Finite Group Databasebeta

← C23×#34375 #19182001

Subgroups — C23×#34375 #19182001

Order 5888.

1898 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#533C211yesmaximal, characteristic
2C23×#2381C214yesmaximal, complemented
3C23×#2371C214yesmaximal, complemented
4C23×#2355C218yesmaximal, complemented
5C23×#2332C218yesmaximal
6C23×#2314C214yesmaximal
7C23×#2283C212yesmaximal
8C23×#533C4116yescomplemented
9C2×C23×C4⋊Q8C2212yes—
10C2×C23×#125C2212yes—
11C2×C23×C22⋊Q8C2214yes—
12C2×C4×Q8×C23C2214yes—
13C2×C23×#117C2212yes—
14C22×C42×C23C2211yescharacteristic
15C23×#391C22116yes—
16C23×#388C22116yes—
17C23×#387C22116yes—
18C23×#387C22116yes—
19C23×#387C22116yes—
20C23×#384C2214yes—
21C23×#383C2214yes—
22C23×#382C2214yes—
23C23×#382C2214yes—
24C23×#381C2214yes—
25C23×#380C2214yes—
26C4×C4⋊C4×C23C2214yes—
27C4×C4⋊C4×C23C2218yes—
28C4×C4⋊C4×C23C22116yes—
29C4×C22⋊C4×C23C2214yes—
30C23×#376C2214yes—
31C22×Q8×C23C2314yes—
32C23×C4×C23C2311yescharacteristic
33C23×C4×C23C2312yes—
34C23×C4⋊Q8C2×C4116yes—
35C23×#125C2×C4116yes—
36C23×C22⋊Q8C2×C4132yes—
37C4×Q8×C23C2×C4132yes—
38C23×#117C2×C4116yes—
39C2×C4⋊C4×C23C2318yes—
40C2×C4⋊C4×C23C23116yes—
41C2×C4⋊C4×C23C2318yes—
42C2×C4⋊C4×C23C2314yes—
43C2×C4⋊C4×C23C2314yes—
44C2×C22⋊C4×C23C2318yes—
45C2×C22⋊C4×C23C2314yes—
46C2×C42×C23C2×C418yes—
47C2×C42×C23C2318yes—
48C2×C42×C23C23116yes—
49C2×C42×C23C2311yescharacteristic
50C2×C42×C23C2312yes—
51C2×C42×C23C2314yes—
52C2×C42×C23C2311yescharacteristic
53C23×#95C2318yes—
54C23×#95C2318yes—
55C23×#95C2318yes—
56C23×#95C2318yes—
57C23×#95C23116yes—
58C23×#95C23116yes—
59C24×C23C2411yescharacteristic
60C2×Q8×C23—44no—
61C2×Q8×C23—44no—
62C2×Q8×C23C22×C4116yes—
63C22×C4×C23—24no—
64C22×C4×C23—44no—
65C22×C4×C23C22×C414yes—
66C22×C4×C23C22×C418yes—
67C22×C4×C23C24116yes—
68C22×C4×C23C2412yes—
69C22×C4×C23C2418yes—
70C22×C4×C23C2411yescharacteristic
71C22×C4×C23C2412yes—
72C22×C4×C23C2411yescharacteristic
73C4⋊C4×C23—216no—
74C4⋊C4×C23—416no—
75C4⋊C4×C23C22×C4116yes—
76C4⋊C4×C23C22×C4132yes—
77C4⋊C4×C23C22×C4116yes—
78C22⋊C4×C23—48no—
79C22⋊C4×C23C22×C4116yes—
80C42×C23—48no—
81C42×C23—416no—
82C42×C23C22×C418yes—
83C42×C23C22×C414yes—
84C42×C23C22×C414yes—
85C42×C23C22×C4116yes—
86#34375C2311yesmaximal, characteristic, direct product factor, Sylow
87C23×C23—22no—
88C23×C23—24no—
89C23×C23C23×C412yes—
90C23×C23C2511yescharacteristic
91Q8×C23—416no—
92C2×C4×C23—22no—
93C2×C4×C23—24no—
94C2×C4×C23—28no—
95C2×C4×C23—28no—
96C2×C4×C23—24no—
97C2×C4×C23—28no—
98C2×C4×C23—216no—
99C2×C4×C23—216no—
100C2×C4×C23—216no—
101C2×C4×C23—48no—
102C2×C4×C23C23×C412yes—
103C2×C4×C23C23×C414yes—
104C2×C4×C23C23×C414yes—
105C2×C4×C23C23×C4116yes—
106C2×C4×C23C23×C412yes—
107C2×C4×C232−1+414yes—
108C2×#533C2×C2311yescharacteristic
109#2381C2×C2314yescomplemented
110#2371C2×C2314yescomplemented
111#2355C2×C2318yescomplemented
112#2332C2×C2318yes—
113#2314C2×C2314yes—
114#2283C2×C2312yes—
115C22×C23—22no—
116C22×C23—24no—
117C22×C23—44no—
118C22×C23C24×C411yescharacteristic
119C22×C23C2×2−1+412yes—
120C22×C23#58514yes—
121C4×C23—22no—
122C4×C23—24no—
123C4×C23—24no—
124C4×C23—216no—
125C4×C23—416no—
126C4×C23#51914yes—
127#533C4×C23116yescomplemented
128C2×C4⋊Q8C22×C2312yes—
129C2×#125C22×C2312yes—
130C2×C22⋊Q8C22×C2314yes—
131C2×C4×Q8C22×C2314yes—
132C2×#117C22×C2312yes—
133C22×C42C22×C2311yescharacteristic
134#391C22×C23116yes—
135#388C22×C23116yes—
136#387C22×C23116yes—
137#387C22×C23116yes—
138#387C22×C23116yes—
139#384C22×C2314yes—
140#383C22×C2314yes—
141#382C22×C2314yes—
142#382C22×C2314yes—
143#381C22×C2314yes—
144#380C22×C2314yes—
145C4×C4⋊C4C22×C2318yes—
146C4×C4⋊C4C22×C23116yes—
147C4×C4⋊C4C22×C2314yes—
148C4×C22⋊C4C22×C2314yes—
149#376C22×C2314yes—
150C2×C23—42no—
151C2×C23C2×#51911yescharacteristic
152C2×C23#342812yes—
153C2×C23#355714yes—
154C22×Q8C23×C2314yes—
155C23×C4C23×C2311yescharacteristic
156C23×C4C23×C2312yes—
157C4⋊Q8C2×C4×C23116yes—
158#125C2×C4×C23116yes—
159C22⋊Q8C2×C4×C23132yes—
160C4×Q8C2×C4×C23132yes—
161#117C2×C4×C23116yes—
162C2×C4⋊C4C23×C2318yes—
163C2×C4⋊C4C23×C23116yes—
164C2×C4⋊C4C23×C2314yes—
165C2×C4⋊C4C23×C2318yes—
166C2×C4⋊C4C23×C2314yes—
167C2×C22⋊C4C23×C2318yes—
168C2×C22⋊C4C23×C2314yes—
169C2×C42C2×C4×C2318yes—
170C2×C42C23×C2318yes—
171C2×C42C23×C23116yes—
172C2×C42C23×C2314yes—
173C2×C42C23×C2312yes—
174C2×C42C23×C2311yescharacteristic
175C2×C42C23×C2311yescharacteristic
176#95C23×C2318yes—
177#95C23×C2318yes—
178#95C23×C2318yes—
179#95C23×C2318yes—
180#95C23×C23116yes—
181#95C23×C23116yes—
182C23#3437511yescharacteristic, direct product factor, Sylow
183C24C24×C2311yescharacteristic
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8C22×C4×C23116yes—
187C22×C4—24no—
188C22×C4—44no—
189C22×C4C22×C4×C2314yes—
190C22×C4C22×C4×C2318yes—
191C22×C4C24×C23116yes—
192C22×C4C24×C2311yescharacteristic
193C22×C4C24×C2312yes—
194C22×C4C24×C2312yes—
195C22×C4C24×C2318yes—
196C22×C4C24×C2311yescharacteristic
197C4⋊C4—216no—
198C4⋊C4—416no—
199C4⋊C4C22×C4×C23116yes—
200C4⋊C4C22×C4×C23132yes—
201C4⋊C4C22×C4×C23116yes—
202C22⋊C4—48no—
203C22⋊C4C22×C4×C23116yes—
204C42—48no—
205C42—416no—
206C42C22×C4×C2318yes—
207C42C22×C4×C2314yes—
208C42C22×C4×C2314yes—
209C42C22×C4×C23116yes—
210C23—22no—
211C23—24no—
212C23C23×C4×C2312yes—
213C23C25×C2311yescharacteristic
214Q8—416no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—28no—
218C2×C4—28no—
219C2×C4—24no—
220C2×C4—28no—
221C2×C4—216no—
222C2×C4—216no—
223C2×C4—216no—
224C2×C4—48no—
225C2×C4C23×C4×C2312yes—
226C2×C4C23×C4×C2314yes—
227C2×C4C23×C4×C2314yes—
228C2×C4C23×C4×C23116yes—
229C2×C4C23×C4×C2312yes—
230C2×C4C23×2−1+414yes—
231C22—22no—
232C22—24no—
233C22—44no—
234C22C24×C4×C2311yescharacteristic
235C22C2×C23×2−1+412yes—
236C22C23×#58514yes—
237C4—22no—
238C4—24no—
239C4—24no—
240C4—216no—
241C4—416no—
242C4C23×#51914yes—
243C2—42no—
244C2C2×C23×#51911yescharacteristic
245C2C23×#342812yes—
246C2C23×#355714yes—
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×#2381C2semidirect
3. C23×#2371C2semidirect
4. C23×#2355C2semidirect
8. C23×#533C4semidirect
86. #34375C23direct (N × H)
109. #2381C2×C23semidirect
110. #2371C2×C23semidirect
111. #2355C2×C23semidirect
127. #533C4×C23semidirect
182. C23#34375direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5332
C23×#23812
C23×#23712
C23×#23552
C23×#23322
C23×#23142
C23×#22832
#3437523