Finite Group Databasebeta

← C23×#8526 #19156152

Subgroups — C23×#8526 #19156152

Order 5888.

784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 784 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×#1555C212yesmaximal, complemented
3C23×#1534C212yesmaximal, complemented
4C23×#1464C212yesmaximal, complemented
5C23×#540—24no—
6C23×#540—24no—
7C2×C23×#127C2211yescharacteristic
8C2×C23×#125C2211yescharacteristic
9C2×C23×#123—22no—
10C2×C23×#121—22no—
11C2×C23×#121C412yes—
12C2×C4×D4×C23C412yes—
13C2×C23×#117C2211yescharacteristic
14C23×#432C2214yes—
15C23×#331—24no—
16C23×#330—24no—
17C23×#324—22no—
18C23×#323—22no—
19C22×D4×C23—22no—
20C22×D4×C23—22no—
21C22×D4×C23C2×C412yes—
22C23×C4×C23—22no—
23C23×C4×C23C2×C411yescharacteristic
24C23×#127—22no—
25C23×#127D414yes—
26C23×#125—22no—
27C23×#125D414yes—
28C23×#123—216no—
29C23×#121—216no—
30C23×#121—24no—
31C23×#121—24no—
32C4×D4×C23—24no—
33C4×D4×C23—24no—
34C23×#117—22no—
35C23×#117—22no—
36C2×C4⋊C4×C23—22no—
37C2×C4⋊C4×C23C2×C412yes—
38C2×C4⋊C4×C23D412yes—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23D412yes—
42C2×C42×C23C2×C411yescharacteristic
43C2×C42×C23C2311yescharacteristic
44C23×C4⋊C8—24no—
45C23×C22⋊C8—24no—
46C23×#97—24no—
47C24×C23—22no—
48C24×C23C22⋊C411yescharacteristic
49C2×D4×C23—28no—
50C2×D4×C23—22no—
51C2×D4×C23—28no—
52C2×D4×C23—44no—
53C2×D4×C23—84no—
54C2×D4×C23—84no—
55C2×D4×C23C22⋊C414yes—
56C22×C4×C23—22no—
57C22×C4×C23—28no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—42no—
61C22×C4×C23—84no—
62C22×C4×C23C22⋊C411yescharacteristic
63C22×C4×C23C22⋊C412yes—
64C22×C4×C23C22×C411yescharacteristic
65C22×C4×C23C2×D412yes—
66C2×C8×C23—44no—
67C4⋊C4×C23—22no—
68C4⋊C4×C23—28no—
69C4⋊C4×C23—24no—
70C4⋊C4×C23C22⋊C414yes—
71C22⋊C4×C23—28no—
72C22⋊C4×C23—28no—
73C22⋊C4×C23—24no—
74C42×C23—21no—
75C42×C23—21no—
76C42×C23C2×D412yes—
77C42×C23C2×D412yes—
78#8526C2311yesmaximal, characteristic, direct product factor, Sylow
79C23×C23—21no—
80C23×C23—24no—
81C23×C23—22no—
82C23×C23—44no—
83C23×C23—44no—
84C23×C23—42no—
85C23×C23—44no—
86C23×C23C2×C22⋊C411yescharacteristic
87D4×C23—44no—
88D4×C23—44no—
89D4×C23—88no—
90D4×C23—88no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—21no—
94C2×C4×C23—24no—
95C2×C4×C23—22no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—44no—
100C2×C4×C23—42no—
101C2×C4×C23—88no—
102C2×C4×C23C2×C22⋊C411yescharacteristic
103C2×C4×C23C2×C22⋊C411yescharacteristic
104C2×C4×C23#12014yes—
105C8×C23—84no—
106C2×#540C2×C2311yescharacteristic
107#1555C2×C2312yescomplemented
108#1534C2×C2312yescomplemented
109#1464C2×C2312yescomplemented
110C22×C23—22no—
111C22×C23—44no—
112C22×C23—42no—
113C22×C23—44no—
114C22×C23—41no—
115C22×C23—44no—
116C22×C23—82no—
117C22×C23—88no—
118C22×C23#37911yescharacteristic
119C22×C23#41312yes—
120C4×C23—24no—
121C4×C23—21no—
122C4×C23—21no—
123C4×C23—44no—
124C4×C23—42no—
125C4×C23—41no—
126#540—24no—
127#540—24no—
128C2×#127C22×C2311yescharacteristic
129C2×#125C22×C2311yescharacteristic
130C2×#123—22no—
131C2×#121—22no—
132C2×#121C4×C2312yes—
133C2×C4×D4C4×C2312yes—
134C2×#117C22×C2311yescharacteristic
135#432C22×C2314yes—
136#331—24no—
137#330—24no—
138#324—22no—
139#323—22no—
140C2×C23—22no—
141C2×C23—81no—
142C2×C23—84no—
143C2×C23#178511yescharacteristic
144C2×C23#179512yes—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C2×C4×C2312yes—
148C23×C4—22no—
149C23×C4C2×C4×C2311yescharacteristic
150#127—22no—
151#127D4×C2314yes—
152#125—22no—
153#125D4×C2314yes—
154#123—216no—
155#121—216no—
156#121—24no—
157#121—24no—
158C4×D4—24no—
159C4×D4—24no—
160#117—22no—
161#117—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C2×C4×C2312yes—
164C2×C4⋊C4D4×C2312yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4D4×C2312yes—
168C2×C42C2×C4×C2311yescharacteristic
169C2×C42C23×C2311yescharacteristic
170C4⋊C8—24no—
171C22⋊C8—24no—
172#97—24no—
173C23#852611yescharacteristic, direct product factor, Sylow
174C24—22no—
175C24C22⋊C4×C2311yescharacteristic
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—44no—
180C2×D4—84no—
181C2×D4—84no—
182C2×D4C22⋊C4×C2314yes—
183C22×C4—22no—
184C22×C4—28no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—84no—
189C22×C4C22⋊C4×C2311yescharacteristic
190C22×C4C22⋊C4×C2312yes—
191C22×C4C22×C4×C2311yescharacteristic
192C22×C4C2×D4×C2312yes—
193C2×C8—44no—
194C4⋊C4—22no—
195C4⋊C4—28no—
196C4⋊C4—24no—
197C4⋊C4C22⋊C4×C2314yes—
198C22⋊C4—28no—
199C22⋊C4—28no—
200C22⋊C4—24no—
201C42—21no—
202C42—21no—
203C42C2×D4×C2312yes—
204C42C2×D4×C2312yes—
205C23—21no—
206C23—24no—
207C23—22no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—44no—
212C23C2×C22⋊C4×C2311yescharacteristic
213D4—44no—
214D4—44no—
215D4—88no—
216D4—88no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—88no—
228C2×C4C2×C22⋊C4×C2311yescharacteristic
229C2×C4C2×C22⋊C4×C2311yescharacteristic
230C2×C4C23×#12014yes—
231C8—84no—
232C22—22no—
233C22—41no—
234C22—44no—
235C22—44no—
236C22—44no—
237C22—42no—
238C22—82no—
239C22—88no—
240C22C23×#37911yescharacteristic
241C22C23×#41312yes—
242C4—21no—
243C4—21no—
244C4—24no—
245C4—41no—
246C4—44no—
247C4—42no—
248C2—22no—
249C2—84no—
250C2—81no—
251C2C23×#178511yescharacteristic
252C2C23×#179512yes—
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#540C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1555C2semidirect
3. C23×#1534C2semidirect
4. C23×#1464C2semidirect
78. #8526C23direct (N × H)
107. #1555C2×C23semidirect
108. #1534C2×C23semidirect
109. #1464C2×C23semidirect
173. C23#8526direct (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×#15552
C23×#15342
C23×#14642
#852623