Finite Group Databasebeta

← C23×#8518 #19156144

Subgroups — C23×#8518 #19156144

Order 5888.

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×C23×#540C211yesmaximal, characteristic
2C23×#1822C211yesmaximal, characteristic, complemented
3C23×#1820C211yesmaximal, characteristic, complemented
4C23×#1510C214yesmaximal, complemented
5C23×#540—28no—
6C2×C23×#127C411yescharacteristic, complemented
7C2×C23×#125C411yescharacteristic, complemented
8C2×C23×#123—22no—
9C2×C23×#121—22no—
10C2×C23×#121C2212yes—
11C2×C4×D4×C23C412yescomplemented
12C2×C23×#117C2211yescharacteristic
13C23×#407C2212yes—
14C23×#381C2212yes—
15C23×#343—22no—
16C23×#342—22no—
17C23×#327—28no—
18C22×D4×C23—22no—
19C22×D4×C23—22no—
20C22×D4×C23C2×C412yes—
21C23×C4×C23—22no—
22C23×C4×C23C2311yescharacteristic
23C2×M16×C23—22no—
24C23×#127—24no—
25C23×#125—24no—
26C23×#123—216no—
27C23×#121—216no—
28C23×#121—24no—
29C23×#121D418yes—
30C4×D4×C23—24no—
31C4×D4×C23—24no—
32C23×#117—24no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23C2×C412yes—
35C2×C4⋊C4×C23D412yes—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23D412yes—
40C2×C42×C23C2×C411yescharacteristic
41C2×C42×C23C2×C411yescharacteristic
42C23×C22⋊C8—24no—
43C23×#95—24no—
44C24×C23—22no—
45C24×C23C2×D411yescharacteristic
46C2×D4×C23—28no—
47C2×D4×C23—28no—
48C2×D4×C23—22no—
49C2×D4×C23—44no—
50C2×D4×C23—84no—
51C2×D4×C23—84no—
52C2×D4×C23C22⋊C414yes—
53C22×C4×C23—22no—
54C22×C4×C23—28no—
55C22×C4×C23—42no—
56C22×C4×C23—42no—
57C22×C4×C23—84no—
58C22×C4×C23C22⋊C412yes—
59C22×C4×C23C22⋊C412yes—
60C22×C4×C23C22×C411yescharacteristic
61C22×C4×C23C2×D411yescharacteristic
62C22×C4×C23C2×D414yes—
63M16×C23—44no—
64C2×C8×C23—42no—
65C4⋊C4×C23—28no—
66C4⋊C4×C23—22no—
67C4⋊C4×C23—24no—
68C4⋊C4×C23C22⋊C414yes—
69C22⋊C4×C23—28no—
70C22⋊C4×C23—24no—
71C22⋊C4×C23—28no—
72C22⋊C4×C23—44no—
73C42×C23—22no—
74C42×C23—22no—
75#8518C2311yesmaximal, characteristic, direct product factor, Sylow
76C23×C23—22no—
77C23×C23—24no—
78C23×C23—42no—
79C23×C23—44no—
80C23×C23—44no—
81C23×C23—44no—
82C23×C23C2×C22⋊C411yescharacteristic
83C23×C23#12012yes—
84D4×C23—44no—
85D4×C23—44no—
86D4×C23—88no—
87D4×C23—88no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—24no—
93C2×C4×C23—44no—
94C2×C4×C23—44no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—88no—
98C2×C4×C23—82no—
99C2×C4×C23—84no—
100C2×C4×C23C2×C22⋊C412yes—
101C2×C4×C23#12012yes—
102C8×C23—82no—
103C2×#540C2×C2311yescharacteristic
104#1822C2×C2311yescharacteristic, complemented
105#1820C2×C2311yescharacteristic, complemented
106#1510C2×C2314yescomplemented
107C22×C23—22no—
108C22×C23—24no—
109C22×C23—22no—
110C22×C23—44no—
111C22×C23—44no—
112C22×C23—44no—
113C22×C23—82no—
114C22×C23—88no—
115C22×C23#37911yescharacteristic
116C22×C23#41012yes—
117C4×C23—22no—
118C4×C23—22no—
119C4×C23—44no—
120C4×C23—42no—
121C4×C23—42no—
122C4×C23—162no—
123#540—28no—
124C2×#127C4×C2311yescharacteristic, complemented
125C2×#125C4×C2311yescharacteristic, complemented
126C2×#123—22no—
127C2×#121—22no—
128C2×#121C22×C2312yes—
129C2×C4×D4C4×C2312yescomplemented
130C2×#117C22×C2311yescharacteristic
131#407C22×C2312yes—
132#381C22×C2312yes—
133#343—22no—
134#342—22no—
135#327—28no—
136C2×C23—22no—
137C2×C23—42no—
138C2×C23—84no—
139C2×C23#178211yescharacteristic
140C2×C23#179512yes—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C2×C4×C2312yes—
144C23×C4—22no—
145C23×C4C23×C2311yescharacteristic
146C2×M16—22no—
147#127—24no—
148#125—24no—
149#123—216no—
150#121—216no—
151#121—24no—
152#121D4×C2318yes—
153C4×D4—24no—
154C4×D4—24no—
155#117—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C2×C4×C2312yes—
158C2×C4⋊C4D4×C2312yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4D4×C2312yes—
163C2×C42C2×C4×C2311yescharacteristic
164C2×C42C2×C4×C2311yescharacteristic
165C22⋊C8—24no—
166#95—24no—
167C23#851811yescharacteristic, direct product factor, Sylow
168C24—22no—
169C24C2×D4×C2311yescharacteristic
170C2×D4—28no—
171C2×D4—28no—
172C2×D4—22no—
173C2×D4—44no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4C22⋊C4×C2314yes—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—84no—
182C22×C4C22⋊C4×C2312yes—
183C22×C4C22⋊C4×C2312yes—
184C22×C4C22×C4×C2311yescharacteristic
185C22×C4C2×D4×C2311yescharacteristic
186C22×C4C2×D4×C2314yes—
187M16—44no—
188C2×C8—42no—
189C4⋊C4—28no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4C22⋊C4×C2314yes—
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×C2311yescharacteristic
206C23C23×#12012yes—
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×C2312yes—
224C2×C4C23×#12012yes—
225C8—82no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—82no—
233C22—88no—
234C22C23×#37911yescharacteristic
235C22C23×#41012yes—
236C4—22no—
237C4—22no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—162no—
242C2—22no—
243C2—42no—
244C2—84no—
245C2C23×#178211yescharacteristic
246C2C23×#179512yes—
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#540C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1822C2semidirect
3. C23×#1820C2semidirect
4. C23×#1510C2semidirect
6. C2×C23×#127C4semidirect
7. C2×C23×#125C4semidirect
11. C2×C4×D4×C23C4semidirect
75. #8518C23direct (N × H)
104. #1822C2×C23semidirect
105. #1820C2×C23semidirect
106. #1510C2×C23semidirect
124. C2×#127C4×C23semidirect
125. C2×#125C4×C23semidirect
129. C2×C4×D4C4×C23semidirect
167. C23#8518direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5402
C23×#18222
C23×#18202
C23×#15102
#851823