Finite Group Databasebeta

← C23×#34348 #19181974

Subgroups — C23×#34348 #19181974

Order 5888.

1898 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#533C211yesmaximal, characteristic
2C23×#2382C214yesmaximal, complemented
3C23×#2381C214yesmaximal, complemented
4C23×#2378C218yesmaximal
5C23×#2298C214yesmaximal
6C23×#2295C218yesmaximal, complemented
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×#389C22116yes—
17C23×#388C22116yes—
18C23×#387C22116yes—
19C23×#387C22116yes—
20C23×#387C22116yes—
21C23×#383C2214yes—
22C23×#382C2214yes—
23C23×#381C2214yes—
24C23×#380C2214yes—
25C4×C4⋊C4×C23C2214yes—
26C4×C4⋊C4×C23C2218yes—
27C4×C22⋊C4×C23C2214yes—
28C23×#376C2214yes—
29C23×#376C2214yes—
30C43×C23C2214yes—
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×C23C2314yes—
42C2×C4⋊C4×C23C2314yes—
43C2×C4⋊C4×C23C2318yes—
44C2×C22⋊C4×C23C2318yes—
45C2×C22⋊C4×C23C2314yes—
46C2×C42×C23C2×C418yes—
47C2×C42×C23C2318yes—
48C2×C42×C23C23116yes—
49C2×C42×C23C2311yescharacteristic
50C2×C42×C23C2311yescharacteristic
51C2×C42×C23C2314yes—
52C2×C42×C23C2312yes—
53C23×#95C2318yes—
54C23×#95C2318yes—
55C23×#95C23116yes—
56C23×#95C2318yes—
57C23×#95C2318yes—
58C23×#95C23116yes—
59C24×C23C2411yescharacteristic
60C2×Q8×C23—48no—
61C2×Q8×C23C22×C4116yes—
62C22×C4×C23—24no—
63C22×C4×C23—44no—
64C22×C4×C23C22×C414yes—
65C22×C4×C23C22×C418yes—
66C22×C4×C23C24116yes—
67C22×C4×C23C2418yes—
68C22×C4×C23C2412yes—
69C22×C4×C23C2411yescharacteristic
70C22×C4×C23C2412yes—
71C22×C4×C23C2411yescharacteristic
72C4⋊C4×C23—416no—
73C4⋊C4×C23—48no—
74C4⋊C4×C23C22×C4116yes—
75C4⋊C4×C23C22×C4132yes—
76C4⋊C4×C23C22×C4116yes—
77C22⋊C4×C23—48no—
78C22⋊C4×C23C22×C4116yes—
79C42×C23—216no—
80C42×C23—416no—
81C42×C23C22×C418yes—
82C42×C23C22×C414yes—
83C42×C23C22×C414yes—
84C42×C23C22×C4116yes—
85#34348C2311yesmaximal, characteristic, direct product factor, Sylow
86C23×C23—22no—
87C23×C23—24no—
88C23×C23C23×C412yes—
89C23×C23C2511yescharacteristic
90Q8×C23—416no—
91C2×C4×C23—22no—
92C2×C4×C23—216no—
93C2×C4×C23—24no—
94C2×C4×C23—24no—
95C2×C4×C23—28no—
96C2×C4×C23—216no—
97C2×C4×C23—216no—
98C2×C4×C23—216no—
99C2×C4×C23—48no—
100C2×C4×C23C23×C412yes—
101C2×C4×C23C23×C414yes—
102C2×C4×C23C23×C414yes—
103C2×C4×C23C23×C4116yes—
104C2×C4×C23C23×C412yes—
105C2×C4×C232−1+414yes—
106C2×#533C2×C2311yescharacteristic
107#2382C2×C2314yescomplemented
108#2381C2×C2314yescomplemented
109#2378C2×C2318yes—
110#2298C2×C2314yes—
111#2295C2×C2318yescomplemented
112#2283C2×C2312yes—
113C22×C23—22no—
114C22×C23—24no—
115C22×C23—44no—
116C22×C23C24×C411yescharacteristic
117C22×C23C2×2−1+412yes—
118C22×C23#58514yes—
119C4×C23—22no—
120C4×C23—24no—
121C4×C23—24no—
122C4×C23—216no—
123C4×C23—416no—
124C4×C23#51914yes—
125#533C4×C23116yescomplemented
126C2×C4⋊Q8C22×C2312yes—
127C2×#125C22×C2312yes—
128C2×C22⋊Q8C22×C2314yes—
129C2×C4×Q8C22×C2314yes—
130C2×#117C22×C2312yes—
131C22×C42C22×C2311yescharacteristic
132#391C22×C23116yes—
133#389C22×C23116yes—
134#388C22×C23116yes—
135#387C22×C23116yes—
136#387C22×C23116yes—
137#387C22×C23116yes—
138#383C22×C2314yes—
139#382C22×C2314yes—
140#381C22×C2314yes—
141#380C22×C2314yes—
142C4×C4⋊C4C22×C2314yes—
143C4×C4⋊C4C22×C2318yes—
144C4×C22⋊C4C22×C2314yes—
145#376C22×C2314yes—
146#376C22×C2314yes—
147C43C22×C2314yes—
148C2×C23—42no—
149C2×C23C2×#51911yescharacteristic
150C2×C23#342812yes—
151C2×C23#353114yes—
152C22×Q8C23×C2314yes—
153C23×C4C23×C2311yescharacteristic
154C23×C4C23×C2312yes—
155C4⋊Q8C2×C4×C23116yes—
156#125C2×C4×C23116yes—
157C22⋊Q8C2×C4×C23132yes—
158C4×Q8C2×C4×C23132yes—
159#117C2×C4×C23116yes—
160C2×C4⋊C4C23×C2318yes—
161C2×C4⋊C4C23×C23116yes—
162C2×C4⋊C4C23×C2314yes—
163C2×C4⋊C4C23×C2318yes—
164C2×C4⋊C4C23×C2314yes—
165C2×C22⋊C4C23×C2318yes—
166C2×C22⋊C4C23×C2314yes—
167C2×C42C2×C4×C2318yes—
168C2×C42C23×C2318yes—
169C2×C42C23×C23116yes—
170C2×C42C23×C2314yes—
171C2×C42C23×C2312yes—
172C2×C42C23×C2311yescharacteristic
173C2×C42C23×C2311yescharacteristic
174#95C23×C2318yes—
175#95C23×C2318yes—
176#95C23×C23116yes—
177#95C23×C2318yes—
178#95C23×C2318yes—
179#95C23×C23116yes—
180C23#3434811yescharacteristic, direct product factor, Sylow
181C24C24×C2311yescharacteristic
182C2×Q8—48no—
183C2×Q8C22×C4×C23116yes—
184C22×C4—24no—
185C22×C4—44no—
186C22×C4C22×C4×C2314yes—
187C22×C4C22×C4×C2318yes—
188C22×C4C24×C23116yes—
189C22×C4C24×C2311yescharacteristic
190C22×C4C24×C2312yes—
191C22×C4C24×C2318yes—
192C22×C4C24×C2312yes—
193C22×C4C24×C2311yescharacteristic
194C4⋊C4—416no—
195C4⋊C4—48no—
196C4⋊C4C22×C4×C23116yes—
197C4⋊C4C22×C4×C23132yes—
198C4⋊C4C22×C4×C23116yes—
199C22⋊C4—48no—
200C22⋊C4C22×C4×C23116yes—
201C42—216no—
202C42—416no—
203C42C22×C4×C2318yes—
204C42C22×C4×C2314yes—
205C42C22×C4×C2314yes—
206C42C22×C4×C23116yes—
207C23—22no—
208C23—24no—
209C23C23×C4×C2312yes—
210C23C25×C2311yescharacteristic
211Q8—416no—
212C2×C4—22no—
213C2×C4—216no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—28no—
217C2×C4—216no—
218C2×C4—216no—
219C2×C4—216no—
220C2×C4—48no—
221C2×C4C23×C4×C2312yes—
222C2×C4C23×C4×C2314yes—
223C2×C4C23×C4×C2314yes—
224C2×C4C23×C4×C23116yes—
225C2×C4C23×C4×C2312yes—
226C2×C4C23×2−1+414yes—
227C22—22no—
228C22—24no—
229C22—44no—
230C22C24×C4×C2311yescharacteristic
231C22C2×C23×2−1+412yes—
232C22C23×#58514yes—
233C4—22no—
234C4—24no—
235C4—24no—
236C4—216no—
237C4—416no—
238C4C23×#51914yes—
239C2—42no—
240C2C2×C23×#51911yescharacteristic
241C2C23×#342812yes—
242C2C23×#353114yes—
subgroupG/H
centerC22×C4×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC24×C23C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2382C2semidirect
3. C23×#2381C2semidirect
6. C23×#2295C2semidirect
8. C23×#533C4semidirect
85. #34348C23direct (N × H)
107. #2382C2×C23semidirect
108. #2381C2×C23semidirect
111. #2295C2×C23semidirect
125. #533C4×C23semidirect
180. C23#34348direct (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×#23822
C23×#23812
C23×#23782
C23×#22982
C23×#22952
C23×#22832
#3434823