Finite Group Databasebeta

← C23×#11448 #19159074

Subgroups — C23×#11448 #19159074

Order 5888.

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

normal subgroups only

This group has 454 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
1C23×#2317C211yesmaximal, characteristic
2C2×C23×#324C212yesmaximal
3C23×#1325C212yesmaximal, complemented
4C23×#1324C212yesmaximal, complemented
5C2×C23×#117C411yescharacteristic
6C2×C23×#117C411yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C2×C23×C22⋊C8—22no—
9C23×#389—22no—
10C23×#385—22no—
11C23×#384C411yescharacteristic
12C23×#384C2211yescharacteristic
13C23×#382C411yescharacteristic
14C23×#382C2211yescharacteristic
15C23×#324—22no—
16C23×#324C2214yes—
17C23×C4×C23C2×C411yescharacteristic
18C23×C4×C23D412yes—
19C22×C8×C23—42no—
20C23×#117—22no—
21C23×#117—42no—
22C23×#117C814yescomplemented
23C2×C4⋊C4×C23—21no—
24C2×C4⋊C4×C23—22no—
25C2×C4⋊C4×C23—22no—
26C2×C4⋊C4×C23—42no—
27C2×C4⋊C4×C23C2×C411yescharacteristic
28C2×C4⋊C4×C23C2×C411yescharacteristic
29C2×C4⋊C4×C23C2×C411yescharacteristic
30C2×C4⋊C4×C23D412yes—
31C2×C4⋊C4×C23C2311yescharacteristic
32C2×C22⋊C4×C23—21no—
33C2×C22⋊C4×C23—21no—
34C2×C42×C23—21no—
35C2×C42×C23C2×C411yescharacteristic
36C2×C42×C23C2×C411yescharacteristic
37C23×C22⋊C8—24no—
38C23×C22⋊C8—42no—
39C23×#95—22no—
40C23×#95—22no—
41C24×C23C22⋊C411yescharacteristic
42C22×C4×C23—22no—
43C22×C4×C23—22no—
44C22×C4×C23—22no—
45C22×C4×C23—21no—
46C22×C4×C23—21no—
47C22×C4×C23—22no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—82no—
51C22×C4×C23C22⋊C411yescharacteristic
52C22×C4×C23C22⋊C412yes—
53C22×C4×C23C2×C812yes—
54C22×C4×C23M1612yes—
55C22×C4×C23C22×C411yescharacteristic
56C22×C4×C23C2×D412yes—
57C2×C8×C23—44no—
58C2×C8×C23—84no—
59C4⋊C4×C23—24no—
60C4⋊C4×C23—41no—
61C4⋊C4×C23—42no—
62C4⋊C4×C23—41no—
63C4⋊C4×C23—41no—
64C4⋊C4×C23—41no—
65C4⋊C4×C23—82no—
66C4⋊C4×C23—82no—
67C22⋊C4×C23—24no—
68C22⋊C4×C23—42no—
69C42×C23—21no—
70C42×C23—21no—
71C42×C23—42no—
72C42×C23C2×C812yescomplemented
73C42×C23C2×C812yescomplemented
74#11448C2311yesmaximal, characteristic, direct product factor, Sylow
75C23×C23—41no—
76C23×C23—41no—
77C23×C23C22⋊C812yes—
78C23×C23#9912yes—
79C23×C23#10112yes—
80C23×C23C2×C22⋊C411yescharacteristic
81C2×C4×C23—24no—
82C2×C4×C23—24no—
83C2×C4×C23—21no—
84C2×C4×C23—21no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—41no—
92C2×C4×C23—41no—
93C2×C4×C23—44no—
94C2×C4×C23—41no—
95C2×C4×C23—41no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—42no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—82no—
103C2×C4×C23C22⋊C812yes—
104C2×C4×C23C22×C811yescharacteristic
105C2×C4×C23C2×M1611yescharacteristic
106C8×C23—84no—
107#2317C2×C2311yescharacteristic
108C2×#324C2×C2312yes—
109#1325C2×C2312yescomplemented
110#1324C2×C2312yescomplemented
111C22×C23—21no—
112C22×C23—22no—
113C22×C23—21no—
114C22×C23—22no—
115C22×C23—42no—
116C22×C23—42no—
117C22×C23#32414yes—
118C22×C23C2×C22⋊C811yescharacteristic
119C22×C23C2×#9911yescharacteristic
120C22×C23C2×#10111yescharacteristic
121C4×C23—22no—
122C4×C23—21no—
123C4×C23—21no—
124C4×C23—42no—
125C4×C23—42no—
126C4×C23—42no—
127C4×C23—42no—
128C4×C23—81no—
129C4×C23—81no—
130C2×#117C4×C2311yescharacteristic
131C2×#117C4×C2311yescharacteristic
132C22×C4⋊C4C22×C2311yescharacteristic
133C2×C22⋊C8—22no—
134#389—22no—
135#385—22no—
136#384C4×C2311yescharacteristic
137#384C22×C2311yescharacteristic
138#382C4×C2311yescharacteristic
139#382C22×C2311yescharacteristic
140#324—22no—
141#324C22×C2314yes—
142C2×C23—21no—
143C2×C23—24no—
144C2×C23—21no—
145C2×C23C2×#32411yescharacteristic
146C2×C23#212711yescharacteristic
147C2×C23#213311yescharacteristic
148C23×C4C2×C4×C2311yescharacteristic
149C23×C4D4×C2312yes—
150C22×C8—42no—
151#117—22no—
152#117—42no—
153#117C8×C2314yescomplemented
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C2×C4×C2311yescharacteristic
159C2×C4⋊C4C2×C4×C2311yescharacteristic
160C2×C4⋊C4C2×C4×C2311yescharacteristic
161C2×C4⋊C4D4×C2312yes—
162C2×C4⋊C4C23×C2311yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C42—21no—
166C2×C42C2×C4×C2311yescharacteristic
167C2×C42C2×C4×C2311yescharacteristic
168C22⋊C8—24no—
169C22⋊C8—42no—
170#95—22no—
171#95—22no—
172C23#1144811yescharacteristic, direct product factor, Sylow
173C24C22⋊C4×C2311yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—82no—
183C22×C4C22⋊C4×C2311yescharacteristic
184C22×C4C22⋊C4×C2312yes—
185C22×C4C2×C8×C2312yes—
186C22×C4M16×C2312yes—
187C22×C4C22×C4×C2311yescharacteristic
188C22×C4C2×D4×C2312yes—
189C2×C8—44no—
190C2×C8—84no—
191C4⋊C4—24no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—42no—
204C42C2×C8×C2312yescomplemented
205C42C2×C8×C2312yescomplemented
206C23—41no—
207C23—41no—
208C23C23×C22⋊C812yes—
209C23C23×#9912yes—
210C23C23×#10112yes—
211C23C2×C22⋊C4×C2311yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4C23×C22⋊C812yes—
235C2×C4C22×C8×C2311yescharacteristic
236C2×C4C2×M16×C2311yescharacteristic
237C8—84no—
238C22—21no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22C23×#32414yes—
245C22C2×C23×C22⋊C811yescharacteristic
246C22C2×C23×#9911yescharacteristic
247C22C2×C23×#10111yescharacteristic
248C4—21no—
249C4—21no—
250C4—22no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—24no—
259C2—21no—
260C2C2×C23×#32411yescharacteristic
261C2C23×#212711yescharacteristic
262C2C23×#213311yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC2×C4C22×C8×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1325C2semidirect
4. C23×#1324C2semidirect
22. C23×#117C8semidirect
72. C42×C23C2×C8semidirect
73. C42×C23C2×C8semidirect
74. #11448C23direct (N × H)
109. #1325C2×C23semidirect
110. #1324C2×C23semidirect
153. #117C8×C23semidirect
172. C23#11448direct (N × H)
204. C42C2×C8×C23semidirect
205. C42C2×C8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#23172
C2×C23×#3242
C23×#13252
C23×#13242
#1144823