Finite Group Databasebeta

← C23×#8543 #19156169

Subgroups — C23×#8543 #19156169

Order 5888.

1266 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×C23×#123C211yesmaximal, characteristic
2C23×#1819C214yesmaximal, complemented
3C23×#1781C212yesmaximal
4C23×D4×C23C2211yescharacteristic
5C24×C4×C23C411yescharacteristic, complemented
6C2×C23×#123—24no—
7C2×C23×#123—28no—
8C22×C4⋊C4×C23C412yescomplemented
9C22×C22⋊C4×C23C411yescharacteristic, complemented
10C22×C22⋊C4×C23C2212yes—
11C2×C23×#99—24no—
12C23×#379C2214yes—
13C23×#342—28no—
14C25×C23C2×C411yescharacteristic
15C25×C23C2311yescharacteristic
16C22×D4×C23—24no—
17C22×D4×C23—44no—
18C22×D4×C23D414yes—
19C23×C4×C23—22no—
20C23×C4×C23—24no—
21C23×C4×C23—44no—
22C23×C4×C23C2×C412yes—
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23C2×C412yes—
25C23×#123—232no—
26C2×C4⋊C4×C23—24no—
27C2×C4⋊C4×C23—24no—
28C2×C4⋊C4×C23—24no—
29C2×C22⋊C4×C23—22no—
30C2×C22⋊C4×C23—24no—
31C2×C22⋊C4×C23—24no—
32C2×C22⋊C4×C23—24no—
33C2×C22⋊C4×C23—28no—
34C2×C22⋊C4×C23—24no—
35C2×C22⋊C4×C23D418yes—
36C23×#99—48no—
37C24×C23—24no—
38C24×C23—22no—
39C24×C23—42no—
40C24×C23—44no—
41C24×C23—44no—
42C24×C23C22⋊C412yes—
43C24×C23C22×C411yescharacteristic
44C24×C23C2×D412yes—
45C24×C23C2×D414yes—
46C2×D4×C23—28no—
47C2×D4×C23—28no—
48C2×D4×C23—44no—
49C2×D4×C23—416no—
50C22×C4×C23—28no—
51C22×C4×C23—22no—
52C22×C4×C23—22no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—22no—
56C22×C4×C23—42no—
57C22×C4×C23—416no—
58C22×C4×C23—44no—
59C22×C4×C23—44no—
60C22×C4×C23—44no—
61C22×C4×C23—44no—
62C22×C4×C23—44no—
63C22×C4×C23C22⋊C412yes—
64C22×C4×C23C22⋊C414yes—
65C22×C4×C23C22⋊C414yes—
66C4⋊C4×C23—216no—
67C22⋊C4×C23—28no—
68C22⋊C4×C23—216no—
69C22⋊C4×C23—48no—
70C22⋊C4×C23—416no—
71#8543C2311yesmaximal, characteristic, direct product factor, Sylow
72C23×C23—22no—
73C23×C23—24no—
74C23×C23—28no—
75C23×C23—22no—
76C23×C23—22no—
77C23×C23—24no—
78C23×C23—28no—
79C23×C23—44no—
80C23×C23—44no—
81C23×C23—44no—
82C23×C23—44no—
83C23×C23—416no—
84C23×C23—44no—
85C23×C23—44no—
86C23×C23—88no—
87C23×C23#9914yes—
88C23×C23C2×C22⋊C411yescharacteristic
89C23×C23C2×C22⋊C412yes—
90C23×C23#12014yes—
91D4×C23—416no—
92C2×C4×C23—24no—
93C2×C4×C23—24no—
94C2×C4×C23—22no—
95C2×C4×C23—24no—
96C2×C4×C23—44no—
97C2×C4×C23—44no—
98C2×C4×C23—416no—
99C2×C4×C23—44no—
100C2×C4×C23—44no—
101C2×C4×C23—44no—
102C2×C4×C23—44no—
103C2×C4×C23—44no—
104C2×C4×C23—84no—
105C2×C4×C23—88no—
106C22×#123C2×C2311yescharacteristic
107#1819C2×C2314yescomplemented
108#1781C2×C2312yes—
109C22×C23—28no—
110C22×C23—22no—
111C22×C23—22no—
112C22×C23—22no—
113C22×C23—24no—
114C22×C23—24no—
115C22×C23—28no—
116C22×C23—44no—
117C22×C23—44no—
118C22×C23—416no—
119C22×C23—44no—
120C22×C23—84no—
121C22×C23—88no—
122C22×C23#37911yescharacteristic
123C22×C23C2×#9912yes—
124C22×C23#41014yes—
125C4×C23—42no—
126C4×C23—44no—
127C4×C23—44no—
128C4×C23—164no—
129C23×D4C22×C2311yescharacteristic
130C24×C4C4×C2311yescharacteristic, complemented
131C2×#123—24no—
132C2×#123—28no—
133C22×C4⋊C4C4×C2312yescomplemented
134C22×C22⋊C4C4×C2311yescharacteristic, complemented
135C22×C22⋊C4C22×C2312yes—
136C2×#99—24no—
137#379C22×C2314yes—
138#342—28no—
139C2×C23—22no—
140C2×C23—24no—
141C2×C23—44no—
142C2×C23—44no—
143C2×C23#178111yescharacteristic
144C2×C23#178212yes—
145C25C2×C4×C2311yescharacteristic
146C25C23×C2311yescharacteristic
147C22×D4—24no—
148C22×D4—44no—
149C22×D4D4×C2314yes—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—44no—
153C23×C4C2×C4×C2312yes—
154C23×C4C2×C4×C2312yes—
155C23×C4C2×C4×C2311yescharacteristic
156#123—232no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4D4×C2318yes—
167#99—48no—
168C23#854311yescharacteristic, direct product factor, Sylow
169C24—24no—
170C24—22no—
171C24—42no—
172C24—44no—
173C24—44no—
174C24C22⋊C4×C2312yes—
175C24C22×C4×C2311yescharacteristic
176C24C2×D4×C2312yes—
177C24C2×D4×C2314yes—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—44no—
181C2×D4—416no—
182C22×C4—28no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—416no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4C22⋊C4×C2312yes—
196C22×C4C22⋊C4×C2314yes—
197C22×C4C22⋊C4×C2314yes—
198C4⋊C4—216no—
199C22⋊C4—28no—
200C22⋊C4—216no—
201C22⋊C4—48no—
202C22⋊C4—416no—
203C23—22no—
204C23—24no—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—28no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—416no—
215C23—44no—
216C23—44no—
217C23—88no—
218C23C23×#9914yes—
219C23C2×C22⋊C4×C2311yescharacteristic
220C23C2×C22⋊C4×C2312yes—
221C23C23×#12014yes—
222D4—416no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—44no—
228C2×C4—416no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—84no—
236C2×C4—88no—
237C22—28no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—24no—
242C22—24no—
243C22—28no—
244C22—44no—
245C22—44no—
246C22—416no—
247C22—44no—
248C22—84no—
249C22—88no—
250C22C23×#37911yescharacteristic
251C22C2×C23×#9912yes—
252C22C23×#41014yes—
253C4—42no—
254C4—44no—
255C4—44no—
256C4—164no—
257C2—22no—
258C2—24no—
259C2—44no—
260C2—44no—
261C2C23×#178111yescharacteristic
262C2C23×#178212yes—
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC25C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1819C2semidirect
5. C24×C4×C23C4semidirect
8. C22×C4⋊C4×C23C4semidirect
9. C22×C22⋊C4×C23C4semidirect
71. #8543C23direct (N × H)
107. #1819C2×C23semidirect
130. C24×C4C4×C23semidirect
133. C22×C4⋊C4C4×C23semidirect
134. C22×C22⋊C4C4×C23semidirect
168. C23#8543direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1232
C23×#18192
C23×#17812
#854323