175. C23#34372direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1252
C23×#23822
C23×#23712
C23×#23452
C23×#23362
C23×#23132
C23×#22832
#3437223